# Valuations, completions, and hyperbolic actions of metabelian groups

Carolyn R. Abbott  Sahana Balasubramanya  Alexander J. Rasmussen  Appendix A
co-authored with Sam Payne

###### Abstract

Actions on hyperbolic metric spaces are an important tool for studying groups,
and so it is natural, but difficult, to attempt to classify all such actions
of a fixed group. In this paper, we build strong connections between
hyperbolic geometry and commutative algebra in order to classify the cobounded
hyperbolic actions of numerous metabelian groups up to a coarse equivalence.
In particular, we turn this classification problem into the problems of
classifying ideals in the completions of certain rings and calculating
invariant subspaces of matrices. We use this framework to classify the
cobounded hyperbolic actions of many abelian-by-cyclic groups associated to
expanding integer matrices. Each such action is equivalent to an action on a
tree or on a Heintze group (a classically studied class of negatively curved
Lie groups). Our investigations incorporate number systems, factorization in
formal power series rings, completions, and valuations.

###### Contents

  1. 1 Introduction
    1. 1.1 About the proofs
  2. 2 Background
    1. 2.1 Posets
    2. 2.2 Comparing generating sets and group actions
    3. 2.3 Actions on hyperbolic spaces
  3. 3 Axioms for commutative rings
    1. 3.1 Number systems and completions
  4. 4 Generalities on confining subsets
  5. 5 Examples
    1. 5.1 Lamplighter groups
    2. 5.2 Torsion-free finitely presented abelian-by-cyclic groups
      1. 5.2.1 Solvable Baumslag-Solitar groups
      2. 5.2.2 Other abelian-by-cyclic groups
  6. 6 Structure of $\mathcal{P}_{+}(G)$: ideals of $\widehat{R}$
    1. 6.1 Constructing confining subsets from ideals
    2. 6.2 Constructing ideals from confining subsets
    3. 6.3 Saturated ideals and the poset $\mathcal{P}_{+}(G)$
  7. 7 Elements of $\mathcal{P}_{+}(G)$: valuations and actions on trees
    1. 7.1 Proof of Theorem 1.1
    2. 7.2 Valuations and actions of $G(R,\gamma)$ on trees
      1. 7.2.1 Bass-Serre trees for lamplighter groups
      2. 7.2.2 Bass-Serre trees for Baumslag-Solitar groups
  8. 8 Preliminaries on torsion-free finitely presented abelian-by-cyclic groups
    1. 8.1 Ring axioms for abelian-by-cyclic groups
    2. 8.2 The structure of $\mathcal{P}_{+}(G)$ for abelian-by-cyclic groups
  9. 9 The structure of $\mathcal{P}_{-}(G$): invariant subspaces
    1. 9.1 Tiles
    2. 9.2 Invariant subspaces of $\mathbb{R}^{n}$
    3. 9.3 The poset of invariant subspaces
    4. 9.4 Corollaries to Theorem 1.2
  10. 10 Elements of $\mathcal{P}_{-}(G)$: actions on Heintze groups
    1. 10.1 The word metric for the main Heintze group
    2. 10.2 Quotients of Heintze groups
    3. 10.3 Proof of Theorems 1.2 and 1.8
  11. A Appendix: Classification of ideals and factorization of formal power series  Co-authored with Sam Payne
  12. B Appendix: Pseudo-valuations and confining subsets
  13. References

##  1 Introduction

Actions on hyperbolic metric spaces (which we call hyperbolic actions) are
among the most important tools in geometric group theory. Hyperbolic actions
may be used to study the coarse geometry of a wide variety of groups and are
crucial tools for studying bounded cohomology ([9]), quotients of groups
([17]), and decision problems for groups ([13]). It is thus a natural, but
typically very difficult, problem to attempt to classify all of the hyperbolic
actions of a group.

To solve this problem, one is forced to study _cobounded_ hyperbolic actions
up to coarse (quasi-isometric) equivalence. These are minor and natural
restrictions, especially as hyperbolicity is preserved under quasi-isometries.
However, even after these restrictions, many groups of geometric interest have
uncountably many distinct equivalence classes of hyperbolic actions (see,
e.g., [1, Theorem 2.6]), making the classification problem seem intractable.
It is thus natural to attempt to solve this classification problem for groups
with fewer hyperbolic actions. The authors made the first steps towards this
goal in [8, 3, 2] by completely classifying the hyperbolic actions of certain
classically studied solvable groups. See also [7] for a recent classification
of hyperbolic actions of certain lattices. The present paper was born from a
realization that all of the classification results of [8, 3, 2] fit within a
common framework. By identifying this common framework and developing new
tools for working with groups that fit within it, we are able to classify the
hyperbolic actions of numerous metabelian, and specifically abelian-by-cyclic,
groups. The key idea that we introduce in this paper is a tool for converting
this classification problem into a commutative algebra problem for certain
groups. Our analysis unites radix representations, completions, and valuations
on commutative rings and relates them to properties of hyperbolic actions.

The set $\mathcal{H}(G)$ of equivalence classes of cobounded hyperbolic
actions of a group $G$ is naturally a poset, as shown in [1]. The partial
order is roughly given by considering one action to be smaller than another
when the smaller action may be obtained by an operation of collapsing an
equivariant family of subspaces in the larger action. The equivalence classes
in $\mathcal{H}(G)$ are called _hyperbolic structures on $G$_. The results of
[8, 3, 2] all give a complete description of the poset $\mathcal{H}(G)$ for
the group $G$ in question. In this paper, for any group $G$ fitting into our
common framework, we give a partial description of the poset $\mathcal{H}(G)$,
while for groups $G$ satisfying certain additional properties we give a
complete description of $\mathcal{H}(G)$. Before stating our main theorems, we
describe the common algebraic framework in which the results apply.

Consider a ring $R$ that is generated as a $\mathbb{Z}$–algebra by an element
$\gamma$. Assuming that $\gamma$ is neither a unit nor a zero divisor, we may
define an ascending HNN extension $G\vcentcolon=G(R,\gamma)$ of $R$ by the
endomorphism of (the abelian group) $R$ defined by multiplication by $\gamma$.
Further assuming some natural restrictions on the algebra of $R$ (see axioms
(A1)–(A4) in Section 3), one may write elements of $R$ in “base-$\gamma$” as
infinite sums $a_{0}+a_{1}\gamma+a_{2}\gamma^{2}+\cdots$ where the “digits”
$a_{i}$ can be chosen to lie in a finite set. We introduce one more axiom
(A5), which ensures that these _radix representations_ are somewhat well-
behaved. Whenever these axioms hold, we derive a description of the poset of
hyperbolic actions $\mathcal{H}(G)$. This description is in terms of the
_$(\gamma)$ -adic completion $\widehat{R}$ of $R$_, a standard tool from
commutative algebra. To the best of our knowledge, this introduces completions
of rings for the first time as a tool in geometric group theory.

######  Theorem 1.1.

Under the assumptions on $G$ above, the poset $\mathcal{H}(G)$ consists of the
following structures (see Figure 1(a)): a single elliptic structure, which is
dominated by a single lineal structure, and two subposets $\mathcal{P}_{-}(G)$
and $\mathcal{P}_{+}(G)$ that intersect in the single lineal structure. The
subposet $\mathcal{P}_{+}(G)$ is isomorphic to the opposite of the poset of
ideals of the $(\gamma)$-adic completion $\widehat{R}$ considered up to
multiplication by $\gamma$. The poset $\mathcal{P}_{-}(G)$ is a lattice.
Moreover, each element of $\mathcal{P}_{+}(G)$ contains an action on a
simplicial tree.

Theorem 1.1 broadly expands our understanding of $\mathcal{H}(G)$ for a large
class of groups $G$. In particular, roughly half the work of describing
$\mathcal{H}(G)$, a geometric problem, is reduced to the algebraic problem of
classifying the ideals of the completion $\widehat{R}$ up to equivalence of
ideals under multiplication by $\gamma$. (See Section 6.3 for the precise
definition of the equivalence relation on ideals of $\widehat{R}$.) However,
the lattice $\mathcal{P}_{-}(G)$ described in Theorem 1.1 is mysterious in
general. We dispel some of the mystery around the lattice $\mathcal{P}_{-}(G)$
by describing it completely for a wide class of metabelian groups $G$.

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$*$$\mathbb{R}$$T$$\mathcal{P}_{+}(G)$$\mathcal{P}_{-}(G)$ (a)
$\mathcal{H}(G)$

$*$$\mathbb{R}$$T_{+}$$T_{-}$$\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$$\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$
(b) $\mathcal{H}((\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z})$

$*$$\mathbb{R}$$\mathbb{H}^{2}$$T$$2^{\\{1,\ldots,r\\}}$ (c)
$\mathcal{H}(BS(1,n))$

Figure 1: Posets of hyperbolic actions. Here $2^{\\{1,\ldots,k\\}}$ denotes
the power set with inclusion and $\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$
denotes the poset of subgroups with inclusion.

Recall that an _abelian-by-cyclic_ group is one of the form
$G=A\rtimes\mathbb{Z}$ where $A$ is abelian. We focus on torsion-free,
finitely presented abelian-by-cyclic groups. These groups are of interest in
dynamics and coarse geometry (see, e.g., [19]). Crucially for applying our
methods, such an abelian-by-cyclic group $G$ can be described as an ascending
HNN extension of a free abelian group $\mathbb{Z}^{n}$ [10]. That is, there is
an endomorphism of $\mathbb{Z}^{n}$, represented by a matrix $\gamma\in
M_{n}(\mathbb{Z})$ with $\det\gamma\neq 0$, such that:

$G=\left\langle\mathbb{Z}^{n},t:tzt^{-1}=\gamma z,\,\forall
z\in\mathbb{Z}^{n}\right\rangle.$

In this case, we completely describe $\mathcal{H}(G)$ under a few hypotheses
on the matrix $\gamma$. We require that all eigenvalues of $\gamma$ lie
outside the unit disk in $\mathbb{C}$ (i.e., $\gamma$ is _expanding_) and
$\mathbb{Z}^{n}$ is a cyclic $\mathbb{Z}[x]$-module, where the action of $x$
is induced by that of $\gamma$. We call a matrix $\gamma$ with these
properties _admissible_. The expanding hypothesis on $\gamma$ is the most
necessary. That $\mathbb{Z}^{n}$ is a cyclic $\mathbb{Z}[x]$-module is always
satisfied up to finite index, and we expect this hypothesis can be removed in
future work.

To classify the cobounded hyperbolic actions of $G$, we consider the prime
factorization of the characteristic polynomial (which, in this case, is also
the minimal polynomial [24, Section 7.1]) $p=up_{1}^{n_{1}}\cdots
p_{r}^{n_{r}}$ in the formal power series ring $\mathbb{Z}[[x]]$. Here
$u\in\mathbb{Z}[[x]]$ is a unit and each $p_{i}\in\mathbb{Z}[[x]]$ is an
irreducible power series. Let $[n]$ be the set $\\{0,\ldots,n-1\\}$ endowed
with the standard partial order $\leq$ on $\mathbb{Z}$, and let
$\operatorname{Div}(n_{1},\ldots,n_{r})$ be the poset
$[n_{1}+1]\times\cdots\times[n_{r}+1]$, which is isomorphic to the poset of
divisors of $p=up_{1}^{n_{1}}\cdots p_{r}^{n_{r}}$ considered up to
multiplication by a unit.

######  Theorem 1.2.

Let $G$ be an ascending HNN extension of $\mathbb{Z}^{n}$ by an admissible
matrix $\gamma$. Then $\mathcal{H}(G)$ is as described in Theorem 1.1 and:

  1. 1.

$\mathcal{P}_{+}(G)$ is isomorphic to
$\operatorname{Div}(n_{1},\ldots,n_{r})$;

  2. 2.

$\mathcal{P}_{-}(G)$ is isomorphic to the poset of subspaces of
$\mathbb{R}^{n}$ that are invariant under $\gamma$; and

  3. 3.

each element of $\mathcal{P}_{-}(G)$ contains an action on a quasi-convex
subspace of a Heintze group.

_Heintze groups_ are negatively-curved Lie groups of the form
$N\rtimes\mathbb{R}$, where $N$ is a nilpotent Lie group. Under the
assumptions of Theorem 1.2, the group $G$ admits actions on Heintze groups of
the form $\mathbb{C}^{k}\rtimes\mathbb{R}$, where $\mathbb{R}$ acts on
$\mathbb{C}^{k}$ via a 1-parameter subgroup of
$\operatorname{GL}_{k}(\mathbb{C})$; see Section 10. However, these actions
are never cobounded, and so do not represent elements of $\mathcal{H}(G)$. We
show that the subspace $\mathbb{R}^{k}\times\mathbb{R}$ (which is not
generally a subgroup) is quasi-isometrically embedded in this group
$\mathbb{C}^{k}\rtimes\mathbb{R}$. Hence the subspace
$\mathbb{R}^{k}\times\mathbb{R}$ is itself Gromov hyperbolic, and admits a
cobounded action of $G$. The elements of $\mathcal{P}_{-}(G)$ are represented
by such cobounded hyperbolic actions.

Theorem 1.2 reduces the classification of hyperbolic structures of abelian-by-
cyclic groups based on admissible matrices $\gamma$ to two algebraic
computations: computing the invariant subspaces of $\gamma$ and computing the
prime factorization of $p$ in the formal power series ring $\mathbb{Z}[[x]]$.
Computing the invariant subspaces is straightforward using Jordan normal forms
(see, e.g., [21]), while computing factorizations in formal power series rings
can be solved algorithmically (see, e.g., [18]).

Theorems 1.1 and 1.2 completely recover the main theorems of [3] and [8].
Moreover, they can be applied to describe the poset $\mathcal{H}(G)$ for
numerous new abelian-by-cyclic groups. We give some examples below; see
Section 5 for details. The methods of Section 5 can be extended to many other
groups.

######  Example 1.3.

Consider a lamplighter group $(\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z}$, where
$n\geq 2$. Then $\mathcal{H}((\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z})$ is as
pictured in Figure 1(b): there is a unique lineal structure and two copies of
the poset $\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$ of subgroups of
$\mathbb{Z}/n\mathbb{Z}$ meeting in the unique lineal structure. Every
equivalence class contains an action on a tree or on a point. This description
of $\mathcal{H}((\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z})$ was first obtained in
[8]. See Section 5.1 for a new, much shorter proof using Theorem 1.1.

######  Example 1.4.

Consider a solvable Baumslag-Solitar group $BS(1,n)=\langle
t,a:tat^{-1}=a^{n}\rangle$ for $n\in\mathbb{Z}\setminus\\{-1,0,1\\}$, and let
$n=\pm p_{1}^{n_{1}}\cdots p_{r}^{n_{r}}$ be the prime factorization of $n$.
Then $\mathcal{H}(BS(1,n))$ is as pictured in Figure 1(c): there is a unique
lineal structure, a copy of the poset $2^{\\{1,\ldots,r\\}}$ of subsets of
$\\{1,\ldots,r\\}$ containing the single lineal structure, and a single
additional structure containing an action on the hyperbolic plane
$\mathbb{H}^{2}$. Every element of $2^{\\{1,\ldots,r\\}}$ contains an action
on a tree. This description of $\mathcal{H}(BS(1,n))$ was first obtained in
[3]. See Section 5.1 for a new, much shorter proof using Theorem 1.2.

|  |
---|---|---

$\mathcal{P}_{+}$$\mathcal{P}_{-}$ (a) $\gamma=\begin{pmatrix}2&0&0\\\
0&3&1\\\ 0&0&3\end{pmatrix}$

$\mathcal{P}_{+}$$\mathcal{P}_{-}$ (b) $\gamma=\begin{pmatrix}0&0&-210\\\
1&0&-1\\\ 0&1&0\end{pmatrix}$

$\mathcal{P}_{+}$$\mathcal{P}_{-}$ (c) $\gamma=\begin{pmatrix}0&0&0&2\\\
1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\end{pmatrix}$

Figure 2: Posets $\mathcal{H}(G(\gamma))$ for various admissible $\gamma$.
Every equivalence class to the left in one of the figures contains an action
on a Heintze group, while every class to the right contains an action on a
tree.

######  Example 1.5.

Consider a group $G(\gamma)=\langle\mathbb{Z}^{n},t:tzt^{-1}=\gamma z\text{
for all }z\in\mathbb{Z}^{n}\rangle$, where $\gamma\in M_{n}(\mathbb{Z})$ is
one of the following matrices:

$\gamma=\begin{pmatrix}2&0&0\\\ 0&3&1\\\ 0&0&3\end{pmatrix},\ \
\gamma=\begin{pmatrix}0&0&-210\\\ 1&0&-1\\\ 0&1&0\end{pmatrix},\text{ or
}\gamma=\begin{pmatrix}0&0&0&2\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\end{pmatrix}.$

Theorem 1.2 can be used to show that $\mathcal{H}(G(\gamma))$ is as pictured
in Figure 2(a), 2(b), or 2(c), respectively. See Section 5.2.2 for details.

It is not immediately clear how common admissible matrices are. The following
corollaries show that Theorem 1.2 applies to an abundant family of examples.
Moreover, in the special cases covered by these corollaries, we give a simple,
explicit description of $\mathcal{P}_{-}(G)$.

######  Corollary 1.6.

Let $p(x)\in\mathbb{Z}[x]$ be monic and irreducible with all roots outside the
unit disk in $\mathbb{C}$. Set $R=\mathbb{Z}[x]/(p)$, $\gamma=x+(p)$, and
$G=G(R,\gamma)$. Then $\mathcal{H}(G)$ is described by Theorem 1.2 and,
moreover, $\mathcal{P}_{-}(G)$ is isomorphic to
$\operatorname{Div}(1,\ldots,1)$ where the number of $1$’s between the
parentheses is the number of real roots of $p$ plus half the number of complex
roots of $p$.

If the constant term of $p$ is square-free, then the problem of computing the
prime factorization of $p$ in $\mathbb{Z}[[x]]$ is straightforward, leading
to:

######  Corollary 1.7.

Suppose, in addition to to the hypotheses of Corollary 1.6, that the constant
term of $p$ is square-free. Then $\mathcal{H}(G)$ is described by Theorem 1.2
and, moreover,

  1. 1.

$\mathcal{P}_{+}(G)$ is isomorphic to $\operatorname{Div}(1,\ldots,1)$ where
the number of $1$’s between the parentheses is the number of prime factors of
the constant term of $p$; and

  2. 2.

$\mathcal{P}_{-}(G)$ is isomorphic to $\operatorname{Div}(1,\ldots,1)$ where
the number of $1$’s between the parentheses is the number of real roots of $p$
plus half the number of complex roots of $p$.

Part of the description in Theorem 1.2 holds in greater generality. In
particular, we can drop the requirement that $\mathbb{Z}^{n}$ is a cyclic
module.

######  Theorem 1.8.

Let $G$ be an ascending HNN extension of $\mathbb{Z}^{n}$ by a matrix
$\gamma\in M_{n}(\mathbb{Z})$ which is expanding and whose characteristic and
minimal polynomials are equal. The poset $\mathcal{H}(G)$ consists of the
following structures: a single elliptic structure, which is dominated by a
single lineal structure, and two subposets $\mathcal{P}_{-}(G)$ and
$\mathcal{P}_{+}(G)$ that intersect in the single lineal structure. The poset
$\mathcal{P}_{-}(G)$ is isomorphic to the poset of invariant subspaces of
$\mathbb{R}^{n}$ under $\gamma$. Each element of $\mathcal{P}_{-}(G)$ contains
an action on a quasi-convex subspace of a Heintze group. Moreover,
$\mathcal{P}_{+}(G)$ is a lattice.

We expect that the theory developed in this paper will lay the groundwork for
future, more general classifications of hyperbolic actions of metabelian
groups. A first direction would be to solve:

######  Problem 1.9.

Complete the classification of hyperbolic actions of abelian-by-cyclic groups
defined by expanding integer matrices.

This will necessitate describing $\mathcal{P}_{+}(G)$ in the case that
$\mathbb{Z}^{n}$ is not a cyclic $\mathbb{Z}[x]$-module. We anticipate that
this can be done by completing $\mathbb{Z}^{n}$ _as a module_ and classifying
the hyperbolic structures of $G$ in terms of closed sub-modules of the
completion. A next direction would be:

######  Problem 1.10.

Classify hyperbolic actions of finitely generated metabelian groups with
abelianizations of rank $>1$.

