# Pattern preserving quasi-isometries in lamplighter groups and other related
groups

Tullia Dymarz  Tullia Dymarz, Department of Mathematics, University of
Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706<EMAIL_ADDRESS>,  Beibei Liu  Beibei Liu, Department of Mathematics, Ohio State University,
100 Math Tower, 231 W 18th Ave, Columbus, OH 43210<EMAIL_ADDRESS>,
Nataša Macura  Nataša Macura (Emerita) Trinity University 1 Trinity Place San
Antonio TX 78212<EMAIL_ADDRESS>and  Rose Morris-Wright  Rose
Morris-Wright, Department of Mathematics and Statistics, Middlebury College,
Warner Hall, Middlebury, VT 05753<EMAIL_ADDRESS>
###### Abstract.

In this paper we explore the interplay between aspects of the geometry and
algebra of three families of groups of the form $B\rtimes{\mathbb{Z}}$, namely
Lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL. In
particular we examine what kind of maps are induced on $B$ by quasi-isometries
that coarsely permute cosets of the ${\mathbb{Z}}$ subgroup. By the results of
Schwartz(1996) and Taback(2000) in the lattice in SOL and solvable Baumslag-
Solitar cases respectively such quasi-isometries induce affine maps of $B$. We
show that this is no longer true in the lamplighter case but the induced maps
do share some features with affine maps.

##  1\. Introduction

A quasi-isometry is a map between metric spaces that preserves the large scale
geometry, only allowing for bounded discontinuities or for distances to be
scaled by a bounded amount. In geometric group theory, we aim to study groups
via this large scale geometry. We consider a finitely generated group as a
metric space by endowing each group element with a length corresponding to the
minimal number of generators in a word representing the given group element.
Two different presentations for the same group induce different word metrics
but all such metrics are quasi-isometric. This means that differences in large
scale geometry correspond to non-isomorphic groups. However quasi-isomorphic
metric spaces do not always come from isomorphic groups. Gromov’s program aims
to classify all finitely generated groups up to quasi-isometry (See Section 2
for technical details on quasi-isometries and word metrics.)

Often a first step in understanding which groups are quasi-isometric to a
given group $G$ is to study the structure of self quasi-isometries of $G$. A
priori, self quasi-isometries may seem unwieldy so to get a handle on them one
looks for structures that they preserve. For example, given a subgroup $H$, we
say that a self quasi-isometry $\Psi:G\to G$ _coarsely permutes_ the cosets of
$H$ if there exists a permutation $\sigma:G/H\to G/H$ and a $C^{\prime}\geq 0$
such that $d_{\mathcal{H}}(\Psi(gH),\sigma(gH))<C^{\prime}$ where
$d_{\mathcal{H}}$ denotes Hausdorff distance. We call such quasi-isometries
_pattern preserving_. Even if not all self quasi-isometries coarsely permute
the cosets of a given subgroup (i.e. are pattern preserving), one can try to
understand how much smaller is the set of self quasi-isometries that do.
Answering these types of questions can often lead to rigidity results for
groups built out of $G$ through graphs of group constructions (see [CM17]) and
has also become a question of independent interest. For a nice survey on some
results and questions in this direction see [HMPS21].

In this paper we focus on three classes of groups of the form
$G=B\rtimes{\mathbb{Z}}$ that have similar geometry:

  1. (1)

Lamplighter group. $L_{n}:=\bigoplus{\mathbb{Z}}_{n}\rtimes{\mathbb{Z}}$ where
semi-direct product is given by the action shifting the index.

  2. (2)

Lattices in SOL. $\Gamma_{A}:={\mathbb{Z}}^{2}\rtimes_{A}{\mathbb{Z}}$ where
the action is by multiplication by a hyperbolic matrix $A\in
SL(2,{\mathbb{Z}})$.

  3. (3)

Solvable Baumslag-Solitar group.
$BS(1,n):={\mathbb{Z}}[\frac{1}{n}]\rtimes{\mathbb{Z}}$ where the action is by
multiplication by $n$.

In each case we call $B$ the _base group_ and the subgroup generated by
${\mathbb{Z}}$ the _vertical subgroup_. By theorems of Farb-Mosher [FM98,
FM99] in the solvable Baumslag-Solitar group case and Eskin-Fisher-Whyte
[EFW12, EFW13] in the other two cases, all quasi-isometries coarsely permute
left cosets of $B$.