The machinery developed in [2] may be useful here. For a general metabelian
group $G$ with (abelian) commutator subgroup $A=[G,G]$ and abelianization
$Q=G/A$, the abelian group $A$ is a $\mathbb{Z}[Q]$-module. It may be possible
to classify (certain) hyperbolic structures on $G$ in terms of closed sub-
modules of certain completions of $A$.

A direction slightly tangential to the methods of this paper would be the
following:

######  Problem 1.11.

Classify hyperbolic actions of polycyclic groups.

Some cases of this problem were solved in [4], and it is possible that the
general case may be approached using invariant subspaces and possibly methods
from algebraic number theory.

###  1.1 About the proofs

The proof of Theorem 1.1 utilizes the description of $G=G(R,\gamma)$ as a
semidirect product $G=\gamma^{-1}R\rtimes\mathbb{Z}$, where $\gamma^{-1}R$ is
the localization of $R$ with respect to the powers of $\gamma$. This
localization is a larger ring containing $R$ in which division by $\gamma$ is
allowed. The proof uses the correspondence, first described in [16], of
_confining subsets_ of $\gamma^{-1}R$ and certain hyperbolic actions of $G$;
such subsets are attracting under multiplication by $\gamma$ and nearly closed
under addition. Another crucial ingredient in the proof is the _$(\gamma)$
-adic completion_ $\widehat{R}$ of $R$. This is the inverse limit of the
quotients $R/(\gamma^{n})$; see Section 3.1 for details.

Theorem 1.1 is then proven by showing the following correspondences:

$\text{ideals of }\widehat{R}\leftrightarrow\text{confining subsets of
}\gamma^{-1}R\text{ under }\gamma\leftrightarrow\text{valuations on
}\gamma^{-1}R\leftrightarrow\text{actions of }G(R,\gamma)\text{ on trees.}$

For the first correspondence, we write an element of $\gamma^{-1}R$ with a
(possibly infinite) base-$\gamma$ address

$a_{-k}\gamma^{-k}+a_{-k+1}\gamma^{-k+1}+\cdots+a_{0}+a_{1}\gamma+\cdots.$

The key step is to show that in a confining subset $Q$ of $\gamma^{-1}R$, the
fractional parts $a_{-k}\gamma^{-k}+\cdots+a_{-1}\gamma^{-1}$ of elements are
governed by an ideal $\mathfrak{a}$ of $\widehat{R}$. In particular we show
that, up to a small amount of ambiguity, the fractional part will agree with
the first $k$ digits of an element

$a_{-k}+a_{-k+1}\gamma+\cdots+a_{-1}\gamma^{k-1}+b_{k}\gamma^{k}+b_{k+1}\gamma^{k+1}+\cdots\in\mathfrak{a},$

giving the correspondence between confining subsets of $\gamma^{-1}R$ and
ideals of $\widehat{R}$.

The proof of Theorem 1.2 involves two major steps: classifying ideals in the
completion of the ring $R=\mathbb{Z}[x]/(p)$ and classifying confining subsets
of $\gamma^{-1}R$ under multiplication by $\gamma^{-1}$. For the first step,
we show that the localization $\gamma^{-1}\widehat{R}$ is a product of rings
with a simple structure: each ring in the product has a single finite chain of
ideals, corresponding to a chain of prime power divisors
$1,p_{i},p_{i}^{2},\ldots,p_{i}^{n_{i}}$ of $p$. This uses recent work of
McDonough [28] and Elliott [18] on quotients and factorizations in formal
power series rings.

For the second step, we study invariant subspaces of $\gamma$ in
$\mathbb{R}^{n}$. We construct a confining subset of $\gamma^{-1}R$ from such
an invariant subspace by (essentially) looking at a neighborhood of the
subspace. Conversely, to recover an invariant subspace from a confining subset
we reverse engineer the construction by taking limits after repeatedly
applying $\gamma^{-1}$ to a confining subset.

For groups $G$ considered in Theorem 1.2, we use valuations to show that every
element of $\mathcal{P}_{+}(G)$ contains an actions on a tree. There is a
natural way to construct a tree associated to a valuation on a set, and this
construction can be modified to incorporate a group action (see Section 7 and
[15]). For a more general class of metabelian groups, hyperbolic structures in
general correspond to a certain form of generalized valuation on an abelian
group that we call a _pseudo-valuation_. This correspondence, described in
Appendix B, could be useful for exploring actions of solvable groups with
higher rank abelianizations.

##### Structure of the paper.

Section 2 gives the required preliminary information for this paper. Section 3
explains the conditions we consider on the element $\gamma$ and delves into
the details of radix representations in base $\gamma$ as well as
$(\gamma)-$adic completions. Section 4 considers the structure of the poset of
hyperbolic structures of a general abelian-by-cyclic group. Section 5
classifies the hyperbolic actions of many groups using Theorem 1.1, some of
which were obtained previously in other papers – this is meant to demonstrate
the power of the theorem, as well as provide some intuition to the reader
regarding the technical results of the following sections. Sections 6 and 9
describe the subposets $\mathcal{P}_{+}(G)$ and $\mathcal{P}_{-}(G)$ in terms
of ideals and invariant subspaces, respectively. Sections 7 and 10 describe
the elements of $\mathcal{P}_{+}(G)$ and $\mathcal{P}_{-}(G)$ via actions on
trees and Heintze groups, respectively. Appendix A deals with the proof of
Theorem 1.2. Lastly, Appendix B describes the correspondence between confining
subsets and pseudo-valuations.

##### Acknowledgements.

The authors thank Mladen Bestvina, Yair Minsky, Manuel Reyes, and Daniel Erman
for helpful conversations. The first author was supported by NSF grants
DMS-1803368 and DMS-2106906. The second author was supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 427320536
– SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044 390685587,
Mathematics Münster: Dynamics–Geometry–Structure. The third author was
partially supported by NSF grants DMS-1840190 and DMS-2202986.

##  2 Background

This section describes some necessary background for this paper. We start with
a more detailed explanation of our notation for posets and then delve into a
discussion of $\mathcal{H}(G)$.

###  2.1 Posets

For $n\in\mathbb{Z}_{\geq 0}$, let $[n]=\\{0,1,\ldots,n-1\\}$. The set $[n]$
is a poset with the usual partial order $\leq$ on $\mathbb{Z}$. For
$n_{1},\ldots,n_{r}\in\mathbb{Z}_{>0}$, denote by
$\operatorname{Div}(n_{1},\ldots,n_{r})$ the poset
$\operatorname{Div}(n_{1},\ldots,n_{r})\vcentcolon=\prod_{i=1}^{r}[n_{i}+1]$
endowed with the product order. Thus, for
$(i_{1},\ldots,i_{r}),(j_{1},\ldots,j_{r})\in\operatorname{Div}(n_{1},\ldots,n_{r})$
we have $(i_{1},\ldots,i_{r})\leq(j_{1},\ldots,j_{r})$ if and only if
$i_{k}\leq j_{k}$ for each $k$. The poset
$\operatorname{Div}(n_{1},\ldots,n_{r})$ is isomorphic to its opposite poset
via the isomorphism sending
$(i_{1},\ldots,i_{r})\in\operatorname{Div}(n_{1},\ldots,n_{r})$ to
$(n_{1}-i_{1},\ldots,n_{r}-i_{r})$. Moreover,
$\operatorname{Div}(n_{1},\ldots,n_{r})$ is a lattice with the meet operation
defined by taking the infimum in each entry and the join operation defined by
taking the supremum in each entry.

######  Example 2.1.

The posets $\operatorname{Div}(n_{1},\ldots,n_{r})$ have many different
guises.

  * •

If $n\in\mathbb{Z}_{>0}$ and $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ is the
prime factorization of $n$, then the poset of (positive) divisors of $n$
ordered by $m_{1}\leq m_{2}$ if $m_{1}$ divides $m_{2}$ is isomorphic to
$\operatorname{Div}(k_{1},\ldots,k_{r})$.

  * •

If $n\in\mathbb{Z}_{>0}$ and $\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$ is
the poset of subgroups of $\mathbb{Z}/n\mathbb{Z}$ ordered by inclusion, then
$\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$ is isomorphic to the poset of
(positive) divisors of $n$, hence isomorphic to
$\operatorname{Div}(k_{1},\ldots,k_{r})$ where $n=p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}}$ is the prime factorization as in the last example.

  * •

If $r\in\mathbb{Z}_{>0}$, $X$ is an $r$ element set, and $2^{X}$ denotes the
power set of $X$ ordered by inclusion, then $2^{X}$ is isomorphic to
$\operatorname{Div}(1,\ldots,1)$ (with $r$ entries between the parentheses).
The isomorphism can be described as follows: denote by $x_{1},\ldots,x_{r}$
the elements of $X$. The isomorphism $2^{X}\to\operatorname{Div}(1,\ldots,1)$
is given by sending a subset $Y\in 2^{X}$ to its _indicator sequence_
$(i_{1},\ldots,i_{r})$ where $i_{j}=1$ if $x_{j}\in Y$ and $i_{j}=0$ if
$x_{j}\notin Y$.

###  2.2 Comparing generating sets and group actions

Throughout this paper, all group actions on metric spaces are assumed to be
isometric. Given a metric space $X$, we denote by $d_{X}$ the distance
function on $X$. The $X$ will frequently be dropped from $d_{X}$ if it is
understood by context. Moreover, if $G$ is a group and $S$ is a generating set
of $G$, then we denote by $\|\cdot\|_{S}$ the word norm on $G$ and by $d_{S}$
the word metric $d_{S}(g,h)=\|gh^{-1}\|_{S}$.

######  Definition 2.2 ([1, Definition 1.1]).

Let $S$, $T$ be two (possibly infinite) generating sets of a group $G$. We say
that $S$ is _dominated_ by $T$, written $S\preceq T$, if the identity map on
$G$ induces a Lipschitz map of metric spaces $(G,d_{T})\to(G,d_{S})$. This is
equivalent to requiring $\sup_{t\in T}\|t\|_{S}<\infty$. The relation
$\preceq$ is a preorder on the set of generating sets of $G$, and therefore it
induces an equivalence relation in the standard way:

$S\sim T\;\;\Leftrightarrow\;\;S\preceq T\;{\rm and}\;T\preceq S.$

This is equivalent to the condition that the Cayley graphs $\Gamma(G,S)$ and
$\Gamma(G,T)$ with respect to $S$ and $T$ are $G$–equivariantly quasi-
isometric. We denote by $[S]$ the equivalence class of a generating set $S$,
and by $\mathcal{G}(G)$ the set of all equivalence classes of generating sets
of $G$. The preorder $\preceq$ induces a partial order $\preccurlyeq$ on
$\mathcal{G}(G)$ in the standard way:

$[S]\preccurlyeq[T]\;\;\Leftrightarrow\;\;S\preceq T.$

For example, all finite generating sets of a finitely generated group are
equivalent and the equivalence class containing any finite generating set is
the largest element of $\mathcal{G}(G)$. For every group $G$, the smallest
element of $\mathcal{G}(G)$ is $[G]$. Note also that this order is “inclusion
reversing”: if $S$ and $T$ are generating sets of $G$ such that $S\subseteq
T$, then $T\preceq S$.

To define a hyperbolic structure on a group, we first recall the definition of
a hyperbolic space. In this paper we employ the definition of hyperbolicity
via the Rips condition.

######  Definition 2.3.

A metric space $X$ is called _$\delta$ –hyperbolic_ if it is geodesic and for
any geodesic triangle $\Delta$ in $X$, each side of $\Delta$ is contained in
the union of the closed $\delta$–neighborhoods of the other two sides.

######  Definition 2.4 ([1, Definition 1.2]).

A _hyperbolic structure_ on $G$ is an equivalence class $[S]\in\mathcal{G}(G)$
such that the Cayley graph $\Gamma(G,S)$ with respect to $S$. Since
hyperbolicity of geodesic metric spaces is quasi-isometry invariant, this
definition is independent of the choice of the representative $S$. We denote
the set of hyperbolic structures by $\mathcal{H}(G)$. It is a sub-poset of
$\mathcal{G}(G)$ with the restriction of the partial order on
$\mathcal{G}(G)$.

The poset $\mathcal{H}(G)$ classifies the _cobounded_ hyperbolic actions of
$G$ up to coarsely equivariant quasi-isometry, as we now summarize.

######  Definition 2.5.

The action $G\curvearrowright X$ is _cobounded_ if for some (equivalently any)
$x\in X$ there exists $R>0$ such that every point of $X$ is distance at most
$R$ from some point of the orbit $Gx$. Given two cobounded hyperbolic actions
$G\curvearrowright X$ and $G\curvearrowright Y$, a map $f:X\to Y$ is _coarsely
equivariant_ if for any $x\in X$ we have

$\sup_{g\in G}d_{Y}(f(gx),gf(x))<\infty.$

Given $C>0$, the map $f$ is _$C$ –coarsely Lipschitz_ if

$d_{Y}(f(x),f(y))\leq Cd_{X}(x,y)+C$

for all $x,y\in X$. It is a _$C$ –quasi-isometry_ if it is $C$–coarsely
Lipschitz and also satisfies

$\frac{1}{C}d_{X}(x,y)-C\leq d_{Y}(f(x),f(y)).$

Given a cobounded hyperbolic action $G\curvearrowright X$, there is an
associated hyperbolic structure given by the following Schwarz-Milnor Lemma:

######  Lemma 2.6 ([1, Lemma 3.11]).

Let $G\curvearrowright X$ be a cobounded hyperbolic action of $G$. Let
$B\subseteq X$ be a bounded subset such that $\displaystyle\bigcup_{g\in
G}gB=X$. Let $D=\operatorname{diam}(B)$ and let $x\in B$. Then $G$ is
generated by the set

$S=\\{g\in G:d_{X}(x,gx)\leq 2D+1\\},$

and $X$ is $G$–equivariantly quasi-isometric to $\Gamma(G,S)$.

Thus, up to equivariant quasi-isometries, hyperbolic actions of $G$ correspond
to actions of $G$ on its hyperbolic Cayley graphs. There is an equivalence
relation on hyperbolic actions given by $G$–coarsely equivariant quasi-
isometry and a preorder on hyperbolic actions defined by $(G\curvearrowright
X)\preceq(G\curvearrowright Y)$ if there is a coarsely equivariant coarsely
Lipschitz map $Y\to X$. In this case we say that the action $G\curvearrowright
X$ is _dominated by_ the action $G\curvearrowright Y$. With these relations,
the set of _equivalence classes of cobounded hyperbolic actions_ of $G$
becomes a poset. This poset is isomorphic to $\mathcal{H}(G)$ [1, Proposition
3.12].

We will frequently speak about representatives of hyperbolic structures. If
$[S]\in\mathcal{H}(G)$, we say that $[S]$ is represented by a cobounded
hyperbolic action $G\curvearrowright X$ if $G\curvearrowright X$ is equivalent
to the action $G\curvearrowright\Gamma(G,S)$; that is, if there is a coarsely
$G$-equivariant quasi-isometry $X\to\Gamma(G,S)$.

###  2.3 Actions on hyperbolic spaces

Let $G$ be a group acting on a hyperbolic space $X$, and denote the Gromov
boundary of $X$ by $\partial X$. In general, $X$ is not assumed to be proper,
and its boundary is defined as the set of equivalence classes of sequences
convergent at infinity. We also denote by $\Lambda(G)$ the set of limit points
of $G$ on $\partial X$. That is, $\Lambda(G)\vcentcolon=\partial
X\cap\overline{Gx},$ where $\overline{Gx}$ denotes the closure of a $G$–orbit
in $X\cup\partial X$, for any choice of basepoint $x\in X$. This definition is
independent of the choice of $x\in X$. The following theorem summarizes the
standard classification of group actions on hyperbolic spaces due to Gromov
[22, Section 8.2] and the results [16, Propositions 3.1 and 3.2].

######  Theorem 2.7.

Let $G$ be a group acting on a hyperbolic space $X$. Then exactly one of the
following conditions holds.

  1. 1)

$|\Lambda(G)|=0$. Equivalently, $G$ has bounded orbits. In this case the
action of $G$ is called _elliptic_.

  2. 2)

$|\Lambda(G)|=1$. In this case the action of $G$ is called _parabolic_. A
parabolic action cannot be cobounded.

  3. 3)

$|\Lambda(G)|=2$. Equivalently, $G$ contains a loxodromic element and any two
loxodromic elements have the same limit points on $\partial X$. In this case
the action of $G$ is called _lineal_. A lineal action $G\curvearrowright X$ is
said to be _orientable_ if $G$ fixes its limit points on $\partial X$
_pointwise_.

  4. 4)

$|\Lambda(G)|=\infty$. Then $G$ always contains loxodromic elements. In turn,
this case breaks into two subcases.

    1. (a)

$G$ fixes a point of $\partial X$. Equivalently, any two loxodromic elements
of $G$ have a common limit point on the boundary. In this case the action of
$G$ is called _quasi-parabolic_ or _focal_.

    2. (b)

$G$ does not fix any point of $\partial X$. In this case the action of $G$ is
said to be of _general type_.

It follows from the above that for any group $G$,

$\mathcal{H}(G)=\mathcal{H}_{e}(G)\sqcup\mathcal{H}_{\ell}(G)\sqcup\mathcal{H}_{qp}(G)\sqcup\mathcal{H}_{gt}(G)$

where the sets of elliptic, lineal, quasi-parabolic, and general type
hyperbolic structures on $G$ are denoted by $\mathcal{H}_{e}(G)$,
$\mathcal{H}_{\ell}(G)$, $\mathcal{H}_{qp}(G)$, and $\mathcal{H}_{gt}(G)$
respectively. Namely, $[S]\in\mathcal{H}(G)$ lies in $\mathcal{H}_{\ell}(G)$
if the action $G\curvearrowright\Gamma(G,S)$ is lineal, and similar
definitions hold for the posets $\mathcal{H}_{e},\mathcal{H}_{qp},$ and
$\mathcal{H}_{gt}$. We direct the reader to [1, Section 4] and [1, Theorem
4.6] for further explanation. We also note that
$\mathcal{H}_{gt}(G)=\emptyset$ for a solvable group $G$, as $G$ contains no
free subgroups (which can be produced from a general type action using the
Ping-Pong lemma).

##### The Busemann pseudocharacter.

A function $q\colon G\to\mathbb{R}$ is a _quasi-character_ (or _quasi-
morphism_) if there exists a constant $D$ such that

$|q(gh)-q(g)-q(h)|\leq D$

for all $g,h\in G$. We say that $q$ has _defect at most $D$_. If, in addition,
the restriction of $q$ to every cyclic subgroup of $G$ is a homomorphism, then
$q$ is called a _pseudocharacter_ (or _homogeneous quasi-morphism_).

Given any action of a group $G$ on a hyperbolic space $X$ fixing a point on
the boundary, one can associate a natural pseudocharacter $\beta$ called the
_Busemann pseudocharacter_. If $\beta$ is a homomorphism, then the action
$G\curvearrowright X$ is called _regular_. As we do not require the exact
definition of $\beta$ in this paper, we refer the reader to [22, Sec. 7.5.D]
and [27, Sec. 4.1] for details. An element $g\in G$ is loxodromic with respect
to the action of $G$ on $X$ if and only if $\beta(g)\neq 0$. In particular,
$\beta$ is not identically zero whenever $G\curvearrowright X$ is quasi-
parabolic or orientable lineal.

Conversely, given a pseudocharacter on a group $G$, one can always construct
an orientable lineal action:

######  Lemma 2.8 ([1, Lemma 4.15]).

Let $p\colon G\to\mathbb{R}$ be a non-zero pseudocharacter. Let $C$ be any
constant such that the defect of $p$ is at most $C/2$ and there exists a value
of $p$ in the interval $(0,C/2)$. Let

$X=X_{p,C}=\\{g\in G:|p(g)|<C\\}.$

Then $X$ generates $G$ and the map $p\colon(G,d_{X})\to\mathbb{R}$ is a quasi-
isometry. In particular, $[X]$ is orientable lineal.

Further, using the Busemann pseudocharacter of a quasi-parabolic action in the
above lemma yields the following relation between quasi-parabolic and lineal
structures.

######  Lemma 2.9 ([1, Corollary 4.26]).

For any $[Y]\in\mathcal{H}_{qp}(G)$, there exists an (orientable)
$[Z]\in\mathcal{H}_{\ell}(G)$ such that $[Z]\preccurlyeq[Y]$. In particular,
if $\mathcal{H}_{qp}(G)\neq\emptyset$, then
$\mathcal{H}_{\ell}(G)\neq\emptyset$.

##### Relation to confining subsets.

Consider a group $G=H\rtimes_{\alpha}\mathbb{Z}$ where $\alpha\in Aut(H)$ and
the generator $t\in\mathbb{Z}$ acts on $H$ by conjugation via
$tht^{-1}=\alpha(h)$ for any $h\in H$. Let $Q$ be a symmetric subset of $H$.
The following definition, developed by Caprace-Cornulier-Monod-Tessera in [16,
Section 4], forms the main tool used in the work in [8, 3, 4]. Here $Q\cdot Q$
denotes the set of products $\\{g\cdot h\in H:g,h\in Q\\}$.

######  Definition 2.10.

Let $(H,\cdot)$ be a group, let $Q$ be a symmetric subset of $H$, and let
$\alpha$ be an automorphism of $H$. The action of $\alpha$ is confining $H$
into $Q$ if it satisfies the following conditions$\colon$

  * (a)

$\alpha(Q)\subseteq Q$

  * (b)

$H=\displaystyle\bigcup_{n\geq 0}\hskip 5.0pt\alpha^{-n}(Q)$; and

  * (c)

$\alpha^{k_{0}}(Q\cdot Q)\subseteq Q$ for some non-negative integer $k_{0}$.

We also call the set $Q$ _confining under $\alpha$_. If $\alpha(Q)\subsetneq
Q$, then $Q$ is called _strictly confining._

The definition of a confining subset given in [16] does not require symmetry
of the subset $Q\subseteq H$. However, according to [16, Theorem 4.1], to
classify regular quasi-parabolic structures on the group
$G=H\rtimes_{\alpha}\mathbb{Z}$, it suffices to consider only confining
subsets which are symmetric; see also [3, Proposition 2.6].

######  Proposition 2.11 ([16, Proposition 4.6, Theorem 4.1]).

Let $H$ be a group and $\alpha$ an automorphism of $H$ which confines $H$ into
some subset $Q\subseteq H$. Consider the group
$G=H\rtimes_{\alpha}\mathbb{Z}$, and let $t$ denote a generator of
$\mathbb{Z}$. Define $S=Q\cup\\{t^{\pm 1}\\}\subseteq G$. Then $\Gamma(G,S)$
is Gromov hyperbolic. If $Q$ is strictly confining then the action
$G\curvearrowright\Gamma(G,S)$ is regular quasi-parabolic. If $Q$ is not
strictly confining then the action $G\curvearrowright\Gamma(G,S)$ is lineal.

In general, confining subsets may not completely describe all the quasi-
parabolic actions of the group $G$. However, under certain conditions,
including the case when $H$ is abelian, there is a one-to-one correspondence
between strictly confining subsets (considered up to a natural equivalence)
and quasi-parabolic actions of $G$:

######  Proposition 2.12 ([16, Proposition 4.5],[2, Theorem 1.2]).

Let $G=H\rtimes_{\alpha}\mathbb{Z}$ be a semidirect product and let
$G\curvearrowright X$ be a cobounded hyperbolic action with a fixed point on
$\partial X$. Let $p:G\to\mathbb{R}$ be the Busemann pseudocharacter
associated to this action and suppose that $p(H)=0$. Then there exists a
subset $Q\subset H$ which is confining under the action of $\alpha$ or
$\alpha^{-1}$ such that $X$ is $G$-equivariantly quasi-isometric to
$\Gamma(G,Q\cup\\{t^{\pm 1}\\})$.

##  3 Axioms for commutative rings

In this section we consider groups associated to certain rings. We will
introduce a convenient way of parametrizing the elements of such a ring $R$
and its completion $\widehat{R}$.

Consider a ring $R$ together with an element $\gamma\in R$. We assume the
following properties, which will naturally lead to our parametrization:

  1. (A1)

$\gamma$ is neither a unit nor a zero divisor in $R$;

  2. (A2)

$R$ is generated as a $\mathbb{Z}$-algebra by $\gamma$;

  3. (A3)

the ideal $(\gamma)$ generated by $\gamma$ is finite index in $R$;

  4. (A4)

the intersection of ideals $\bigcap_{n=1}^{\infty}(\gamma^{n})$, is zero.

By the universal property of the polynomial ring $\mathbb{Z}[x]$, there is a
unique homomorphism $\mathbb{Z}[x]\twoheadrightarrow R$ sending $x$ to
$\gamma$. Therefore there is a natural isomorphism
$R\cong\mathbb{Z}[x]/\mathfrak{a}$ for some ideal $\mathfrak{a}$ in
$\mathbb{Z}[x]$, so $R$ is a commutative ring. As a quotient of the Noetherian
ring $\mathbb{Z}[x]$, it is also necessarily Noetherian.

######  Remark 3.1.

If $R$ happens to be an integral domain, then axiom (A4) follows from the
other axioms; see [6, Corollary 10.18]. However, we will not generally assume
that $R$ is a domain.

Consider $R$ with its additive abelian group structure. Left multiplication by
$\gamma$ defines an injective endomorphism _of abelian groups_ $\alpha\colon
R\to R$. We define a group $G(R,\gamma)$ to be the ascending HNN extension of
(the abelian group) $R$ by the endomorphism $\alpha$:

$G(R,\gamma)=\langle R,t:trt^{-1}=\gamma r\text{ for }r\in R\rangle.$

We will denote this group simply by $G$ in the case that $R$ and $\gamma$ are
clear from context. There is another description of this group in terms of
localizations of rings. The powers $\\{1,\gamma,\gamma^{2},\ldots\\}$ form a
multiplicatively closed subset $S$ of $R$. Consider the _localization_
$S^{-1}R$ of $R$ by $S$, denoted simply by $\gamma^{-1}R$ for short. Recall
that $\gamma^{-1}R$ consists of the quotients $r/\gamma^{k}$ for $r\in R$ and
$k\geq 0$. There is an equivalence relation on such quotients defined by
$r/\gamma^{k}=s/\gamma^{l}$ if $r\gamma^{l}=s\gamma^{k}$ (a more complicated
definition is required if $S$ contains zero divisors, but this is not
necessary in our case). The multiplication-by-$\gamma$ endomorphism
$\alpha\colon R\to R$ extends to an automorphism (of abelian groups) of
$\gamma^{-1}R$ which we will continue to denote by $\alpha$.

The localization $\gamma^{-1}R$ can be used to give another description of
$G(R,\gamma)$. We first state the following ubiquitously used and easy to
prove lemma:

######  Lemma 3.2 ([3, Lemma 2.20]).

Let $G=A\rtimes_{\alpha}\mathbb{Z}=\langle A,t:tat^{-1}=\alpha(a)\text{ for
}a\in A\rangle$. Suppose that $Q\subset A$ is a subset with the property that
$Q\cup\\{t^{\pm 1}\\}$ is a generating set of $G$ and $\alpha(Q)\subset Q$.
Then any element $w\in G$ can be written as

$w=t^{-r}x_{1}\cdots x_{m}t^{\ell}$

where $r,\ell\geq 0$, $x_{i}\in Q$ for all $i$, and
$r+\ell+m=\|w\|_{Q\cup\\{t^{\pm 1}\\}}$. Moreover, if $w\in A$ then $r=\ell$.

######  Lemma 3.3.

The ascending HNN extension $G(R,\gamma)$ is isomorphic to the semidirect
product $\gamma^{-1}R\rtimes_{\alpha}\mathbb{Z}$.

###### Proof.

Let $s$ be a generator of $\mathbb{Z}$ in the semidirect product, so that $s$
acts on $\gamma^{-1}R$ by conjugation via multiplication by $\gamma$. As
above, $t$ denotes the stable letter of $G=G(R,\gamma)$. Sending $t$ to $s$
and $R$ to $\gamma^{-1}R$ by the natural inclusion defines a homomorphism
$F\colon G\to\gamma^{-1}R\rtimes_{\alpha}\mathbb{Z}$. To see that this
homomorphism is surjective, it suffices to show that
$\gamma^{-n}r\in\gamma^{-1}R$ (with $r\in R$) is in the image of $F$. We may
alternatively write this as $s^{-n}rs^{n}$, which is clearly in the image of
$F$ since $s$ and $r$ are. Applying Lemma 3.2 with $A$ the normal subgroup of
$G$ generated by $R$ and $Q=R$, we may write an element of $G$ as
$t^{-k}rt^{l}$ with $r\in R$ and $k,l\geq 0$. To prove injectivity, consider
such an element $t^{-k}rt^{l}$ of $G$. The image under $F$ is
$s^{-k}rs^{l}=(\gamma^{-k}r)s^{-k+l}$. This is equal to the identity if and
only if $-k+l=0$ and $\gamma^{-k}r=0$. Since $\gamma$ is not a zero divisor,
we have $r=0$. Thus $t^{-k}rt^{l}=t^{-k}rt^{k}$ is the identity of $G$. ∎

######  Example 3.4.

The following are examples of rings $R$ and elements $\gamma$ along with the
resulting groups $G$.

  * •

Taking $R=\mathbb{Z}$ and $\gamma=n\in\mathbb{Z}\setminus\\{-1,0,1\\}$ yields
$G(R,\gamma)=\langle
a,t:tat^{-1}=a^{n}\rangle=BS(1,n)\cong\mathbb{Z}[1/n]\rtimes\mathbb{Z}$. Note
that $R$ and $\gamma$ satisfy the axioms (A1)–(A4) since $R$ contains no zero
divisors, $R/(\gamma)\cong\mathbb{Z}/n\mathbb{Z}$, and $R$ is trivially
generated as a $\mathbb{Z}$-algebra by $\gamma$.

  * •

Taking $R=(\mathbb{Z}/n\mathbb{Z})[x]$, $\gamma=x$ yields
$G(R,\gamma)\cong(\mathbb{Z}/n\mathbb{Z})[x^{\pm
1}]\rtimes\mathbb{Z}=(\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z}$. Again, the reader
may check that $R$ and $\gamma$ satisfy (A1)–(A4).

When discussing confining subsets, it will usually be convenient to think of
$G$ using the semidirect product description given by Lemma 3.3.

###  3.1 Number systems and completions

In this subsection we parametrize elements of $R$. Specifically, we will write
elements of $R$ in “base-$\gamma$.” Since the ideal $(\gamma)$ is finite index
in $R$, we may choose a finite _transversal_ for the cosets of $(\gamma)$ in
$R$. That is, $T$ contains exactly one element from each coset of $(\gamma)$.
In fact, we will choose a standard transversal as follows:

######  Lemma 3.5.

There is a number $d\in\mathbb{Z}_{>0}$ such that $T=[d]=\\{0,1,\ldots,d-1\\}$
forms a transversal for $(\gamma)$.

###### Proof.

Recall that $R=\mathbb{Z}[x]/\mathfrak{a}$. There is a surjective homomorphism
of rings $\mathbb{Z}\cong\mathbb{Z}[x]/(x)\twoheadrightarrow R/(\gamma)$. If
$d$ is the order of the cyclic group $R/(\gamma)$, then $[d]$ is a
transversal. ∎

The transversal $T=[d]$ defines a base-$\gamma$ expansion for each element of
$R$ as follows. Consider an element $r\in R$. There exists a unique element
$a_{0}\in T$ such that $r-a_{0}\in(\gamma)$, and so $r-a_{0}=r_{1}\gamma$ for
some $r_{1}\in R$. Since $\gamma$ is not a zero divisor, the element $r_{1}$
is uniquely determined by $r-a_{0}$ and hence by $r$ itself. There is in turn
a unique $a_{1}\in T$ with $r_{1}-a_{1}\in(\gamma)$, and so
$r_{1}-a_{1}=r_{2}\gamma$ for some $r_{2}\in R$. We have

$r=a_{0}+r_{1}\gamma=a_{0}+(a_{1}+r_{2}\gamma)\gamma=a_{0}+a_{1}\gamma+r_{2}\gamma^{2}.$

Suppose for induction that we have found $a_{0},a_{1},\ldots,a_{n-1}\in T$ and
$r_{n}\in R$, each uniquely determined by $r$, such that

$r=a_{0}+a_{1}\gamma+\cdots+a_{n-1}\gamma^{n-1}+r_{n}\gamma^{n}.$

Then $r_{n}=a_{n}+r_{n+1}\gamma$ for some $a_{n}\in T$ and $r_{n+1}\in R$.
Since $r_{n}$ is uniquely determined by $r$, so are $a_{n}$ and $r_{n+1}$. We
have as before,

$r=a_{0}+\cdots+a_{n}\gamma^{n}+r_{n+1}\gamma^{n+1}.$

We will call the formal power series
$a_{0}+a_{1}\gamma+a_{2}\gamma^{2}+\cdots$ the _$\gamma$ -adic address_ for
$r$ or the base-$\gamma$ expansion of $r$. If $r,s\in R$ have the same
$\gamma$-adic address $a_{0}+a_{1}\gamma+\cdots$, then they must in fact be
equal, since for any $n$, there are $r_{n},s_{n}\in R$ with

$r=a_{0}+a_{1}\gamma+\cdots+a_{n-1}\gamma^{n-1}+r_{n}\gamma^{n},\ \ \
s=a_{0}+a_{1}\gamma+\cdots+a_{n-1}\gamma^{n-1}+s_{n}\gamma^{n}.$

We thus have $r-s=(r_{n}-s_{n})\gamma^{n}\in(\gamma^{n})$. Since
$\bigcap_{n=1}^{\infty}(\gamma^{n})=0$, this shows that $r=s$. To summarize:

######  Lemma 3.6.

Each $r\in R$ has a unique $\gamma$-adic address. If $r,s\in R$ have the same
$\gamma$-adic address, then $r=s$.

Such addresses lead us naturally to consider _completions_. Completions will
also be crucial to discussing hyperbolic structures on $G(R,\gamma)$. The
$(\gamma)$-adic completion $\widehat{R}$ is the inverse limit $\varprojlim
R/(\gamma^{n})$. Here the morphisms $R/(\gamma^{n})\to R/(\gamma^{m})$ for
$n\geq m$ are the natural quotient maps. The elements of $\widehat{R}$ are
sequences $(x_{n}+(\gamma^{n}))_{n=0}^{\infty}$ with $x_{n}\in R$ for each $n$
and $x_{n}+(\gamma^{m})=x_{m}+(\gamma^{m})$ for $n\geq m$. Entry-by-entry
addition and multiplication endow $\widehat{R}$ with well-defined ring
operations. Furthermore, each element of $\widehat{R}$ has a well-defined
$\gamma$-adic address. To see this, choose
$x=(x_{n}+(\gamma^{n}))\in\widehat{R}$ and fix $m\geq 0$. The element $x_{m}$
has a $\gamma$-adic address $a_{0}+a_{1}\gamma+\cdots$. We have in particular
that

$x_{m}+(\gamma^{m})=a_{0}+a_{1}\gamma+\cdots+a_{m-1}\gamma^{m-1}+(\gamma^{m}).$

If $n\geq m$ and $b_{0}+b_{1}\gamma+\cdots$ is the $\gamma$-adic address of
$x_{n}$ then

$b_{0}+\cdots+b_{m-1}\gamma^{m-1}+(\gamma^{m})=x_{n}+(\gamma^{m})=x_{m}+(\gamma^{m})=a_{0}+\cdots+a_{m-1}\gamma^{m-1}+(\gamma^{m}).$

We must have $b_{i}=a_{i}$ for each $i\leq m-1$.

Consequently, if we choose $a_{0},\ldots,a_{n}$ to agree with the coefficients
of the first $n+1$ terms in the $\gamma$-adic address of $x_{n+1}$, then the
sequence $a_{0},a_{1},\ldots$ is well-defined, independent of $n$, and we
define $a_{0}+a_{1}\gamma+\cdots$ to be the $\gamma$-adic address of $x$. If
$x=(x_{n}+(\gamma^{n})),y=(y_{n}+(\gamma^{n}))\in\widehat{R}$ have the same
$\gamma$-adic address, then we see by using addresses that
$x_{n}+(\gamma^{n})=y_{n}+(\gamma^{n})$ for each $n$, so that $x=y$. Finally,
given any sequence $a_{0},a_{1},\ldots$ of elements of $T$, the sequence
$(a_{0}+\cdots+a_{n-1}\gamma^{n-1}+(\gamma^{n}))_{n=0}^{\infty}$ is an element
of $\widehat{R}$. This yields:

######  Lemma 3.7.

Each $x\in\widehat{R}$ has a unique $\gamma$-adic address. If
$x,y\in\widehat{R}$ have the same $\gamma$-adic address, then $x=y$. Finally,
given any sequence $a_{0},a_{1},\ldots$ of elements of $T$, there is an
element of $\widehat{R}$ with $\gamma$-adic address
$a_{0}+a_{1}\gamma+a_{2}\gamma^{2}+\cdots$.

The ring $\widehat{R}$ is endowed with the inverse limit topology. That is, as
the inverse limit of the rings $R/(\gamma^{n})$ (considered as discrete sets)
it is a subset of the product $\prod_{n=0}^{\infty}R/(\gamma^{n})$ and
inherits the subspace topology. In this topology a sequence
$\\{a_{0}^{i}+a_{1}^{i}\gamma+a_{2}^{i}\gamma^{2}+\ldots\\}_{i=1}^{\infty}\subset\widehat{R}$
converges to $b_{0}+b_{1}\gamma+b_{2}\gamma^{2}+\ldots$ if and only if for
each $j$, $a_{j}^{i}$ is eventually equal to $b_{j}$ (as $i\to+\infty$).

Finally we discuss the operations on $\widehat{R}$ in terms of $\gamma$-adic
addresses. Consider $x=\sum_{i=0}^{\infty}a_{i}\gamma^{i}$ and
$y=\sum_{i=0}^{\infty}b_{i}\gamma^{i}$. Then
$x+y=\sum_{i=0}^{\infty}c_{i}\gamma^{i}$ where the $c_{i}$ are determined
inductively as follows. First of all, $c_{0}$ is the unique element of $T$ in
the same coset as $a_{0}+b_{0}$. We have $a_{0}+b_{0}=c_{0}+r_{0}\gamma$ for
some $r_{0}\in R$. If $c_{0},\ldots,c_{n}\in T$ and $r_{0},\ldots,r_{n}\in R$
have been determined such that $a_{i}+b_{i}+r_{i-1}=c_{i}+r_{i}\gamma$ for
each $i\leq n$, then $c_{n+1}\in T$ and $r_{n+1}\in R$ are determined uniquely
by the equation $a_{n+1}+b_{n+1}+r_{n}=c_{n+1}+r_{n+1}\gamma$. This formula
for $x+y$ may be verified by summing

$\left(\sum_{i=0}^{n}a_{i}\gamma^{i}+\left(\gamma^{n+1}\right)\right)+\left(\sum_{i=0}^{n}b_{i}\gamma^{i}+\left(\gamma^{n+1}\right)\right)$

for each $n\geq 0$ and verifying that this is equal to
$\sum_{i=0}^{n}c_{i}\gamma^{i}+(\gamma^{n+1})$. Similarly,
$xy=\sum_{i=0}^{\infty}d_{i}\gamma^{i}$ where the $d_{i}$ are determined
inductively as follows. First, $a_{0}b_{0}=d_{0}+s_{0}\gamma$ with $d_{0}\in
T$ and $s_{0}\in R$. If $d_{0},\ldots,d_{n}\in T$ and $s_{0},\ldots,s_{n}\in
R$ have been determined such that

$\sum_{i=0}^{k}a_{i}b_{k-i}+s_{k-1}=d_{k}+s_{k}\gamma$

for $k\leq n$, then $d_{n+1}\in T$ and $s_{n+1}\in R$ are determined uniquely
by

$\sum_{i=0}^{n+1}a_{i}b_{n+1-i}+s_{n}=d_{n+1}+s_{n+1}\gamma.$

This formula for $xy$ may be verified analogously.