In this paper we are interested in quasi-isometries that coarsely permute the
left cosets of the vertical subgroup ${\mathbb{Z}}$. Note that the space of
cosets of the vertical subgroup can be identified with the base group $B$ and
by assumption any pattern preserving quasi-isometry $\Psi$ induces a
permutation $\psi:B\to B$. A priori, this induced map does not need to have
any structure. However by work of Schwartz [Sch96] in the case of $\Gamma_{A}$
and Taback [Tab00] in the case of $BS(1,n)$ it turns out that the induced maps
on $B={\mathbb{Z}}^{2}$ and $B={\mathbb{Z}}[\frac{1}{n}]$ respectively are
algebraic in nature. In particular they satisfy the property that for any
$a,b,c,d\in B$ with $a+c=b+d$ the image under the induced map $\psi$ satisfies

$\psi(a)+\psi(c)=\psi(b)+\psi(d)$

where ‘+’ represents the operation in the group $B$. If we imagine the four
group elements $a,b,c,d$ as the sides of a quadrilateral then we can view the
condition that $a+c=b+d$ as implying that the quadrilateral is a
parallelogram. We therefore call the above condition on $\psi$ _parallelogram
preserving_ since it states that $\psi$ takes parallelograms in $B$ to
parallelograms.

In the case of $BS(1,n)$ and $\Gamma_{A}$, the map $\psi$ being parallelogram
preserving is equivalent to $\psi$ being affine on ${\mathbb{Z}}[1/n]$ or
${\mathbb{Z}}^{2}$ respectively. If we consider parallelogram preserving maps
on B in the lamplighter case where $B=\bigoplus{\mathbb{Z}}_{n}$ we already
encounter different behavior from the other two groups. For both $BS(1,n)$ and
$\Gamma_{A}$ the action of the vertical subgroup ${\mathbb{Z}}$ on their
respective base groups $B$ is affine ($x\to nx$ in the $BS(1,n)$ case and
$x\to Ax$ in the $\Gamma_{A}$ case) while in the lamplighter group the shift
map is not. We are interested in studying a set of maps that includes all
isometries induced by the group action, so in order to allow for left
multiplication by group elements in all three cases, we call a map of
$\bigoplus{\mathbb{Z}}_{n}$ generalized affine if it is the composition of an
affine map and a vertical shift map $(x_{i})\mapsto(x_{i+k})$ for some
constant $k\in{\mathbb{Z}}$. With this definition, left multiplication by a
group element in $L_{n}$ will induce a generalized affine map on
$\bigoplus{\mathbb{Z}}_{n}$.

Our main question is whether or not all pattern preserving quasi-isometries
are generalized affine. This is true for lattices in SOL and for Baumslag-
Solitar groups by [Sch96, Tab00] but the situation is more complicated for
Lamplighter groups. This paper looks at the proofs of these known results from
a geometric standpoint and discusses how the shared geometry of these three
groups might lead one to expect that the lamplighter group has similar
behavior. It turns out, however, that pattern preserving quasi-isometries
behave differently in lamplighter groups and this paper describes their
behavior and explains why it is different from the first two cases.

In order to have any hope of finding conditions in that lamplighter group
$L_{n}$ that will guarantee generalized affine behavior, we must first
restrict our attention to $n=2$. For $L_{n}$ with $n\geq 3$ there are examples
of pattern preserving quasi-isometries (and even isometries if we chose our
generating set correctly) that do not induce generalized affine maps on
$B=\bigoplus{\mathbb{Z}}_{n}$. For example at any index simply take a
permutation of ${\mathbb{Z}}_{n}$ that does not come from an affine map of
${\mathbb{Z}}_{n}$. The situation for $n=2$ is more interesting. First we
examine pattern preserving isometries. Since different word metrics are not
isometrically equivalent we fix a generating set for $L_{2}$ whose Cayley
graph is the Diestel-Leader graph $DL(2,2)$ (see Section 2 for details).

######  Theorem 1.1.

If $\Psi:L_{2}\to L_{2}$ is a pattern preserving isometry then the induced map
$\psi$ comes from left multiplication by a group element (up to composition
with the map induced by inversion $(x_{i})_{i}\mapsto(x_{-i})_{i}$), and hence
is a generalized affine map.

When we generalize to arbitrary quasi-isometries on $L_{2}$, we show that the
induced maps are much less rigid than in $BS(1,n)$ or $\Gamma_{A}$.

######  Theorem 1.2.

There are pattern preserving quasi-isometries of $L_{2}$ that do not induce
generalized affine maps of $\bigoplus{\mathbb{Z}}_{2}$.

In Section 4.3, we explicitly construct a family of maps
$\psi:\bigoplus{\mathbb{Z}}_{2}\to\bigoplus{\mathbb{Z}}_{2}$ that are not
affine, but which induce pattern preserving quasi-isometries on $L_{2}$.
Despite the less rigid structure implied by Theorem 1.2, we are still able to
show that quasi-isometries that preserve cosets of the vertical subgroup are
“affine on the large scale.” That is they are affine when only applied to
“large” enough parallelograms. These notions are developed in Section 2.5 and
the result is proved as Proposition 4.1 but the key ingredient, captured in
the Lemma 1.3 below, boils down to the fact that all large enough
quadrilaterals must be parallelograms.