Finally, we introduce one more property of the pair $(R,\gamma)$ which will be
crucial for investigating hyperbolic structures on $G(R,\gamma)$. It says that
$\gamma$-adic addresses of elements of $R$ are “well-behaved.” The importance
of this property is for proving Lemma 6.3. We say that a $\gamma$-adic address
$a_{0}+a_{1}\gamma+\cdots$ is _eventually periodic_ if there exists $N\geq 0$
and $k>0$ with $a_{n+k}=a_{n}$ for all $n\geq N$. That is, the $a_{j}$
eventually repeat in a sequence of period $k$. Recall that the transversal
$T=[d]$ defines the $\gamma$-adic address of any element of $R$ or
$\widehat{R}$. We introduce one further axiom for the pair $(R,\gamma)$:

  1. (A5)

for every $r\in R$, the $\gamma$-adic address of $r$ is eventually periodic.

The eventually periodic property of addresses occurs immediately and naturally
in simple examples:

######  Example 3.8.

Consider the ring $R=\mathbb{Z}$ with $\gamma=2$. The $\gamma$-adic address of
$-1$ is $1+2+2^{2}+2^{3}+\cdots$. This would normally be written in base 2 as
$-1=111\cdots$. Thus the $\gamma$-adic address for $-1$ is eventually
periodic. In fact, writing $-1=a_{0}+a_{1}2+a_{2}2^{2}+\cdots$ we have
$a_{n}=a_{n+1}$ so that the eventually periodic property holds with $k=1$ and
for all values of $n$. For a (slightly) more complicated example, the
$\gamma$-adic address of $-13$ is

$1+2+2^{4}+2^{5}+2^{6}+2^{7}+\cdots.$

This is eventually periodic with $a_{n}=a_{n+1}$ for all $n\geq 4$.

######  Example 3.9.

Consider the ring $R=\mathbb{Z}[x]/(x^{2}-2)$ with $\gamma=x+(x^{2}-2)$. This
is isomorphic to $\mathbb{Z}[\sqrt{2}]$ via the isomorphism sending $\gamma$
to $\sqrt{2}$. A transversal is given by $[2]=\\{0,1\\}$. One may check that
the $\gamma$-adic address of $-1$ in this ring is

$-1=1+\gamma^{2}+\gamma^{4}+\gamma^{6}+\cdots$

(in fact this essentially follows from the address $-1=1+2+2^{2}+\cdots$ from
the last example). This is eventually periodic with period $k=2$.

In the ring $R=\mathbb{Z}[x]/(x^{3}-x-2)$ with $\gamma=x+(x^{3}-x-2)$, we
again have that $[2]$ is a transversal. The $\gamma$-adic address of $-1$ is

$-1=1+\gamma+\gamma^{3}+\gamma^{4}+\gamma^{6}+\gamma^{7}+\gamma^{9}+\cdots.$

This is eventually periodic with period $k=3$.

##  4 Generalities on confining subsets

In this section we consider the hyperbolic structures of a general abelian-by-
cyclic group $G=A\rtimes_{\alpha}\mathbb{Z}$. Under certain hypotheses about
abelianizations, we describe the structure of $\mathcal{H}(G)$ which holds
without any of the additional machinery introduced later in the paper. We will
use this structure extensively when proving our main theorems.

######  Proposition 4.1.

Let $G=A\rtimes_{\alpha}\mathbb{Z}$ with $A$ abelian, and denote by $t$ a
generator of $\mathbb{Z}$. Suppose that the abelianization of $G$ is virtually
cyclic and that the same holds for its finite index subgroup $A\rtimes\langle
t^{2}\rangle$. Then $\mathcal{H}(G)$ has the following structure: every
hyperbolic structure on $G$ is either elliptic, lineal, or quasi-parabolic.
The union $\mathcal{H}_{qp}(G)\cup\mathcal{H}_{\ell}(G)$ consists of two
subposets $\mathcal{P}_{-}(G)$ and $\mathcal{P}_{+}(G)$ which are both
lattices and intersect in the unique lineal structure of $G$. Every other
element of $\mathcal{P}_{-}(G)\cup\mathcal{P}_{+}(G)$ is quasi-parabolic. The
elements of $\mathcal{P}_{-}(G)\setminus\mathcal{H}_{\ell}(G)$ are
incomparable to the elements of
$\mathcal{P}_{+}(G)\setminus\mathcal{H}_{\ell}(G)$. Finally, the unique lineal
structure of $G$ dominates the unique elliptic structure. See Figure 3.

The rather technical condition that the abelianizations of $G=A\rtimes\langle
t\rangle$ and $A\rtimes\langle t^{2}\rangle$ are both virtually cyclic is
meant to rule out non-orientable lineal actions, as will become apparent in
the proof. The proposition does not hold, e.g., when $G=\langle
a,t:tat^{-1}=a^{-1}\rangle$ (the fundamental group of the Klein bottle), for
which the finite index subgroup $\langle a\rangle\rtimes\langle t^{2}\rangle$
is $\mathbb{Z}^{2}$.

$*$$\mathbb{R}$$\mathcal{P}_{+}(G)$$\mathcal{P}_{-}(G)$ Figure 3: The poset of
hyperbolic structures for a general abelian-by-cyclic group
$A\rtimes\mathbb{Z}$ under basic hypotheses on abelianizations. The subposets
$\mathcal{P}_{-}(G)$ and $\mathcal{P}_{+}(G)$ are lattices.

The remainder of this section is devoted to proving Proposition 4.1. We show
that we can identify $\mathcal{P}_{+}(G)$ with the poset of equivalence
classes of confining subsets of $A$ under $\alpha$, and, in Corollary 4.6, we
show that this latter poset is a lattice. An analogous argument holds for
$\mathcal{P}_{-}(G)$ with $\alpha^{-1}$ in place of $\alpha$. For the rest of
the section let $G=A\rtimes_{\alpha}\mathbb{Z}$, where $A$ is abelian and
$\alpha$ is an automorphism of $A$, and let $t$ denote the generator of the
$\mathbb{Z}$ factor of $G$ satisfying $tat^{-1}=\alpha(a)$ for all $a\in A$.
Finally, assume that the abelianizations of $G$ and the subgroup
$A\rtimes\langle t^{2}\rangle$ are virtually cyclic. Recall that subsets which
are confining under $\alpha$ are defined in Definition 2.10.

######  Lemma 4.2.

If $P,Q\subset A$ are confining under $\alpha$, then $P\cup\\{t^{\pm
1}\\}\preceq Q\cup\\{t^{\pm 1}\\}$ if and only if there exists
$N\in\mathbb{Z}$ such that $\alpha^{N}(Q)\subset P$.

###### Proof.

If $\alpha^{N}(Q)\subset P$, then $t^{N}qt^{-N}\in P$ for every $q\in Q$. Thus
$\|q\|_{P\cup\\{t^{\pm 1}\\}}\leq 2|N|+1$, which shows that $P\cup\\{t^{\pm
1}\\}\preceq Q\cup\\{t^{\pm 1}\\}$.

If $P\cup\\{t^{\pm 1}\\}\preceq Q\cup\\{t^{\pm 1}\\}$, then there exists $k>0$
such that $\|q\|_{P\cup\\{t^{\pm 1}\\}}\leq k$ for all $q\in Q$. By Lemma 3.2
we may write any such $q$ as $q=t^{-a}u_{1}\cdots u_{b}t^{a}$ where $u_{i}\in
P$ for all $i$ and $0\leq a,b\leq k$. Consequently $\alpha^{a}(q)=u_{1}\cdots
u_{b}\in P^{b}\subset P^{k}$. Choose $d$ large enough that
$\alpha^{d}(P^{k})\subset P$; such $d$ exists by part (c) of Definition 2.10.
Then $\alpha^{a+d}(q)\in P$. We have $a+d\leq k+d$, so choosing $N=k+d$
completes the proof. ∎

Lemma 4.2 implies that there is a preorder on confining subsets: $P\preceq Q$
if $\alpha^{N}(Q)\subset P$ for some $N\in\mathbb{Z}$. This defines an
equivalence relation $P\sim Q$ if $P\preceq Q$ and $Q\preceq P$. Thus
$P\preceq Q$ if and only if $[P\cup\\{t^{\pm 1}\\}]\preccurlyeq[Q\cup\\{t^{\pm
1}\\}]$ and $P\sim Q$ if and only if $[P\cup\\{t^{\pm 1}\\}]=[Q\cup\\{t^{\pm
1}\\}]$. We also define “meet” and “join” operations that descend to honest
meet and join operations on the set of equivalence classes of confining
subsets:

  * •

$P\vee Q\vcentcolon=P\cap Q$; and

  * •

$P\wedge Q\vcentcolon=P\cdot Q=\\{q_{1}q_{2}\colon q_{1}\in P,q_{2}\in Q\\}$.

######  Lemma 4.3.

The sets $P\vee Q$ and $P\wedge Q$ are confining under $\alpha$.

###### Proof.

First we prove that $P\vee Q$ is confining. Since $\alpha(P)\subset P$ and
$\alpha(Q)\subset Q$, we have $\alpha(P\cap
Q)\subset\alpha(P)\cap\alpha(Q)\subset P\cap Q$. This verifies Definition
2.10(a). If $g\in A$ then there exist $m,n\geq 0$ with $\alpha^{m}(g)\in P$
and $\alpha^{n}(g)\in Q$. Hence $\alpha^{\max(m,n)}(g)\in P\cap Q$. This
verifies Definition 2.10(b). Finally there exist $k_{0}\geq 0$ and $l_{0}\geq
0$ with $\alpha^{k_{0}}(P\cdot P)\subset P$ and $\alpha^{l_{0}}(Q\cdot
Q)\subset Q$. Hence $\alpha^{\max(k_{0},l_{0})}\left((P\cap Q)\cdot(P\cap
Q)\right)\subset P\cap Q$, which verifies Definition 2.10(c).

The fact that $P\wedge Q$ is confining can be similarly verified by using the
following observations. First, $\alpha(P\cdot
Q)=\alpha(P)\cdot\alpha(Q)\subset P\cdot Q$. Secondly, if $g\in A$ then there
exists $m\geq 0$ with $\alpha^{m}(g)\in P$. Thus, $\alpha^{m}(g)\in P\cdot Q$.
Finally, there exist $k_{0}\geq 0$ and $l_{0}\geq 0$ with
$\alpha^{k_{0}}(P\cdot P)\subset P$ and $\alpha^{l_{0}}(Q\cdot Q)\subset Q$.
If $q_{1},q_{2}\in P$ and $u_{1},u_{2}\in Q$, then as $A$ is abelian,

$\alpha^{\max(k_{0},l_{0})}(q_{1}u_{1}\cdot
q_{2}u_{2})=\alpha^{\max(k_{0},l_{0})}(q_{1}q_{2})\alpha^{\max(k_{0},l_{0})}(u_{1}u_{2})\in
P\cdot Q.$

This shows that $\alpha^{\max(k_{0},l_{0})}\left((P\cdot Q)\cdot(P\cdot
Q)\right)\subset P\cdot Q$. ∎

The next lemma shows that $\vee$ and $\wedge$ indeed descend to meet and join
operations on equivalence classes:

######  Lemma 4.4.

If $P\sim P^{\prime}$ and $Q\sim Q^{\prime}$, then $P\vee Q\sim P^{\prime}\vee
Q^{\prime}$ and $P\wedge Q\sim P^{\prime}\wedge Q^{\prime}$.

###### Proof.

It suffices to prove this for the case that $P=P^{\prime}$ and $Q\sim
Q^{\prime}$. Lemma 4.2 provides $N\geq 0$ large enough so that
$\alpha^{N}(Q)\subset Q^{\prime}$ and $\alpha^{N}(Q^{\prime})\subset Q$.

Then

$\alpha^{N}(P\vee Q)=\alpha^{N}(P)\cap\alpha^{N}(Q)\subset P\cap
Q^{\prime}=P\vee Q^{\prime}.$

Similarly $\alpha^{N}(P\vee Q^{\prime})\subset P\vee Q$. Therefore $P\vee
Q\sim P\vee Q^{\prime}$ by Lemma 4.2. The proof that $P\wedge Q\sim P\wedge
Q^{\prime}$ is analogous. ∎

Let $\mathcal{Q}$ denote the set of equivalence classes of confining subsets
under the relation $\sim$ defined above. The preorder $\preceq$ induces a
partial order on $\mathcal{Q}$. We denote by $[P]$ the equivalence class
containing a particular confining subset $P$.

######  Lemma 4.5.

The equivalence class $[Q_{1}\wedge Q_{2}]$ is a greatest lower bound for
$[Q_{1}]$ and $[Q_{2}]$, and $[Q_{1}\vee Q_{2}]$ is a least upper bound for
$[Q_{1}]$ and $[Q_{2}]$.

###### Proof.

Suppose that $P\preceq Q_{1}$ and $P\preceq Q_{2}$. Choose $N\geq 0$ large
enough such that $\alpha^{N}(Q_{1})\subset P$, $\alpha^{N}(Q_{2})\subset P$,
and $\alpha^{N}(P\cdot P)\subset P$. Then $\alpha^{2N}(Q_{1}\cdot
Q_{2})=\alpha^{N}\left(\alpha^{N}(Q_{1})\cdot\alpha^{N}(Q_{2})\right)\subset\alpha^{N}(P\cdot
P)\subset P,$ and so $P\preceq Q_{1}\wedge Q_{2}$.

If $Q_{1}\preceq P$ and $Q_{2}\preceq P$, then choose $N\geq 0$ large enough
that $\alpha^{N}(P)\subset Q_{1}$ and $\alpha^{N}(P)\subset Q_{2}$. It follows
immediately that $\alpha^{N}(P)\subset Q_{1}\cap Q_{2}$, and so $Q_{1}\vee
Q_{2}\preceq P$. ∎

######  Corollary 4.6.

The poset $\mathcal{Q}$ is a lattice.

We will call the poset $\mathcal{Q}$ the _lattice of confining subsets under
$\alpha$_. Finally, we prove Proposition 4.1.

###### Proof of Proposition 4.1.

Since $G$ is solvable, every hyperbolic structure is elliptic, lineal, or
quasi-parabolic. The abelianization of $G$ is virtually cyclic, and so the
natural homomorphism $G\to\mathbb{Z}$ is the unique non-trivial homomorphism
$G\to\mathbb{R}$ up to scaling. Moreover, by amenability, every
pseudocharacter (homogeneous quasi-morphism) on $G$ is a homomorphism to
$\mathbb{R}$ ([25], see also [20, Theorem 6.16]).

We first consider _orientable_ lineal actions $G\curvearrowright X$. That is,
we consider actions where $X$ is quasi-isometric to a line and $G$ fixes both
points of $\partial X$. The Busemann pseudocharacter $\rho$ defined by this
action is a non-trivial homomorphism $\rho\colon G\to\mathbb{R}$ that vanishes
on $A$. Caprace-Cornulier-Monod-Tessera show in [16, Proposition 4.5] that the
action $G\curvearrowright X$ is equivalent to the action
$G\curvearrowright\Gamma(G,Q\cup\\{t^{\pm 1}\\})$, where $Q$ is some confining
subset of $A$ under $\alpha$ or $\alpha^{-1}$. See also the discussion in [2,
Section 3.2]. We note that both [16, Proposition 4.5] and [2, Section 3.2]
consider only quasi-parabolic actions, but the discussion and proofs go
through verbatim for lineal actions: both require only that
$G\curvearrowright\partial X$ has (at least one) global fixed point. By the
discussion after [16, Theorem 4.1] (see also [2, Theorem 1.1]), the confining
subset $Q$ cannot be strictly confining since $G\curvearrowright X$ is not
quasi-parabolic. Thus $Q=A$. We conclude that $G\curvearrowright X\sim
G\curvearrowright\Gamma(G,A\cup\\{t^{\pm 1}\\})$ and hence $G$ has a unique
orientable lineal structure.

Now we consider the case of a general lineal action $G\curvearrowright X$.
That is, we do not assume that $G$ fixes $\partial X$ pointwise. We will show
that in fact the action must be orientable. To do this, we will first show
that $A$ must act elliptically. Suppose for contradiction that $A$ does not
act elliptically. Then the restricted action $A\curvearrowright X$ is a
cobounded lineal action. Since $A$ is abelian, the action $A\curvearrowright
X$ is orientable (see [1, Example 4.23]). Then $A$ and $t^{2}$ both fix
$\partial X$ pointwise and therefore the subgroup that they generate,
$G_{0}=A\rtimes\langle t^{2}\rangle$, also fixes $\partial X$ pointwise. By
hypothesis, the abelianization of $G_{0}$ is virtually cyclic. Hence, we may
apply the reasoning of the previous paragraph to the group $G_{0}$ in place of
$G$, which shows that $A$ acts elliptically, contradicting our assumption. In
any case, $A$ acts elliptically. Therefore the action $G\curvearrowright X$ is
dominated by $G\curvearrowright\Gamma(G,A\cup\\{t^{\pm 1}\\})$. But two
distinct lineal actions are either equivalent or incomparable (see [1,
Corollary 4.12]). Thus, we have $(G\curvearrowright
X)\sim(G\curvearrowright\Gamma(G,A\cup\\{t^{\pm 1}\\}))$. In other words,
$[A\cup\\{t^{\pm 1}\\}]$ is the unique lineal structure on $G$.

Every quasi-parabolic structure on $G$ dominates this single lineal structure
(see the discussion in Section 2.3). By the first paragraph, the Busemann
pseudocharacter $\rho$ of a quasi-parabolic structure is proportional to the
natural homomorphism $G\to\mathbb{Z}$, and so the element $t$ acts
loxodromically. Moreover, the global fixed point of $G$ in this quasi-
parabolic structure is either the attracting fixed point of $t$ (if
$\rho(t)<0$) or the repelling fixed point of $t$ (if $\rho(t)>0$). Thus we may
divide $\mathcal{H}_{\ell}(G)\cup\mathcal{H}_{qp}(G)$ into two subposets:
structures for which $G$ fixes the attracting fixed point of $t$, which we
denote $\mathcal{P}_{-}(G)$, and structures for which $G$ fixes the repelling
fixed point of $t$, which we denote $\mathcal{P}_{+}(G)$. These subposets meet
in the single lineal structure and are otherwise incomparable to each other.
To see this last point, note, for instance, that if an element of
$\mathcal{P}_{+}(G)$ dominates an element of $\mathcal{P}_{-}(G)$, then the
element of $\mathcal{P}_{-}(G)$ must have _both_ the attracting and repelling
fixed points of $t$ as global fixed points. This element can only be the
single lineal structure.

By [2, Theorems 1.1 & 1.2], elements of
$\mathcal{H}_{qp}(G)\cup\mathcal{H}_{\ell}(G)$ coincide with hyperbolic
structures $[Q\cup\\{t^{\pm 1}\\}]$ where $Q$ is confining under $\alpha$ (if
$\rho(t)<0$) or under $\alpha^{-1}$ (if $\rho(t)>0$). We have shown that,
equivalently, the elements of $\mathcal{P}_{-}(G)$ coincide with the
hyperbolic structures $[Q\cup\\{t^{\pm 1}\\}]$ where $Q$ is confining under
$\alpha^{-1}$, and the elements of $\mathcal{P}_{+}(G)$ coincide with the
hyperbolic structures $[Q\cup\\{t^{\pm 1}\\}]$ where $Q$ is confining under
$\alpha$. By Lemma 4.2 the partial order on $\mathcal{P}_{-}(G)$ is given by
$[P\cup\\{t^{\pm 1}\\}]\preccurlyeq[Q\cup\\{t^{\pm 1}\\}]$ if and only if
there exists $N\geq 0$ such that $\alpha^{-N}(Q)\subset P$. By Corollary 4.6,
this poset is a lattice (namely, isomorphic to the lattice of confining
subsets under $\alpha^{-1}$). By the same reasoning interchanging
$\alpha^{-1}$ with $\alpha$, $\mathcal{P}_{+}(G)$ is also a lattice
(isomorphic to the lattice of confining subsets under $\alpha$). ∎

We pull out one fact from the proof of Proposition 4.1 which will be needed in
later sections.

######  Proposition 4.7.

Under the assumptions and notation of Proposition 4.1, the elements of
$\mathcal{P}_{-}(G)$ coincide with the hyperbolic structures $[Q\cup\\{t^{\pm
1}\\}]$ where $Q$ is confining under $\alpha^{-1}$, and the elements of
$\mathcal{P}_{+}(G)$ coincide with the hyperbolic structures $[Q\cup\\{t^{\pm
1}\\}]$ where $Q$ is confining under $\alpha$.

##  5 Examples

Before turning to the proofs of Theorems 1.1 and 1.2, we first illustrate
their utility by classifying the hyperbolic actions of a number of groups. In
addition to describing the poset $\mathcal{H}(G)$ for several new groups, we
also show how the previous work of the authors in [3, 8] fits into the more
general framework of this paper, recovering the main results of [3] and [8]
with significantly shorter proofs.

Theorem 1.1 requires a bit of explanation. Consider the group $G=G(R,\gamma)$
where $R$ and $\gamma$ satisfy axioms (A1)–(A5), and consider the completion
$\widehat{R}$. There is a preorder $\leq$ on ideals of $\widehat{R}$ defined
by $\mathfrak{a}\leq\mathfrak{b}$ if for every $x\in\mathfrak{a}$, there
exists $i\in\mathbb{Z}_{\geq 0}$ with $\gamma^{i}x\in\mathfrak{b}$. This
induces an equivalence relation $\sim$ in the usual way:
$\mathfrak{a}\sim\mathfrak{b}$ if $\mathfrak{a}\leq\mathfrak{b}$ and
$\mathfrak{b}\leq\mathfrak{a}$, and a partial order $\preccurlyeq$ on the set
of equivalence classes. The resulting poset of equivalence classes under
$\sim$ with the partial order $\preccurlyeq$ is what we call the _poset of
ideals of $\widehat{R}$ up to multiplication by $\gamma$_. Theorem 1.1 states
that $\mathcal{P}_{+}(G)$ is isomorphic to this poset.

###  5.1 Lamplighter groups

We first use Theorem 1.1 to give a streamlined proof of the structure of the
poset of hyperbolic actions of the lamplighter group
$(\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z}$, for $n\geq 2$.

######  Theorem 5.1 ([8, Theorem 1.4]).

If $G=(\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z}$ is a lamplighter group for $n\geq
2$, then $\mathcal{H}(G)$ is as pictured in Figure 1(b).

The lamplighter group $(\mathbb{Z}/n\mathbb{Z})\wr\mathbb{Z}$ may be expressed
as $G(R,\gamma)$ for $R=(\mathbb{Z}/n\mathbb{Z})[x]$ and $\gamma=x$, as
described in Example 3.4. For this $R$ and $\gamma$, the axioms (A1)–(A4) are
automatic. Every element of $R$ has a finite base-$\gamma$ address with
respect to the transversal $[n]=\\{0,1,\ldots,n-1\\}$ for $(\gamma)$, which
verifies (A5). We begin by using ideals of
$\widehat{R}=(\mathbb{Z}/n\mathbb{Z})[[x]]$ to classify $\mathcal{P}_{+}(G)$
and then later discuss $\mathcal{P}_{-}(G)$.

An element $m\in\mathbb{Z}/n\mathbb{Z}$ generates the ideal
$(m)=m(\mathbb{Z}/n\mathbb{Z})[[x]]$, and two such ideals $(m_{1})$ and
$(m_{2})$ are equal exactly when $m_{1}$ and $m_{2}$ generate the same
(additive) subgroup of $\mathbb{Z}/n\mathbb{Z}$.

######  Lemma 5.2.

Let $f\in(\mathbb{Z}/n\mathbb{Z})[[x]]$ with
$f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots$. If $a_{0}$ is a unit in the ring
$\mathbb{Z}/n\mathbb{Z}$, then $f$ is a unit in
$(\mathbb{Z}/n\mathbb{Z})[[x]]$.

###### Proof.

Suppose that $f(x)$ is as given with $a_{0}$ a unit. We will find a power
series $g(x)=b_{0}+b_{1}x+b_{2}x^{2}+\cdots$ such that

$a_{0}b_{0}+(a_{0}b_{1}+a_{1}b_{0})x+(a_{0}b_{2}+a_{1}b_{1}+a_{2}b_{0})x^{2}+\cdots=f(x)g(x)=1.$

To this end, choose $b_{0}\in\mathbb{Z}/n\mathbb{Z}$ with $a_{0}b_{0}=1$,
which is possible since $a_{0}$ is a unit. The other coefficients $b_{i}$ will
be defined inductively. Suppose $b_{0},\ldots,b_{k-1}$ have been chosen so
that $a_{0}b_{0}=1$ and the coefficients of $x,x^{2},\dots,x^{k-1}$ in the
product are zero, and set $b_{k}=b_{0}(-a_{1}b_{k-1}-\cdots-a_{k}b_{0})$. Then

$a_{0}b_{k}+a_{1}b_{k-1}+a_{2}b_{k-2}+\cdots+a_{k}b_{0}=0,$

and so the coefficient of $x^{k}$ is zero, as desired. ∎

######  Lemma 5.3.

Let $f\in(\mathbb{Z}/n\mathbb{Z})[[x]]$. If the coefficients of $f$ generate
$\mathbb{Z}/n\mathbb{Z}$ as a group, then $x^{k}$ lies in the ideal generated
by $f$ for some $k$.

###### Proof.

The proof is by induction on the total number of factors in the prime
factorization of $n$. The base case is when $n$ is prime. Then since every
nonzero element of the ring $\mathbb{Z}/n\mathbb{Z}$ is a unit, for some $k$
we have that
$f(x)=a_{k}x^{k}+a_{k+1}x^{k+1}+\cdots=x^{k}(a_{k}+a_{k+1}x+\cdots)$, where
$a_{k}$ is a unit in $\mathbb{Z}/n\mathbb{Z}$. By Lemma 5.2, there exists an
element $g(x)\in(\mathbb{Z}/n\mathbb{Z})[[x]]$ with
$(a_{k}+a_{k+1}x+\cdots)g(x)=1$. Thus, $f(x)g(x)=x^{k}$, and so $x^{k}$ lies
in the ideal generated by $f$.

Now suppose that the prime decomposition of $n$ is $n=p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}}$. Suppose for induction that the lemma has been proven for all
numbers $m$ with at most $k_{1}+\cdots+k_{r}-1$ prime factors. Re-ordering if
necessary, suppose that $p_{r}$ is the smallest number in the set
$\\{p_{1},\ldots,p_{r}\\}$.

Consider the power series $p_{1}^{k_{1}}\cdots
p_{r-1}^{k_{r-1}}p_{r}^{k_{r}-1}f(x)$. Its coefficients all lie in the
additive subgroup of $\mathbb{Z}/n\mathbb{Z}$ generated by
$p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}$. Furthermore, this power series is
nonzero, since some coefficient of $f$ is not divisible by $p_{r}$ (because
the coefficients of $f$ generate $\mathbb{Z}/n\mathbb{Z}$ as an additive
group). Note that as a set, the subgroup of $\mathbb{Z}/n\mathbb{Z}$ generated
by $p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}$ is given by

$\left\\{0,p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1},2p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1},3p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1},\ldots,(p_{r}-1)p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}\right\\}.$

Hence we may write

$p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}f(x)=p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1}x^{l}(a_{0}+a_{1}x+\cdots)$

for some $l\geq 0$, where $a_{0}\neq 0$ and $0\leq a_{i}\leq p_{r}-1$ for all
$i$. In particular, since $a_{0}$ is not divisible by any $p_{i}$, it is a
unit in $\mathbb{Z}/n\mathbb{Z}$. Thus by Lemma 5.2, there is a power series
$g(x)$ with $(a_{0}+a_{1}x+\cdots)g(x)=1$ and

$(p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}f(x))g(x)=p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1}x^{l}.$

Reduce $f(x)$ modulo $p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}$, and denote the
result by $\overline{f}(x)\in(\mathbb{Z}/p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1}\mathbb{Z})[[x]]$. Since the coefficients of $f(x)$ generate
$\mathbb{Z}/n\mathbb{Z}$ as a group, the coefficients of $\overline{f}(x)$
generate $\mathbb{Z}/p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}\mathbb{Z}$ as a
group. Thus, by the induction hypothesis, there exists a power series
$\overline{h}(x)\in(\mathbb{Z}/p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1}\mathbb{Z})[[x]]$ with $\overline{f}(x)\overline{h}(x)=x^{m}$
for some $m\geq 0$.

Choose a power series $h(x)\in(\mathbb{Z}/n\mathbb{Z})[[x]]$ which reduces to
$\overline{h}(x)$ modulo $p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}$. Then

$f(x)h(x)=x^{m}+p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}u(x)$

for some power series $u(x)$. Since $p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}x^{l}$
is in the ideal generated by $f$, so is $p_{1}^{k_{1}}\cdots
p_{r}^{k_{r}-1}x^{l}u(x)$. Thus, both
$f(x)h(x)x^{l}=x^{m+l}+p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}x^{l}u(x)$ and
$p_{1}^{k_{1}}\cdots p_{r}^{k_{r}-1}x^{l}u(x)$ lie in the ideal generated by
$f$, and so their difference $x^{m+l}$ lies in $(f)$, as well. This completes
the inductive step. ∎

The following fact is probably well known and we leave its proof to the
reader.

######  Lemma 5.4.

If $a_{1},\ldots,a_{l}\in\mathbb{Z}/n\mathbb{Z}$ generate the subgroup
$\langle m\rangle$, then there exist elements
$b_{1},\ldots,b_{l}\in\mathbb{Z}/n\mathbb{Z}$ that generate
$\mathbb{Z}/n\mathbb{Z}$ and such that $a_{j}=b_{j}m$ for each $j$.

From this we immediately derive:

######  Corollary 5.5.

Let $f(x)\in(\mathbb{Z}/n\mathbb{Z})[[x]]$ and $m\in\mathbb{Z}/n\mathbb{Z}$ be
such that the coefficients of $f$ generate the subgroup $\langle
m\rangle\leq\mathbb{Z}/n\mathbb{Z}$. Then $mx^{k}$ lies in the ideal of
$(\mathbb{Z}/n\mathbb{Z})[[x]]$ generated by $f$, for some
$k\in\mathbb{Z}_{\geq 0}$.

###### Proof.

Applying Lemma 5.4 to the coefficients of $f$, we may write $f(x)=mg(x)$ where
the coefficients of $g$ generate $\mathbb{Z}/n\mathbb{Z}$. By Lemma 5.3, there
exist $h(x)$ and $k\geq 0$ with $g(x)h(x)=x^{k}$. Thus $mx^{k}\in(f)$. ∎

######  Lemma 5.6.

If $\mathfrak{a}$ is an ideal of $(\mathbb{Z}/n\mathbb{Z})[[x]]$, then there
exists $m\in\mathbb{Z}/n\mathbb{Z}$ such that $\mathfrak{a}$ is equivalent to
the ideal $m(\mathbb{Z}/n\mathbb{Z})[[x]]$ generated by $m$.

###### Proof.

Let $\mathfrak{a}$ be an arbitrary ideal. Let $\langle
m\rangle\leq\mathbb{Z}/n\mathbb{Z}$ be the smallest subgroup containing the
coefficients of _every_ power series in $\mathfrak{a}$. Define
$\mathfrak{b}=m(\mathbb{Z}/n\mathbb{Z})[[x]]$. Then
$\mathfrak{a}\subset\mathfrak{b}$, and we claim that
$\mathfrak{a}\sim\mathfrak{b}$. By the definition of the equivalence relation,
it suffices to show that $mx^{l}\in\mathfrak{a}$ for some $l$.

To see this, choose any $f_{1}\in\mathfrak{a}$. Its coefficients generate a
subgroup $\langle m_{1}\rangle$ of $\langle
m\rangle\leq\mathbb{Z}/n\mathbb{Z}$. By Corollary 5.5, we have that
$m_{1}x^{k_{1}}\in\mathfrak{a}$ for some $k_{1}$. If $\langle
m_{1}\rangle\neq\langle m\rangle$, then since $\langle m\rangle$ is the
_smallest_ subgroup containing the coefficients of $\mathfrak{a}$, there
exists some $f_{2}\in\mathfrak{a}$ whose coefficients generate a subgroup
$\langle m_{2}\rangle\leq\mathbb{Z}/n\mathbb{Z}$ which is not contained in
$\langle m_{1}\rangle$. We have $m_{2}x^{k_{2}}\in\mathfrak{a}$ for some
$k_{2}$. Similarly, if $\langle m_{1},m_{2}\rangle\neq\langle m\rangle$ then
there exists some $f_{3}\in\mathfrak{a}$ whose coefficients generate a
subgroup $\langle m_{3}\rangle\leq\mathbb{Z}/n\mathbb{Z}$ with $\langle
m_{3}\rangle\not\leq\langle m_{1},m_{2}\rangle$. We continue this process
inductively. Since $\langle m\rangle$ is finite, we eventually find numbers
$m_{1},\ldots,m_{r}\in\mathbb{Z}/n\mathbb{Z}$ with $\langle
m_{1},\ldots,m_{r}\rangle=\langle m\rangle$ and elements
$m_{1}x^{k_{1}},\ldots,m_{r}x^{k_{r}}\in\mathfrak{a}$. Setting
$l=\max\\{k_{1},\ldots,k_{r}\\}$ we have
$m_{1}x^{l},\ldots,m_{r}x^{l}\in\mathfrak{a}$. Since $\langle
m_{1},\ldots,m_{r}\rangle=\langle m\rangle$, we have $mx^{l}\in\mathfrak{a}$,
as claimed. ∎

###### Proof of Theorem 5.1.

By Lemma 5.6, any ideal $\mathfrak{a}\subset(\mathbb{Z}/n\mathbb{Z})[[x]]$ is
equivalent to an ideal $m(\mathbb{Z}/n\mathbb{Z})[[x]]$ generated by some
$m\in\mathbb{Z}/n\mathbb{Z}$. The ideal $m(\mathbb{Z}/n\mathbb{Z})[[x]]$
satisfies the property that if $f\in(\mathbb{Z}/n\mathbb{Z})[[x]]$ and
$x^{i}f\in m(\mathbb{Z}/n\mathbb{Z})[[x]]$ for some $i\geq 0$, then $f\in
m(\mathbb{Z}/n\mathbb{Z})[[x]]$. Thus we see that
$m(\mathbb{Z}/n\mathbb{Z})[[x]]\leq m^{\prime}(\mathbb{Z}/n\mathbb{Z})[[x]]$
if and only if $m(\mathbb{Z}/n\mathbb{Z})[[x]]\subset
m^{\prime}(\mathbb{Z}/n\mathbb{Z})[[x]]$. Thus
$m(\mathbb{Z}/n\mathbb{Z})[[x]]\sim m^{\prime}(\mathbb{Z}/n\mathbb{Z})[[x]]$
if and only if
$m(\mathbb{Z}/n\mathbb{Z})[[x]]=m^{\prime}(\mathbb{Z}/n\mathbb{Z})[[x]]$.
Moreover, $m(\mathbb{Z}/n\mathbb{Z})[[x]]\subset
m^{\prime}(\mathbb{Z}/n\mathbb{Z})[[x]]$ exactly if the subgroup
$m(\mathbb{Z}/n\mathbb{Z})$ is contained in
$m^{\prime}(\mathbb{Z}/n\mathbb{Z})$. It follows that the poset of ideals of
$(\mathbb{Z}/n\mathbb{Z})[[x]]$ up to equivalence is isomorphic to
$\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})$, the poset of subgroups of
$\mathbb{Z}/n\mathbb{Z}$ with inclusion. Thus $\mathcal{P}_{+}(G)$ is
isomorphic to the opposite of
$\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})\cong\operatorname{Div}(k_{1},\ldots,k_{r})$,
where $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ is the prime factorization.
Additionally,
$\operatorname{Sub}(\mathbb{Z}/n\mathbb{Z})\cong\operatorname{Div}(k_{1},\ldots,k_{r})$
is isomorphic to its own opposite.

To fully describe $\mathcal{H}(G)$, it remains to describe the lattice
$\mathcal{P}_{-}(G)$. For this we use confining subsets and the partial order
described in Section 4. Recall the automorphism $\alpha$ of
$(\mathbb{Z}/n\mathbb{Z})[x^{\pm 1}]$ defined by $\alpha(p(x))=xp(x)$. The
poset of confining subsets under $\alpha$ (that is, $\mathcal{P}_{+}(G)$) is
the poset induced by the preorder defined by $Q\preceq P$ if
$\alpha^{N}(P)\subset Q$ for some $N\geq 0$. The poset $\mathcal{P}_{-}(G)$ is
isomorphic to the poset of confining subsets under $\alpha^{-1}$ induced by
the preorder $Q\preceq P$ if $\alpha^{-N}(P)\subset Q$ for some $N\geq 0$.
There is an isomorphism from $\mathcal{P}_{+}(G)$ to $\mathcal{P}_{-}(G)$.
Namely, there is a unique automorphism of the ring
$(\mathbb{Z}/n\mathbb{Z})[x^{\pm 1}]$ defined by $x\mapsto x^{-1}$. It is
straightforward to see that this automorphism sends confining subsets under
$\alpha$ to confining subsets under $\alpha^{-1}$ and vice versa. Moreover,
this automorphism respects the preorders, and so $\mathcal{P}_{+}(G)$ is
isomorphic to $\mathcal{P}_{-}(G)$. This completes the classification of
$\mathcal{H}(G)$. ∎

###  5.2 Torsion-free finitely presented abelian-by-cyclic groups

In this section we consider the class of torsion-free finitely presented
abelian-by-cyclic groups. Recall that such groups are ascending HNN extensions
of free abelian groups $\mathbb{Z}^{n}$ by matrices $\gamma\in
M_{n}(\mathbb{Z})$ with non-zero determinant [10]. That is, a torsion-free
finitely presented abelian-by-cyclic group $G$ has the form

$G=\langle\mathbb{Z}^{n},t:txt^{-1}=\gamma x\text{ for
}x\in\mathbb{Z}^{n}\rangle.$

We will denote $G$ by $G(\gamma)$ to make the matrix $\gamma$ explicit when
necessary. We assume that $\gamma$ is admissible, that is:

  * (i)

$\gamma$ is expanding (i.e. all of its eigenvalues lie outside the unit disk
in the complex plane);

  * (ii)

$\mathbb{Z}^{n}$ is a cyclic $\mathbb{Z}[x]$-module (with the structure
induced by letting $x$ act on $\mathbb{Z}^{n}$ by $\gamma$).

Consider the prime factorization $p=up_{1}^{n_{1}}\cdots p_{r}^{n_{r}}$ in
$\mathbb{Z}[[x]]$, where $u\in\mathbb{Z}[[x]]$ is a unit and the $p_{i}$ are
distinct irreducible power series in $\mathbb{Z}[[x]]$. We recall Theorem 1.2
for the convenience of the reader.

See 1.2

In the next three subsections we will apply Theorem 1.2 to classify the
hyperbolic actions of solvable Baumslag-Solitar groups and other torsion-free
finitely presented abelian-by-cyclic groups. Theorem 1.2 will then be proved
in the remaining sections of the paper.

####  5.2.1 Solvable Baumslag-Solitar groups

We now use Theorem 1.2 to give a streamlined proof of the structure of the
poset of hyperbolic actions of the Baumslag-Solitar group $BS(1,n)$ for
$n\in\mathbb{Z}\setminus\\{-1,0,1\\}$:

######  Theorem 5.7 ([3, Theorem 1.1]).

Let $G=BS(1,n)$ be a Baumslag-Solitar group with $n\notin\\{-1,0,1\\}$, and
let $n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ be the prime factorization of $n$.
The poset $\mathcal{H}(G)$ is as pictured in Figure 1(c).

Recall that $BS(1,n)\cong\mathbb{Z}[\frac{1}{n}]\rtimes\mathbb{Z}$, so
$BS(1,n)=G(\gamma)$ where $\gamma$ is the $1\times 1$ matrix
$\begin{pmatrix}n\end{pmatrix}$ acting on $\mathbb{R}=\mathbb{R}^{1}$. The
matrix $\gamma$ is expanding with characteristic polynomial $-n+x$, and
$\mathbb{Z}$ is a cyclic $\mathbb{Z}[x]$-module. Hence we may apply Theorem
1.2. There are only two subspaces of $\mathbb{R}$ invariant under $\gamma$: 0
and $\mathbb{R}$ itself. It then remains to consider the prime factorization
of the monic polynomial $-n+x$ in the formal power series ring
$\mathbb{Z}[[x]]$. Equivalently, we will factor the polynomial $n-x$.

We start by stating some basic results on prime factorization in (the unique
factorization domain) $\mathbb{Z}[[x]]$. These results are standard and
straightforward to prove, so we omit their proofs. Note that since
$\mathbb{Z}[[x]]$ is a unique factorization domain, an element is prime if and
only if it is irreducible.

######  Lemma 5.8.

Let $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots\in\mathbb{Z}[[x]]$. Then $f$ is a
unit if and only if $a_{0}=\pm 1$.

The following allows us to factor any formal power series whose constant term
is not a prime power.

######  Lemma 5.9.

Let $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots\in\mathbb{Z}[[x]]$. If
$a_{0}=b_{0}c_{0}$ with $b_{0},c_{0}\in\mathbb{Z}$ relatively prime, then $f$
may be factored as $f=gh$ where $g(x)=b_{0}+b_{1}x+\cdots$ and
$h(x)=c_{0}+c_{1}x+\cdots$.

An immediate corollary is the following:

######  Corollary 5.10.

Let $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots\in\mathbb{Z}[[x]]$. If
$a_{0}=(-1)^{\delta}p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ is the prime
factorization of $a_{0}$ then $f$ may be factored as
$f=(-1)^{\delta}f_{1}\cdots f_{r}$ where the constant term of $f_{i}$ is
$p_{i}^{k_{i}}$.

However, there is no guarantee that the factors $f_{i}$ given by Corollary
5.10 are themselves prime. The following gives a necessary condition for a
power series to be prime. However this is not sufficient: there are reducible
power series with constant term a power of a prime (e.g.,
$(2+x)^{2}=4+4x+x^{2}$).

######  Corollary 5.11.

Let $f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots\in\mathbb{Z}[[x]]$. If $f$ is prime,
then $a_{0}=\pm p^{n}$ where $p$ is prime and $n\geq 1$.

######  Lemma 5.12.

Let $f(x)=a_{0}+a_{1}x+\cdots$ and $g(x)=b_{0}+b_{1}x+\cdots$ be two formal
power series in $\mathbb{Z}[[x]]$. Then $f$ and $g$ are relatively prime if
and only if $a_{0}$ and $b_{0}$ are relatively prime.

The following lemmas give a partial converse to Corollary 5.11.

######  Lemma 5.13.

Consider a formal power series $f(x)=a_{0}+a_{1}x+\cdots\in\mathbb{Z}[[x]]$.
If $a_{0}$ is (up to sign) a power of a prime $p\in\mathbb{Z}$ and $a_{1}$ is
relatively prime to $p$, then $f$ is prime.

###### Proof.

Multiply by $-1$ if necessary to assume $a_{0}>0$, and write $a_{0}=p^{n}$ for
some $n\geq 1$. Suppose that $f=gh$ where $g=b_{0}+b_{1}x+\cdots$ and
$h=c_{0}+c_{1}x+\cdots$. Up to multiplying both $g$ and $h$ by $-1$, we have
$b_{0}=p^{k}$ and $c_{0}=p^{l}$ where $k,l\geq 0$. Moreover,
$a_{1}=b_{0}c_{1}+b_{1}c_{0}$. Since $a_{1}$ is not divisible by $p$, it must
be the case that $b_{0}=1$ or $c_{0}=1$, and so either $g$ or $h$ is a unit. ∎

######  Lemma 5.14.

Consider a formal power series $f(x)=a_{0}+a_{1}x+\cdots\in\mathbb{Z}[[x]]$.
If $a_{0}$ is (up to sign) a prime $p\in\mathbb{Z}$, then $f$ is prime.

###### Proof.

Multiply by $-1$ if necessary to assume that $a_{0}=p$. If $f=gh$ where
$g=b_{0}+b_{1}x+\cdots$ and $h=c_{0}+c_{1}x+\cdots$, then $b_{0}c_{0}=p$. Up
to swapping $g$ and $h$, $p$ divides $b_{0}$ while $c_{0}=\pm 1$. Thus $h$ is
a unit by Lemma 5.8. ∎

We are now ready to apply these results to the polynomial $n-x$.

######  Lemma 5.15.

Let $n\in\mathbb{Z}\setminus\\{-1,0,1\\}$, and consider the polynomial
$f(x)=n-x$. Let $(-1)^{\delta}p_{1}^{k_{1}}\cdots p_{r}^{k_{r}}$ be the prime
factorization of $n$. The prime factorization of $f$ is given by
$f(x)=(-1)^{\delta}f_{1}\cdots f_{r}$, where the polynomials $f_{i}$ are
prime; the polynomials $f_{i}$ are pairwise relatively prime; and the constant
term of $f_{i}$ is $p_{i}^{k_{i}}$.

###### Proof.

By Corollary 5.10 we may factor $f$ as $(-1)^{\delta}f_{1}\cdots f_{r}$, where
the constant term of $f_{i}$ is $p_{i}^{k_{i}}$ for each $i$. We claim that
each of these factors is prime. Write
$f_{i}=p_{i}^{k_{i}}+a^{i}_{1}x+a^{i}_{2}x^{2}+\cdots$. Then the coefficient
of $x$ in $f_{1}\cdots f_{r}$ is

$\sum_{j=1}^{r}(p_{1}^{k_{1}}\cdots\widehat{p_{j}^{k_{j}}}\cdots
p_{r}^{k_{r}})a^{j}_{1},$

where $\widehat{\cdot}$ denotes omission of the corresponding factor. If some
$a_{1}^{i}$ is divisible by $p_{i}$, then each term of this sum is divisible
by $p_{i}$. However, we have

$\sum_{j=1}^{r}(p_{1}^{k_{1}}\cdots\widehat{p_{j}^{k_{j}}}\cdots
p_{r}^{k_{r}})a^{j}_{1}=\pm 1.$

Thus $a_{1}^{i}$ is relatively prime to $p_{i}$ for each $i$. By Lemma 5.13,
this proves that $f_{i}$ is prime. The power series $f_{i}$ are relatively
prime to each other by Lemma 5.12. ∎

Finally, we can deduce the structure of $\mathcal{H}(BS(1,n))$ from Theorem
1.2. Recall that $G=G(\gamma)$ where $\gamma=\begin{pmatrix}n\end{pmatrix}$.
The matrix $\gamma$ is admissible. Hence, we have
$\mathcal{P}_{+}(G)\cong\operatorname{Div}(1,\ldots,1)\cong
2^{\\{1,\ldots,r\\}}$ where there are $r$ 1’s between the parentheses and
where the number of distinct prime factors of $n$ is $r$. There are two
invariant subspaces of $\mathbb{R}$: 0 and $\mathbb{R}$ itself. Thus
$\mathcal{P}_{-}(G)$ is isomorphic to the lattice $\operatorname{Div}(1)$ with
two elements, one being greater than the other. This matches the description
given in Figure 1(c), proving Theorem 5.7.

####  5.2.2 Other abelian-by-cyclic groups

In this subsection we classify the hyperbolic actions of certain more exotic
abelian-by-cyclic groups $G(\gamma)$. The methods used here generalize easily
to classify the hyperbolic actions of other abelian-by-cyclic groups. As in
the case of $BS(1,n)$, one must compute the prime factorization of the
characteristic polynomial in $\mathbb{Z}[[x]]$ and the invariant subspaces of
the matrix $\gamma$.

######  Example 5.16.

Consider the matrix $\gamma=\begin{pmatrix}2&0&0\\\ 0&3&1\\\
0&0&3\end{pmatrix}$. Its characteristic polynomial is $p(x)=(2-x)(3-x)^{2}$.
The reader may check that $\mathbb{Z}^{3}$ is a cyclic $\mathbb{Z}[x]$-module
generated by $(1,0,1)$. Thus Theorem 1.2 applies to the group
$G=G(\gamma)=\langle\mathbb{Z}^{n},t:txt^{-1}=\gamma x\text{ for
}x\in\mathbb{Z}^{n}\rangle$.

The prime factorization of $p$ in $\mathbb{Z}[[x]]$ is immediate: it is
$p(x)=(2-x)(3-x)^{2}$ since the factors $2-x$ and $3-x$ are prime in
$\mathbb{Z}[[x]]$ by Lemma 5.14. Thus by Theorem 1.2 we have
$\mathcal{P}_{+}(G)\cong\operatorname{Div}(1,2)$.

The invariant subspaces of $\gamma$ are the direct sums of the generalized
eigenspaces $\ker(\gamma-\lambda I)^{i}$ for $i\geq 0$ and $\lambda$ an
eigenvalue of $\gamma$. The possible direct summands are

$\ker(\gamma-2I)=\operatorname{Span}\left\\{\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}\right\\},\quad\ker(\gamma-3I)=\operatorname{Span}\left\\{\begin{pmatrix}0\\\
1\\\ 0\end{pmatrix}\right\\},\quad\textrm{and
}\ker(\gamma-3I)^{2}=\operatorname{Span}\left\\{\begin{pmatrix}0\\\ 1\\\
0\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}\right\\},$

with the second kernel being contained in the third. Thus the poset of
invariant subspaces is $\operatorname{Div}(1,2)$ and this is
$\mathcal{P}_{-}(G)$; see Figure 2(a).

######  Example 5.17.

Consider the matrix $\gamma=\begin{pmatrix}0&0&-210\\\ 1&0&-1\\\
0&1&0\end{pmatrix}$. This is the companion matrix to the irreducible
polynomial $p(x)=x^{3}+x+210$ (see Section 8), and so $p$ is the
characteristic and minimal polynomial of $\gamma$. It has two complex
conjugate roots and one real root all lying outside the unit disk in
$\mathbb{C}$, so $\gamma$ is expanding. That $\mathbb{Z}^{3}$ is a cyclic
$\mathbb{Z}[x]$-module follows automatically from the fact that $\gamma$ is
the companion matrix to a polynomial. A cyclic vector for $\gamma$ is
$(1,0,0)$. Thus Theorem 1.2 applies.

First we calculate the prime factorization of $p$ in $\mathbb{Z}[[x]]$. The
constant term $210$ of $p$ factors as $2\cdot 3\cdot 5\cdot 7$. By Corollary
5.10, we may factor $p$ in $\mathbb{Z}[[x]]$ as
$(2+\cdots)(3+\cdots)(5+\cdots)(7+\cdots)$. Each of these factors is prime by
Lemma 5.14, so $\mathcal{P}_{+}(G)\cong\operatorname{Div}(1,1,1,1)$ by Theorem
1.2.

Now we calculate the invariant subspaces. Either complex eigenvector gives
rise to its real and imaginary parts which span a plane in $\mathbb{R}^{3}$.
This plane is one invariant subspace for $\gamma$. The real eigenvector of
$\gamma$ spans a line in $\mathbb{R}^{3}$ which is another invariant subspace
for $\gamma$. The other invariant subspaces are 0 and $\mathbb{R}^{3}$, which
are direct sums of some sub-collection of the invariant plane and the
invariant line; see Section 9.2 for more details on this. Thus, the poset of
invariant subspaces is $\operatorname{Div}(1,1)$ and this is
$\mathcal{P}_{-}(G)$; see Figure 2(b).

######  Example 5.18.

Consider the matrix $\gamma=\begin{pmatrix}0&0&0&2\\\ 1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\end{pmatrix}$. This is the companion matrix to the irreducible
polynomial $p(x)=x^{4}-2$, which is thus the characteristic and minimal
polynomial of $\gamma$. It has two complex conjugate roots $\pm i\sqrt[4]{2}$
and two real roots $\pm\sqrt[4]{2}$, all outside the unit disk. Again,
$\mathbb{Z}^{4}$ is a cyclic $\mathbb{Z}[x]$-module with generator
$v=(1,0,0,0)$, and so Theorem 1.2 applies.

The polynomial $p$ is prime in $\mathbb{Z}[[x]]$ by Lemma 5.14, so
$\mathcal{P}_{+}(G)\cong\operatorname{Div}(1)$. Now we calculate the invariant
subspaces. Again, either complex eigenvector gives rise to an
$\gamma$-invariant plane in $\mathbb{R}^{4}$ by taking its real and imaginary
parts. The real eigenvectors of $\gamma$ each give rise to a
$\gamma$-invariant eigenline. Every invariant subspace is a direct sum of some
collection of these three subspaces (that is, the plane and two lines); again,
see Section 9.2 for more details. Thus the poset of invariant subspaces
$\mathcal{P}_{-}(G)$ is $\operatorname{Div}(1,1,1)$; see Figure 2(c).

##  6 Structure of $\mathcal{P}_{+}(G)$: ideals of $\widehat{R}$

In this section we study a group $G=G(R,\gamma)$ where $R$ and $\gamma\in R$
satisfy axioms (A1)–(A5). Recall that $G(R,\gamma)$ is isomorphic to
$\gamma^{-1}R\rtimes_{\alpha}\mathbb{Z}$ and we denote a generator of
$\mathbb{Z}$ by $t$. In order to describe $\mathcal{H}(G)$, we wish to apply
Proposition 4.1 to the group $G=G(R,\gamma)$. To do this, we need to
understand the abelianizations of both $G$ and $\gamma^{-1}R\rtimes\langle
t^{2}\rangle$.

######  Proposition 6.1.

Let $(R,\gamma)$ satisfy axioms (A1)–(A5), and set $G=G(R,\gamma)$. Then the
abelianizations of $G$ and $\gamma^{-1}R\rtimes\langle t^{2}\rangle$ are
virtually cyclic.

###### Proof.

We will prove the statement for $G=\gamma^{-1}R\rtimes\langle t\rangle$ first,
developing some tools that will apply to $\gamma^{-1}R\rtimes\langle
t^{2}\rangle$ along the way. We can see from the presentation

$G=\langle R,t:trt^{-1}=\gamma r\text{ for }r\in R\rangle$

that the abelianization of $G$ is $H\times\mathbb{Z}$ where $H$ is the image
of $R$. In particular, $H$ is the quotient of $R$ (as an abelian group) by the
normal subgroup generated by the elements $r-\gamma r$ for $r\in R$. This is
simply the quotient (of rings) $R/(1-\gamma)$. Similarly,
$\gamma^{-1}R\rtimes\langle t^{2}\rangle$ is the HNN extension

$\langle R,t:trt^{-1}=\gamma^{2}r\text{ for }r\in R\rangle$

and hence its abelianization is $K\times\mathbb{Z}$ where
$K=R/(1-\gamma^{2})$. We will show that both $H$ and $K$ are finite.

For use later in the proof, we first note that $1-a\gamma$ is not a zero
divisor in $R$ for any $a\in R$. Indeed, suppose that $(1-a\gamma)r=0$. Then
we have that

$r=a\gamma r=a^{2}\gamma^{2}r=\ldots$

lies in $\bigcap_{k=0}^{\infty}(\gamma^{k})=0$, and hence $r=0$.

Consider the quotient $H=R/(1-\gamma)$; we will show that $H$ is finite
cyclic. Let $\mathbb{Z}[x]\to R$ be the natural homomorphism sending $x$ to
$\gamma$, and let $\mathfrak{a}$ be the kernel. Then there are surjective ring
homomorphisms

$\mathbb{Z}[x]\to\mathbb{Z}[x]/(1-x)\to\mathbb{Z}[x]/(\mathfrak{a}+(1-x))\to
R/(1-\gamma).$

The restriction of the homomorphism $\mathbb{Z}[x]\to R/(1-\gamma)$ to
$\mathbb{Z}$ is surjective, and this shows that $R/(1-\gamma)$ is cyclic. If
the image of some $m\in\mathbb{Z}\setminus\\{0\\}$ in $R/(1-\gamma)$ is zero,
then this will show that $R/(1-\gamma)$ is finite. Note that the image of any
$f(x)\in\mathbb{Z}[x]$ in $R/(1-\gamma)$ is the same as the image of $f(1)$
and that the image of any $f\in\mathfrak{a}\subset\mathbb{Z}[x]$ is zero. Thus
it suffices to show that $f(1)\neq 0$ for some $f\in\mathfrak{a}$.

Suppose for contradiction that $f(1)=0$ for every $f\in\mathfrak{a}$. Then
every element of $\mathfrak{a}$ is divisible by the linear polynomial $1-x$,
and so $\mathfrak{a}\subset(1-x)$. Choose a non-zero $f\in\mathfrak{a}$. We
may factor $f$ as $f(x)=(1-x)^{k}g(x)$ where $k\geq 1$ and $g(1)\neq 0$, so
that $g\notin\mathfrak{a}$. In $R$ we have
$0=f(\gamma)=(1-\gamma)^{k}g(\gamma)$ and $g(\gamma)\neq 0$. Set
$r=g(\gamma)$. The binomial theorem implies that $(1-\gamma)^{k}=1-a\gamma$
for some $a\in R$, and so $(1-a\gamma)r=0$. However, this is a contradiction,
as we have shown that $1-a\gamma$ is not a zero divisor. Therefore
$H=R/(1-\gamma)$ is finite cyclic.

Now consider $K=R/(1-\gamma^{2})$. There is a short exact sequence

$0\to(1-\gamma)/(1-\gamma^{2})\to R/(1-\gamma^{2})\to R/(1-\gamma)\to 0.$

By our previous work, $R/(1-\gamma)$ is finite cyclic. On the other hand there
is an isomorphism of $R$-modules $R/(1+\gamma)\to(1-\gamma)/(1-\gamma^{2})$
defined by $r+(1+\gamma)\mapsto r(1-\gamma)+(1-\gamma^{2})$. By the same
reasoning as for $R/(1-\gamma)$, $R/(1+\gamma)\cong(1-\gamma)/(1-\gamma^{2})$
is also finite cyclic. Thus, $R/(1-\gamma^{2})$ is finite, as desired. ∎

Thus Proposition 4.1 applies and we conclude that
$\mathcal{H}_{qp}(G)\cup\mathcal{H}_{\ell}(G)=\mathcal{P}_{-}(G)\cup\mathcal{P}_{+}(G)$,
where $\mathcal{P}_{+}(G)$ is the poset of subsets of $\gamma^{-1}R$ which are
confining under the action of $\alpha$, while $\mathcal{P}_{-}(G)$ is the
poset of confining subsets under $\alpha^{-1}$. The intersection
$\mathcal{P}_{+}(G)\cap\mathcal{P}_{-}(G)$ is the unique lineal structure of
$\mathcal{H}(G)$.

We now proceed to show that there is a one-to-one correspondence between
elements of $\mathcal{P}_{+}(G)$ and ideals of $\widehat{R}$ up to a certain
equivalence relation. In Section 6.1, we show how to construct a confining
subset of $\gamma^{-1}R$ from an ideal in $\widehat{R}$. In Section 6.2, we
show how to construct an ideal in $\widehat{R}$ from a confining subset of
$\gamma^{-1}R$ and analyze how these two constructions relate to each other.
Finally, in Section 6.3 we prove our main result (Theorem 6.18), which shows
that there is an order-reversing isomorphism between the poset of ideals of
$\widehat{R}$ up to equivalence and $\mathcal{P}_{+}(G)$.

The ideas and proofs in the rest of this section are generalizations of the
corresponding facts from [3], and the proofs presented here mostly follow
those in [3]. We include full proofs here even when they are in essence
identical to those in [3] because of the (significant) change of notation
involved and to keep the paper self-contained.

We first introduce some new notation for elements of $\gamma^{-1}R$ and
$\widehat{R}$ which will be useful for the sake of brevity in this section.
Recall that $T=[d]=\\{0,\ldots,d-1\\}$ denotes a transversal for the ideal
$(\gamma)$ in $R$, where $d$ is a positive integer multiple of 1, provided by
Lemma 3.5. An element $s\in R$ can be written in base-$\gamma$ as
$s=a_{0}+a_{1}\gamma+\cdots,$ where $a_{i}\in T$ for each $i\geq 0$. An
element $r\in\gamma^{-1}R$ has the form $s/\gamma^{k}$ for some $s\in R$ and
some $k\geq 0$. This yields that every element $r\in\gamma^{-1}R$ can be
written as

$r=a_{-k}\gamma^{-k}+\cdots+a_{-1}\gamma^{-1}+a_{0}+a_{1}\gamma+a_{2}\gamma^{2}+\cdots,$

for some $a_{i}\in T$ and some $k\geq 0$.

In this section, and this section only, it will sometimes be convenient to use
decimal notation for the $\gamma$–adic addresses of elements of $\gamma^{-1}R$
and $\widehat{R}$. If
$r=a_{-k}\gamma^{-k}+\cdots+a_{-1}\gamma^{-1}+a_{0}+a_{1}\gamma+a_{2}\gamma^{2}+\cdots\in\gamma^{-1}R$,
we write $r=\dots a_{2}a_{1}a_{0}.a_{-1}\dots a_{-k}$. We say $\dots
a_{2}a_{1}a_{0}$ is the _integral part_ of $r$ and $0.a_{-1}\dots a_{-k}$ is
the _fractional part_ of $r$. Similarly, if
$x=\sum_{i=0}^{\infty}x_{i}\gamma^{i}\in\widehat{R}$, we write $x=\dots
x_{2}x_{1}x_{0}$. We will use both the summation notation and the decimal
notation to represent elements of $\gamma^{-1}R$ and $\widehat{R}$
interchangeably throughout the section. Note that multiplication by $\gamma$
shifts the decimal point one position to the right: if $r=\ldots
a_{1}a_{0}.a_{-1}a_{-2}\ldots a_{-k}$, then $\gamma r=\ldots
a_{1}a_{0}a_{-1}.a_{-2}\ldots a_{-k}.$

We begin by showing that $R$ (as a subset of $\gamma^{-1}R$) is confining
under $\alpha$.

######  Lemma 6.2.

$R\subseteq\gamma^{-1}R$ is confining under $\alpha$.

###### Proof.

Since $R$ is a ring containing $\gamma$, it is closed under addition and under
multiplication by $\gamma$ (which is the automorphism $\alpha$ of
$\gamma^{-1}R$). This verifies Definition 2.10(a) and Definition 2.10(c).
Given any $x$ in $\gamma^{-1}R$, we may write $x=\gamma^{-\ell}r$ for some
$\ell\in\mathbb{Z}_{\geq 0}$ and $r\in R$, and thus $\alpha^{\ell}(x)\in R$,
verifying Definition 2.10(b). ∎

The following lemma is crucial for identifying confining subsets with ideals
of $\widehat{R}$. It is the only place axiom (A5), eventually periodic
addresses, is used. Recall that the preorder $\preceq$ on confining subsets
under $\alpha$ was defined in Section 4.

######  Lemma 6.3.

For any $Q\subseteq\gamma^{-1}R$ which is confining under $\alpha$, we have
$Q\preceq R$ in the preorder $\preceq$ on confining subsets.

###### Proof.

We will show that every element of $R$ has uniformly bounded word length with
respect to the generating set $Q\cup\\{t^{\pm 1}\\}$, which is equivalent to
$Q\preceq R$. As the fixed transversal $T$ is finite, there is some constant
$K$ such that $\alpha^{K}(f)\in Q$ for all $f\in T$. Let
$\overline{Q}=Q\cup\bigcup_{i\geq 0}\alpha^{i}{T}$. By [8, Lemma 4.9 &
Corollary 4.10] (see also [3, Lemma 3.2]), $\overline{Q}$ is confining under
$\alpha$ and $Q\sim\overline{Q}$. Thus it suffices to show that elements of
$R$ have bounded word length with respect to $\overline{Q}\cup\\{t^{\pm
1}\\}$. Note that $\overline{Q}$ contains all terms of the form $a\gamma^{j}$
for $a\in T$ and $j\geq 0$.

We divide the proof into two cases and first consider only those elements of
$R$ whose $\gamma$–adic addresses are finite.

Case 1: Let $R_{0}$ be the subset of $R$ consisting of the elements whose
$\gamma$–adic addresses are finite. We will show that every element of $R_{0}$
has uniformly bounded word length with respect to $\overline{Q}\cup\\{t^{\pm
1}\\}$.

######  Claim 6.4.

Every element of $\alpha^{k_{1}}(R_{0})$ has uniformly bounded word length
$k_{1}$ in $\overline{Q}$, where $k_{1}>0$ is large enough that
$\alpha^{k_{1}}(\overline{Q}+\overline{Q})\subseteq\overline{Q}$ (see
Definition 2.10(c)).

Assuming the claim, the statement about all elements of $R_{0}$ can be deduced
as follows. Let $x\in R_{0}$, and suppose $x$ has degree $n_{x}$, so that

$x=\sum_{i=0}^{n_{x}}a_{i}\gamma^{i}=\sum_{i=0}^{k_{1}-1}a_{i}\gamma^{i}+\sum_{i=k_{1}}^{n_{x}}a_{i}\gamma^{i},$
(1)

where $a_{i}\in T$. The first term in (1) has length $\leq k_{1}$ in
$\overline{Q}$. The second term in (1) is an element of
$\alpha^{k_{1}}(R_{0})$ and so has length $\leq k_{1}$ in $\overline{Q}$ if we
prove the claim. Therefore $x$ will have length at most $2k_{1}$ with respect
to $\overline{Q}\cup\\{t^{\pm 1}\\}$. We now prove the claim.

###### Proof of Claim 6.4.

Fix $y\in\alpha^{k_{1}}(R_{0})$, and let $n_{y}$ be the degree of $y$, so that

$y=\sum_{i=k_{1}}^{n_{y}}a_{i}\gamma^{i}.$

We will prove the statement by induction on $n_{y}$. For the base case,
suppose $n_{y}\in[k_{1},2k_{1})$. Then

$y=\sum_{i=k_{1}}^{2k_{1}-1}a_{i}\gamma^{i},$

where $a_{i}\in T$ for each $i$ and $a_{i}=0$ for each $n_{y}<i\leq 2k_{1}-1$
(if $n_{y}<2k_{1}-1$). Since each term $a_{i}\gamma^{i}$ is an element of
$\overline{Q}$, it follows that $y$ has length at most $k_{1}$ in
$\overline{Q}$.

For the induction step, assume that $y$ has length at most $k_{1}$ in
$\overline{Q}$ whenever $n_{y}\in[k_{1},(l-1)k_{1})$, and suppose
$n_{y}\in[k_{1},lk_{1})$. Then we may write

$y=\sum_{i=k_{1}}^{lk_{1}-1}a_{i}\gamma^{i}=\sum_{i=k_{1}}^{2k_{1}-1}a_{i}\gamma^{i}+\sum_{i=2k_{1}}^{lk_{1}-1}a_{i}\gamma^{i}=g+f,$

where $g,f$ are the two summands after the second equals sign. The element $g$
has $k_{1}$ terms, and

$\displaystyle
g=\alpha^{k_{1}}\left(\sum_{i=0}^{k_{1}-1}b_{i}\gamma^{i}\right),$

where $b_{i}=a_{i+k_{1}}\in T$. Moreover,

$\displaystyle
f=\alpha^{k_{1}}\left(\sum_{i=k_{1}}^{(l-1)k_{1}-1}c_{i}\gamma^{i}\right)=\alpha^{k_{1}}(f^{\prime}),$

where $c_{i}\in T$ and $f^{\prime}\in R_{0}$ has degree at most
$(l-1)k_{1}-1$. By the induction hypothesis, $f^{\prime}$ has length at most
$k_{1}$ in $\overline{Q}$, and so $f=\alpha^{k_{1}}(f_{1}+f_{2}+\cdots+f_{r})$
with $r\leq k_{1}$ and $f_{j}\in\overline{Q}$. Thus

$\displaystyle y$
$\displaystyle=\alpha^{k_{1}}\left(\sum_{i=0}^{k_{1}-1}b_{i}\gamma^{i}\right)+\alpha^{k_{1}}(f_{1}+\cdots+f_{r})$
$\displaystyle=\alpha^{k_{1}}(b_{0}+f_{1})+\alpha^{k_{1}}(b_{1}\gamma+f_{2})+\cdots+\alpha^{k_{1}}(b_{r-1}\gamma^{r-1}+f_{r})+\alpha^{k_{1}}(b_{r}\gamma^{r})+\cdots+\alpha^{k_{1}}(b_{k_{1}-1}\gamma^{k_{1}-1})$

As each $b_{i}\gamma^{i}$ is an element of $\overline{Q}$ and each $f_{j}$ is
an element of $\overline{Q}$, each term is contained in $\overline{Q}$ since
$\alpha^{k_{1}}(\overline{Q}+\overline{Q})\subset\overline{Q}$ and
$\overline{Q}$ is closed under $\alpha$. Thus $y$ has length at most $k_{1}$
in $\overline{Q}$, which completes the induction. ∎

Case 2: We now consider elements of $R$ whose $\gamma$–adic addresses are
infinite. Let $y\in R$ be such an element. By axiom (A5), the $\gamma$–adic
address of $y$ is eventually periodic, and so $y=y_{1}+y_{2}$, where $y_{1}$
has a finite $\gamma$–adic address and $y_{2}$ has a periodic $\gamma$–adic
address. Case 1 implies that $\|y_{1}\|_{\overline{Q}\cup\\{t^{\pm 1}\\}}\leq
2k_{1}$. As $y_{2}\in R$, Definition 2.10(b) ensures that there is some non-
negative integer $N$ such that $\alpha^{i}(y_{2})\in\overline{Q}$ for all
$i\geq N$. Choose $i\geq N$ so that $i$ is a multiple of the period of
$y_{2}$. This ensures that $z=y_{2}-\alpha^{i}(y_{2})$ has a finite
$\gamma$–adic address, and so $\|z\|_{\overline{Q}\cup\\{t^{\pm 1}\\}}\leq
2k_{1}$ by Case 1. Thus $y_{2}=\alpha^{i}(y_{2})+z$, and, using that fact that
$\alpha^{i}(y_{2})\in\overline{Q}$ by our choice of $i$, we see that
$\|y_{2}\|_{\overline{Q}\cup\\{t^{\pm 1}\\}}\leq 2k_{1}+1$. Therefore
$\|y\|_{\overline{Q}\cup\\{t^{\pm 1}\\}}\leq 4k_{1}+1$. ∎

The following lemma will be useful in the next several subsections.

######  Lemma 6.5.

Let $\mathfrak{a}$ be an ideal of $\widehat{R}$. Then $\mathfrak{a}$ is closed
topologically in $\widehat{R}$.

###### Proof.

By [6, Theorem 10.26], $\widehat{R}$ is Noetherian, so every ideal
$\mathfrak{a}$ of $\widehat{R}$ is finitely generated. The ring $\hat{R}$ is
sequentially compact, and every finitely generated ideal in a sequentially
compact ring is closed: if $\mathfrak{a}=(f_{1},\ldots,f_{k})$ and
$x_{i}=a_{1}^{i}f_{1}+\cdots+a_{k}^{i}f_{k}\in\mathfrak{a}$ with $x_{i}\to
x\in\widehat{R}$, then we may pass to a subsequence to assume that the
sequences $\\{a_{j}^{i}\\}_{i=1}^{\infty}$ all converge in $\widehat{R}$, say
$a_{j}^{i}\to a_{j}\in\widehat{R}$. Then $x_{i}\to
a_{1}f_{1}+\cdots+a_{k}f_{k}$, and so
$x=a_{1}f_{1}+\cdots+a_{k}f_{k}\in\mathfrak{a}$. ∎

###  6.1 Constructing confining subsets from ideals

Given an ideal $\mathfrak{a}\subseteq\widehat{R}$, we first show how to
construct a confining subset of $\gamma^{-1}R$. Roughly, given any element
$x\in\mathfrak{a}$, the associated confining subset contains all elements of
$\gamma^{-1}R$ whose fractional part coincides with (some number of) the final
digits of $x$. In particular, infinitely many elements of the confining subset
can be constructed from a single element of the ideal. Formally, we have the
following definition.

######  Definition 6.6.

Given an ideal $\mathfrak{a}\subseteq\widehat{R}$, define

$\mathcal{C}(\mathfrak{a})=\\{\dots a_{2}a_{1}a_{0}.a_{-1}\dots
a_{-k}\in\gamma^{-1}R:\exists x\in\mathfrak{a}\textrm{ such that }x=\dots
x_{1}x_{0}a_{-1}\dots a_{-k}\textrm{ for some }x_{i}\in T\\}.$

It is important to note here that the string of non-negative digits $\ldots
a_{2}a_{1}a_{0}$ is not arbitrary but must define an element of $R$. This is
necessary in order for $\mathcal{C}(\mathfrak{a})$ to be a subset of
$\gamma^{-1}R$.

######  Example 6.7.

If $R=\mathbb{Z}$ and $\gamma=2$, then $\gamma^{-1}R=\mathbb{Z}[\frac{1}{2}]$
and $\widehat{R}=\mathbb{Z}_{2}$, the ring of $2$–adic integers. Consider the
ideal $\mathfrak{a}=2^{4}\mathbb{Z}_{2}$ in $\widehat{R}=\mathbb{Z}_{2}$ and
the element $x=\dots 101010000\in\mathfrak{a}$. Then, for example, the
elements $1111.10101(=1111.101010000),1.1(=1.10000),$ and $0.101(=0.1010000)$
are all elements of $\mathcal{C}(\mathfrak{a})$.

######  Remark 6.8.

Since $0\in\mathfrak{a}$ for every ideal $\mathfrak{a}\subseteq\widehat{R}$,
we have that $R\subseteq\mathcal{C}(\mathfrak{a})$ for every ideal
$\mathfrak{a}$.

######  Lemma 6.9.

If $\mathfrak{a}\subseteq\widehat{R}$ is an ideal, then
$\mathcal{C}(\mathfrak{a})$ is confining under $\alpha$.

###### Proof.

We will check that Definition 2.10 holds. For condition (a), let $r=\ldots
a_{2}a_{1}a_{0}.a_{-1}\ldots a_{-k}\in\mathcal{C}(\mathfrak{a})$. By the
definition of $C(\mathfrak{a})$, there exists $x=\ldots
c_{1}c_{0}a_{-1}a_{-2}\ldots a_{-k}\in\mathfrak{a}$ for some $c_{i}\in T$.
Thus $\alpha(r)=\gamma r=\ldots a_{2}a_{1}a_{0}a_{-1}.a_{-2}\ldots
a_{-k}\in\mathcal{C}(\mathfrak{a}),$ and so
$\alpha(\mathcal{C}(\mathfrak{a}))\subseteq\mathcal{C}(\mathfrak{a})$,
verifying (a). By Remark 6.8, $R\subseteq\mathcal{C}(\mathfrak{a})$, and so
$\bigcup_{i=0}^{\infty}\alpha^{-i}(\mathcal{C}(\mathfrak{a}))=\gamma^{-1}R$,
verifying (b).

To show condition (c) holds, let $r_{1}=\dots a_{2}a_{1}a_{0}.a_{-1}\dots
a_{-k}$ and $r_{2}=\dots b_{2}b_{1}b_{0}.b_{-1}\dots b_{-\ell}$ be elements of
$\mathcal{C}(\mathfrak{a})$. We may assume without loss of generality that
$k\geq\ell$. The sum $r_{1}+r_{2}$ is calculated as follows:

| $\dots$ | $a_{1}$ | $a_{0}.$ | $a_{-1}$ | $\dots$ | $a_{-\ell}$ | $a_{-\ell-1}$ | $\dots$ | $a_{-k}$
---|---|---|---|---|---|---|---|---|---
$+$ | $\dots$ | $b_{1}$ | $b_{0}.$ | $b_{-1}$ | $\dots$ | $b_{-\ell}$ |  |  |
= | $\dots$ | $c_{1}$ | $c_{0}.$ | $c_{-1}$ | $\dots$ | $c_{-\ell}$ | $a_{-\ell-1}$ | $\dots$ |  $a_{-k}$.

That is,

$r_{1}+r_{2}=\dots c_{1}c_{0}.c_{-1}\dots c_{-\ell}a_{-\ell-1}\dots a_{-k}.$
(2)

Since $r_{1},r_{2}\in\mathcal{C}(\mathfrak{a})$, there exist
$x_{1},x_{2}\in\mathfrak{a}$ with $x_{1}=\dots d_{2}d_{1}d_{0}a_{-1}\dots
a_{-k}$ and $x_{2}=\dots f_{2}f_{1}f_{0}b_{-1}\dots b_{-\ell}$ for some
$d_{i},f_{i}\in T$. Since $\mathfrak{a}$ is an ideal,
$x_{1}+\gamma^{k-\ell}x_{2}$ is also in $\mathfrak{a}$. We see that

$\displaystyle x_{1}+\gamma^{k-\ell}x_{2}$ $\displaystyle=\left(\dots
d_{2}d_{1}d_{0}a_{-1}\dots a_{-k}\right)+(\dots f_{2}f_{1}f_{0}b_{-1}\dots
b_{-\ell}\underbrace{0\dots\dots 0}_{k-\ell\textrm{ times}})$
$\displaystyle=\dots z_{1}z_{0}c_{-1}\dots c_{-\ell}a_{-\ell-1}\dots a_{-k}$

for some $z_{i}\in T$. Comparing this with (2), we see that
$r_{1}+r_{2}\in\mathcal{C}(\mathfrak{a})$. Therefore (c) holds with $k_{0}=0$.
∎

In fact, the structure of $\mathcal{C}(\mathfrak{a})$ is even nicer than
suggested by Lemma 6.9.

######  Lemma 6.10.

If $\mathfrak{a}\subseteq\widehat{R}$ is an ideal, then the confining subset
$\mathcal{C}(\mathfrak{a})$ is a subring of $\gamma^{-1}R$ containing $R$.

###### Proof.

By the proof of Lemma 6.9, the set $\mathcal{C}(\mathfrak{a})$ is closed under
addition and contains $R$. We will prove that it is also closed under
multiplication.

Any element of $\gamma^{-1}R$ can be written as $a\gamma^{-k}+x$, where
$a=a_{0}+a_{1}\gamma+\dots+a_{k-1}\gamma^{k-1}$ and $x\in R$, for any
sufficiently large $k$. Here, $x$ is the “integral part” of the element
(involving only non-negative powers of $\gamma$) and $a\gamma^{-k}$ is the
“fractional part” (involving only negative powers). Note that we are free to
take any sufficiently large $k$ because the fractional part of the element can
be made to have as many additional digits as desired simply by adding zero
digits.

In particular, given any $r,s\in\mathcal{C}(\mathfrak{a})$, we may choose $k$
large enough to write $r=a\gamma^{-k}+x$ and $s=b\gamma^{-k}+y$ with $x,y\in
R$ and $a=a_{0}+a_{1}\gamma+\cdots+a_{k-1}\gamma^{k-1}$ and
$b=b_{0}+b_{1}\gamma+\cdots+b_{k-1}\gamma^{k-1}$. Since
$r,s\in\mathcal{C}(\mathfrak{a})$, by definition there must exist elements
$a_{0}+a_{1}\gamma+\cdots+a_{k-1}\gamma^{k-1}+d_{k}\gamma^{k}+\cdots=a+z\gamma^{k}$
and
$b_{0}+b_{1}\gamma+\cdots+b_{k-1}\gamma^{k-1}+e_{k}\gamma^{k}+\cdots=b+w\gamma^{k}$
in $\mathfrak{a}$, where $z,w\in\widehat{R}$. We will show that
$rs\in\mathcal{C}(\mathfrak{a})$.

We have

$rs=(a\gamma^{-k}+x)(b\gamma^{-k}+y)=(ab+ay\gamma^{k}+bx\gamma^{k})\gamma^{-2k}+xy.$

Since $xy\in R$, the fractional part of $rs$ is contained in
$(ab+ay\gamma^{k}+bx\gamma^{k})\gamma^{-2k}$. Hence it suffices to show that
the first $2k$ digits of $ab+ay\gamma^{k}+bx\gamma^{k}$ agree with the first
$2k$ digits of some element of $\mathfrak{a}$.

As $a+z\gamma^{k}\in\mathfrak{a}$ and $b+w\gamma^{k}\in\mathfrak{a}$, the
ideal $\mathfrak{a}$ contains the element

$(a+z\gamma^{k})(b+w\gamma^{k})+(a+z\gamma^{k})(-w\gamma^{k}+y\gamma^{k})+(b+w\gamma^{k})(-z\gamma^{k}+x\gamma^{k})=ab+ay\gamma^{k}+bx\gamma^{k}+\gamma^{2k}(zy+wx-
zw).$

The first $2k$ digits of this element agree with the first $2k$ digits of
$ab+ay\gamma^{k}+bx\gamma^{k}$, completing the proof. ∎

###  6.2 Constructing ideals from confining subsets

We now show how to construct an ideal of $\widehat{R}$ from a confining subset
of $\gamma^{-1}R$. Roughly speaking, the ideal is the set of all elements of
$\widehat{R}$ with the property that every terminal string of digits appears
as the fractional part of _some_ element of the confining subset.
Alternatively, it may be thought of as a kind of limit set of the fractional
parts of elements of $Q$.

######  Definition 6.11.

Given a subset $Q\subset\gamma^{-1}R$ that is confining under $\alpha$, define
$\mathcal{I}(Q)$ by

$\mathcal{I}(Q)\vcentcolon=\\{\ldots a_{2}a_{1}a_{0}\in\widehat{R}:\forall
t\geq 0\,\exists q\in Q\textrm{ such that }q=\dots b_{2}b_{1}b_{0}.a_{t}\dots
a_{0}\textrm{ for some }b_{i}\in T\\}.$

######  Lemma 6.12.

If $Q\subset\gamma^{-1}R$ is confining under $\alpha$, then $\mathcal{I}(Q)$
is an ideal of $\widehat{R}$.

###### Proof.

We first show that $\mathcal{I}(Q)$ is closed under addition. Let $x=\dots
x_{2}x_{1}x_{0}$ and $y=\dots y_{2}y_{1}y_{0}$ be elements of
$\mathcal{I}(Q)$. Then $x+y=z=\dots z_{2}z_{1}z_{0}$ for some $z_{i}\in T$.
Let $k_{0}$ be large enough that $\alpha^{k_{0}}(Q+Q)\subset Q$. By the
definition of $\mathcal{I}(Q)$, for any $t\geq 0$, there exist $r=\dots
a_{2}a_{1}a_{0}.x_{t+k_{0}}\dots x_{0}$ and $s=\dots
b_{2}b_{1}b_{0}.y_{t+k_{0}}\dots y_{0}$ in $Q$ for some $a_{i},b_{i}\in T$.
Thus $r+s=\dots c_{2}c_{1}c_{0}.z_{t+k_{0}}\dots z_{0}$ for some $c_{i}\in T$,
and

$\alpha^{k_{0}}(r+s)=\dots c_{2}c_{1}c_{0}z_{t+k_{0}}\dots z_{t+1}.z_{t}\dots
z_{0}\in Q.$

Since $t$ is arbitrary, it follows that $z\in\mathcal{I}(Q)$.

We next show that $\mathcal{I}(Q)$ is closed under multiplication by elements
of $\widehat{R}$. We do this in three steps. First, since $\mathcal{I}(Q)$ is
closed under addition, $\mathcal{I}(Q)$ is closed under multiplication by any
positive integer multiple $z=z\cdot 1$ of $1$ in $\widehat{R}$: if
$x\in\mathcal{I}(Q)$, then

$zx=\underbrace{x+\cdots+x}_{z\text{ times}}\in\mathcal{I}(Q).$

In particular, since $T=[d]=\\{0,\dots,d-1\\}$, it follows that
$\mathcal{I}(Q)$ is closed under multiplication by elements of $T$. Second, we
show that $\mathcal{I}(Q)$ is closed under multiplication by $\gamma$. If
$x=\dots x_{2}x_{1}x_{0}\in\mathcal{I}(Q)$, then for any $s\geq 0$ there
exists $r=\dots a_{2}a_{1}a_{0}.x_{s}\dots x_{0}\in Q$, for some $a_{i}\in T$.
But

$r=\dots a_{2}a_{1}a_{0}.x_{s}\dots x_{0}=\dots a_{2}a_{1}a_{0}.x_{s}\dots
x_{0}0,$

and since $\gamma x=\dots x_{2}x_{1}x_{0}0$, this shows that $\gamma
x\in\mathcal{I}(Q)$.

Third, we show that $\mathcal{I}(Q)$ is (topologically) closed in
$\widehat{R}$. Fix $x=\dots x_{2}x_{1}x_{0}\in\overline{\mathcal{I}(Q)}$. Then
for any $s\geq 0$, there exists $y=\dots y_{2}y_{1}y_{0}\in\mathcal{I}(Q)$
with $y_{i}=x_{i}$ for all $i\leq s$. By the definition of $\mathcal{I}(Q)$,
there exists $r\in Q$ with $r=\dots a_{2}a_{1}a_{0}.y_{s}\dots y_{0}$ for some
$a_{i}\in T$. But then we also have $r=\dots a_{2}a_{1}a_{0}.x_{s}\dots
x_{0}$. Since $s$ was arbitrary, this shows that $x\in\mathcal{I}(Q)$.

We can now put these three steps together to show that $\mathcal{I}(Q)$ is
closed under multiplication by elements of $\widehat{R}$. Let $x=\dots
x_{2}x_{1}x_{0}\in\mathcal{I}(Q)$ and $r=\dots r_{2}r_{1}r_{0}\in\widehat{R}$,
where $r_{i}\in T$. For any $s\geq 0$, we have

$\left(\sum_{i=0}^{s}r_{i}\gamma^{i}\right)\cdot
x=\sum_{i=0}^{s}r_{i}\gamma^{i}x\in\mathcal{I}(Q).$

Since $\mathcal{I}(Q)$ is topologically closed, this implies that

$rx=\lim_{s\to\infty}\left(\sum_{i=0}^{s}r_{i}\gamma^{i}\right)\cdot
x=\lim_{s\to\infty}\sum_{i=0}^{s}r_{i}\gamma^{i}x\in\mathcal{I}(Q),$

as desired. Therefore $\mathcal{I}(Q)$ is an ideal of $\widehat{R}$. ∎

###  6.3 Saturated ideals and the poset $\mathcal{P}_{+}(G)$

In this subsection, we complete our discussion of the relationship between
ideals and the poset $\mathcal{P}_{+}(G)$. We first define a preorder on
ideals which leads to a notion of equivalence for ideals.

######  Definition 6.13.

Define a preorder on ideals of $\widehat{R}$ by $\mathfrak{a}\leq\mathfrak{b}$
if for all $x\in\mathfrak{a}$, there exists $i\in\mathbb{Z}_{\geq 0}$ such
that $\gamma^{i}x\in\mathfrak{b}$. This induces an equivalence relation $\sim$
on the ideals of $\widehat{R}$ by $\mathfrak{a}\sim\mathfrak{b}$ whenever
$\mathfrak{a}\leq\mathfrak{b}$ and $\mathfrak{b}\leq\mathfrak{a}$.

The preorder $\leq$ induces a partial order $\preccurlyeq$ on the resulting
set of equivalence classes of ideals.

######  Remark 6.14.

Since ideals in $\widehat{R}$ are finitely generated,
$\mathfrak{a}\leq\mathfrak{b}$ exactly when
$\gamma^{i}\mathfrak{a}\subset\mathfrak{b}$ for some $i\geq 0$, i.e., there is
a uniform power $i$ such that $\gamma^{i}x\in\mathfrak{b}$ for all
$x\in\mathfrak{a}$, even though this is not required by the definition.

The following lemma shows that this relation on ideals of $\widehat{R}$
interacts well with the way we constructed confining subsets from ideals.

######  Lemma 6.15.

If $\mathfrak{a}\leq\mathfrak{b}$, then
$\mathcal{C}(\mathfrak{a})\subset\mathcal{C}(\mathfrak{b})$. If
$\mathfrak{a}\sim\mathfrak{b}$, then
$\mathcal{C}(\mathfrak{a})=\mathcal{C}(\mathfrak{b})$.

###### Proof.

Assume $\mathfrak{a}\leq\mathfrak{b}$. By the definition of
$\mathcal{C}(\mathfrak{a})$, elements of the ideal $\mathfrak{a}$ determine
only the coefficients of the negative powers of $\gamma$ in elements of
$\mathcal{C}(\mathfrak{a})$. That is, for each $a=\ldots
a_{1}a_{0}\in\mathfrak{a}$, $n\geq 0$, and $z\in R$, there is an element
$a_{0}\gamma^{-n}+\cdots+a_{n-1}\gamma^{-1}+z$ in $\mathcal{C}(\mathfrak{a})$.
Moreover, if $a=\dots a_{1}a_{0}$ lies in $\mathfrak{a}$, then $a$ and
$\gamma^{i}a=\dots a_{n}\dots a_{0}\underbrace{0\dots 0}_{i\textrm{ times}}$
determine the same elements of $\mathcal{C}(\mathfrak{a})$. Since there exists
$i$ such that $\gamma^{i}\mathfrak{a}\subseteq\mathfrak{b}$, this implies that
$\mathcal{C}(\mathfrak{a})\subseteq\mathcal{C}(\mathfrak{b})$. The second
sentence of the lemma follows immediately. ∎

We now choose a canonical representative of each equivalence class
$[\mathfrak{a}]$ of ideals of $\widehat{R}$ under $\sim$. Since $\gamma$ is
not a zero divisor in $\widehat{R}$, there is a natural injection from
$\widehat{R}$ to the localization $\gamma^{-1}\widehat{R}$, which we denote by
$f$. If $\mathfrak{a}$ is an ideal of $\widehat{R}$, its image
$f(\mathfrak{a})$ generates an ideal denoted by $\mathfrak{a}^{e}$. The ideal
$f^{-1}(\mathfrak{a}^{e})$ contains $\mathfrak{a}$ and is called the
_saturation_ of $\mathfrak{a}$. An ideal is _saturated_ if $\mathfrak{a}$ is
its own saturation: $\mathfrak{a}=f^{-1}(\mathfrak{a}^{e})$. Equivalently,
$\mathfrak{a}\subset\widehat{R}$ is saturated if and only if whenever
$r\in\widehat{R}$ and $\gamma^{i}r\in\mathfrak{a}$ for some $i\geq 0$, we have
$r\in\mathfrak{a}$. Recall that an element of a poset is _largest_ when it is
greater than or equal to every other element of the poset.

######  Lemma 6.16.

The saturation of an ideal $\mathfrak{a}\subseteq\widehat{R}$ is the unique
largest element in the equivalence class $[\mathfrak{a}]$ with respect to the
partial order given by inclusion.

###### Proof.

Let $\mathfrak{m}$ be the saturation of $\mathfrak{a}$. Since
$\mathfrak{a}\subset\mathfrak{m}$, it follows that
$\mathfrak{a}\leq\mathfrak{m}$. For any $r\in\mathfrak{m}$, we have
$\gamma^{i}r\in\mathfrak{a}$ for some $i\geq 0$, and so
$\mathfrak{m}\leq\mathfrak{a}$. Thus $\mathfrak{m}\sim\mathfrak{a}$. To see
that $\mathfrak{m}$ is largest in $[\mathfrak{a}]$, let
$\mathfrak{b}\in[\mathfrak{a}]$. Since $\mathfrak{b}\sim\mathfrak{a}$, for
every $b\in\mathfrak{b}$, there is an $i\in\mathbb{Z}_{\geq 0}$ such that
$\gamma^{i}b\in\mathfrak{a}\subset\mathfrak{m}$. As $\mathfrak{m}$ is
saturated, this implies that $b\in\mathfrak{m}$, and so
$\mathfrak{b}\subseteq\mathfrak{m}$. ∎

We may form the poset of saturated ideals of $\widehat{R}$ with the partial
order given by inclusion. The following general lemma shows that this poset is
isomorphic to the poset of (all) ideals of $\gamma^{-1}\widehat{R}$ with
inclusion.

######  Lemma 6.17 ([12, Ch. 2 Sec. 2 No. 4 Proposition 10]).

Let $A$ be a ring, $S\subset A$ a multiplicatively closed subset of $A$, and
$S^{-1}A$ the localization. Let $f:A\to S^{-1}A$ be the natural homomorphism.
Then the map sending an ideal $\mathfrak{b}\subset S^{-1}A$ to its preimage
$f^{-1}(\mathfrak{b})$ is an isomorphism from the poset of ideals of $S^{-1}A$
with inclusion to the poset of saturated ideals of $A$ with inclusion.

Our goal is to prove the following theorem.

######  Theorem 6.18.

In the notation and terminology of Theorem 1.1, $\mathcal{P}_{+}(G)$ is
isomorphic to the opposite of the poset of saturated ideals of $\widehat{R}$
and hence to the opposite of the poset of ideals of the localization
$\gamma^{-1}\widehat{R}$.

The next two lemmas show the relationship between the order on saturated
ideals and the order on $\mathcal{P}_{+}(G)$. Again we will investigate the
structure of the preorder $\preceq$ on confining subsets under $\alpha$.

######  Lemma 6.19.

If $\mathfrak{a},\mathfrak{b}$ are saturated ideals of $\widehat{R}$ such that
$\mathfrak{a}\not\leq\mathfrak{b}$ and $\mathfrak{b}\not\leq\mathfrak{a}$,
then $\mathcal{C}(\mathfrak{a})$ and $\mathcal{C}(\mathfrak{b})$ are
incomparable with respect to the preorder $\preceq$ on confining subsets.

###### Proof.

Since $\mathfrak{a}\not\leq\mathfrak{b}$, we have that
$\mathfrak{a}\not\subseteq\mathfrak{b}$. Thus there exists $x=\dots
x_{2}x_{1}x_{0}\in\mathfrak{a}$ such that $x\not\in\mathfrak{b}$. Since
$x\not\in\mathfrak{b}$, there exists some non-negative integer $M$ such that
no element $y=\dots y_{2}y_{1}y_{0}\in\mathfrak{b}$ satisfies $y_{i}=x_{i}$
for all $i\leq M$. To see this, notice that if such an element of
$\mathfrak{b}$ existed for each $M\in\mathbb{Z}_{\geq 0}$, then the limit of
these elements would be equal to $x$. Since $\mathfrak{b}$ is topologically
closed by Lemma 6.5, this would contradict $x\notin\mathfrak{b}$.

For any $K\in\mathbb{Z}_{\geq 0}$, there is an element
$q\in\mathcal{C}(\mathfrak{a})$ such that $q=\dots q_{2}q_{1}q_{0}.x_{K}\dots
x_{0}$. Assume $K\geq M$, and consider the smallest $u\in\mathbb{Z}$ such that
$\alpha^{u}(q)\in\mathcal{C}(\mathfrak{b})$. We will show that $u>K-M$, so
suppose for contradiction that $u\leq K-M$. Then the digits of $\alpha^{u}(q)$
to the right of the decimal point are $x_{K-u}x_{K-u-1}\ldots x_{0}$ and
$K-u\geq M$. These digits are uniquely determined by $\alpha^{u}(q)$ up to
possibly adding or deleting zeroes all the way to the right of the expression.
Since $\alpha^{u}(q)\in\mathcal{C}(\mathfrak{b})$, there is an element
$y\in\mathfrak{b}$ determining the same element of $\gamma^{-1}R$. That is,
$y=\ldots y_{M+2}y_{M+1}x_{M}\ldots x_{0}0\ldots 0$ with some number of zeroes
at the end. Since $\mathfrak{b}$ is saturated, we have $y^{\prime}=\ldots
y_{M+2}y_{M+1}x_{M}\ldots x_{0}\in\mathfrak{b}$ (with no extra zeroes). This
contradicts our choice of $M$, and so $u>K-M$. Thus
$\mathcal{C}(\mathfrak{b})\not\preceq\mathcal{C}(\mathfrak{a})$, since $K$ is
arbitrarily large. A symmetric argument shows that
$\mathcal{C}(\mathfrak{a})\not\preceq\mathcal{C}(\mathfrak{b})$, so that the
confining subsets are incomparable. ∎

######  Lemma 6.20.

If $\mathfrak{a},\mathfrak{b}$ are two saturated ideals of $\widehat{R}$ such
that $\mathfrak{a}\lneq\mathfrak{b}$, then $\mathcal{C}(\mathfrak{a})\succeq
C(\mathfrak{b})$ and
$\mathcal{C}(\mathfrak{a})\not\sim\mathcal{C}(\mathfrak{b})$.

###### Proof.

Since $\mathfrak{a}\leq\mathfrak{b}$, we have
$\mathcal{C}(\mathfrak{a})\subset\mathcal{C}(\mathfrak{b})$ by Lemma 6.15.
Thus $\mathcal{C}(\mathfrak{a})\succeq\mathcal{C}(\mathfrak{b})$. Since
$\mathfrak{a}\lneq\mathfrak{b}$, an argument similar to the proof of Lemma
6.19 produces elements of $q\in\mathcal{C}(\mathfrak{b})$ with
$\inf\\{u:\alpha^{u}(q)\in\mathcal{C}(\mathfrak{a})\\}$ arbitrarily large,
which shows that $\mathcal{C}(\mathfrak{b})\not\sim\mathcal{C}(\mathfrak{a})$.
∎

The final lemmas necessary to prove Theorem 6.18 show that, up to equivalence
of hyperbolic structures, $\mathcal{C}(\mathcal{I}(Q))$ is the same as $Q$ for
any confining subset $Q$. First we show that
$Q\preceq\mathcal{C}(\mathcal{I}(Q))$.

######  Lemma 6.21.

If $Q\subseteq\gamma^{-1}R$ is confining under $\alpha$, then
$Q\preceq\mathcal{C}(\mathcal{I}(Q))$.

###### Proof.

It suffices to find $K\geq 0$ with
$\mathcal{C}(\mathcal{I}(Q))\subset\alpha^{-K}(Q)$ by Lemma 4.2. Let $a=\dots
a_{2}a_{1}a_{0}.a_{-1}\dots a_{-k}\in\mathcal{C}(\mathcal{I}(Q))$. By
definition of $\mathcal{C}(\mathcal{I}(Q))$, there exists an element
$x\in\mathcal{I}(Q)$ such that $x=\dots c_{2}c_{1}c_{0}a_{-1}\dots a_{-k}$,
where $c_{i}\in T$. By definition of $\mathcal{I}(Q)$, there is an element
$q\in Q$ such that $q=\dots b_{2}b_{1}b_{0}.a_{-1}\dots a_{-k}$ for some
$b_{i}\in T$. Let $r=a-q\in R$.

By Lemma 6.3, there is a constant $M\geq 0$ such that $\alpha^{M}(R)\subset
Q$, and so $\alpha^{M}(a)=\alpha^{M}(q)+\alpha^{M}(r)\in Q+Q$. Let $k_{0}$ be
large enough that $\alpha^{k_{0}}(Q+Q)\subseteq Q$. Then
$\alpha^{M+k_{0}}(a)=\alpha^{k_{0}}(\alpha^{M}(a))\in\alpha^{k_{0}}(Q+Q)\subseteq
Q$, as desired. Setting $K=M+k_{0}$ concludes the proof. ∎

We now prove the equivalence of $Q$ and $\mathcal{C}(\mathcal{I}(Q))$.

######  Lemma 6.22.

If $Q\subseteq\gamma^{-1}R$ is confining under $\alpha$, then
$\mathcal{C}(\mathcal{I}(Q))\sim Q$.

###### Proof.

It follows from Lemma 6.21 that $\mathcal{C}(\mathcal{I}(Q))\succeq Q$. We
will show that $\mathcal{C}(\mathcal{I}(Q))\preceq Q$. Suppose this is not the
case. Then $Q\not\subseteq\alpha^{-k}(\mathcal{C}(\mathcal{I}(Q)))$ for any
$k\in\mathbb{Z}$ by Lemma 4.2, and so there are elements $a\in Q$ with
$m_{a}\vcentcolon=\inf\\{k:\alpha^{k}(a)\in\mathcal{C}(\mathcal{I}(Q))\\}$
arbitrarily large. Choose such an element $a=\dots a_{0}.a_{-1}\dots
a_{-\ell}\in Q$ with $m=m_{a}>k_{0}$, where $k_{0}$ is as in Definition
2.10(c), which holds for $Q$.

Choose $s\leq\ell$ largest such that no element of $\mathcal{I}(Q)$ has the
form $\dots a_{-s}\dots a_{-\ell}$, that is, such that no element of
$\mathcal{I}(Q)$ agrees with the last $\ell-s+1$ digits of $a$. We must have
$s\geq m$, for $\alpha^{s}(a)=\dots a_{0}a_{-1}\dots a_{-s}.a_{-s-1}\dots
a_{-\ell}\in\mathcal{C}(\mathcal{I}(Q))$, and if $s<m$, then this contradicts