######  Lemma 1.3.

Let $S>1$ and for $(x_{i})\in\bigoplus{\mathbb{Z}}_{2}$ denote $supp((x_{i}))$
to be the smallest interval containing all indices $i$ for which $x_{i}\neq
0$. Suppose that $a,b,c,d\in\bigoplus{\mathbb{Z}}_{2}$ satisfy the following

  1. (1)

$|supp(b-a)|<S,|supp(c-a)|<S$

  2. (2)

$|supp(b-c)|>2S,|supp(d-a)|>2S$

Then $a+b=c+d$.

This result is analogous to the first step of the proofs in [Sch96, Tab00] in
the $\Gamma_{A}$ and $BS(1,n)$ cases and follows from the common geometry of
all three groups. We will describe the common geometry shared by all three
groups in Section 2. In Section 3, we will compare the proofs of the results
from Taback for $BS(1,n)$ [Tab00], and Schwartz for lattices in SOL [Sch96],
with a focus on how these proofs reflect this shared geometry. Finally, in
Section 4 we pinpoint where the arguments of Taback and Schwartz break down in
the lamplighter case and prove the main results, Theorem 1.1, 1.2, and 1.3.

Acknowledgment: The first author was partially supported by NSF CAREER
1552234. The second author is partially supported by the NSF grant
DMS-2203237. The authors would like to thank the organizers of Women in
Groups, Geometry and Dynamics funded by NSF CAREER 1651963 for giving them the
opportunity to come together to work on this project. The first author would
also like to thank Jennifer Taback and Kevin Whyte for many useful
conversations.

##  2\. Background

###  2.1. Lamplighter groups, lattices in SOL, and Baumslag-Solitar groups

We are interested in three different types of groups, all of which are
commonly studied in geometric group theory. Below we establish notation and
define each group as a semi-direct product of some base group $B$ with
${\mathbb{Z}}$.

####  2.1.1. Lamplighter groups

Let ${\mathbb{Z}}_{n}$ be a finite cyclic group with $n$ elements. You can
apply this construction for any finite group, but for simpilicity we will
focus on ${\mathbb{Z}}_{n}$. Suppose that we have a bi-infinite sequence of
lamps, one for each integer on the real line. Each lamp has $n$ settings
corresponding to the $n$ elements in ${\mathbb{Z}}_{n}$. Each element $g$ of
the lamplighter group, $L_{n}$, is written $g=((x_{i}),k)$, where
$(x_{i})\in\bigoplus{\mathbb{Z}}_{n}$ represents the locations of finitely
many lamps on non-zero settings, and $k\in{\mathbb{Z}}$ represents the
location of the lamplighter.

To compose two elements, we think of the element $g=((x_{i}),k)$ as a set of
instructions, telling the lamplighter to start at position 0, go flip the
switch to setting $x_{i}$ at each position $i$, where $x_{i}\neq 0$ and then
end at position $k$. Composition in the group can be thought of as
concatenating two sets of such instructions, such that lamplighter begins the
second set of instructions at position $k$. So if $g=((x_{i}),k)$ and
$h=((y_{i}),\ell)$ then

$gh=((x_{i}+y_{i-k}),k+\ell)$

The group $L_{n}$ is then a semi-direct product
$\bigoplus{\mathbb{Z}}_{n}\rtimes{\mathbb{Z}}$. We will use a presentation of
this group with generators $t,at$ where $t$ represents the instruction for the
lamplighter to move one step in the positive direction without changing the
lamp, each $x$ corresponds to a generator of ${\mathbb{Z}}_{n}$, so $at$
represents the instructions for the lamplighter to change the setting of the
current lamp accordingly and then move one step in the positive direction.
This gives the following presentation.

$L_{n}=\langle t,at\mid a^{n}=1,[t^{i}at^{-i},t^{j}at^{-j}]\quad\forall
i,j\in{\mathbb{Z}}\rangle$

Notice that this presentation has infinitely many relations, and in fact the
lamplighter group is not a finitely presented group. The Cayley graph given by
this presentation is the Diestel-Leader graph discussed below.

####  2.1.2. Lattices in SOL

Consider the semi-direct product ${\mathbb{R}}^{2}\rtimes{\mathbb{R}}$ where
${\mathbb{R}}$ acts on ${\mathbb{R}}^{2}$ by $(x,y)\to(e^{t}x,e^{-t}y)$. This
is the three-dimensional solvable Lie group $SOL$. Let $A\in
SL(2,{\mathbb{Z}})$ be diagonalizable with no eigenvalues on the unit circle,
and $\Gamma_{A}=\mathbb{Z}^{2}\rtimes{\mathbb{Z}}$ where ${\mathbb{Z}}$ acts
on ${\mathbb{Z}}^{2}$ by powers of $A$. We can see that $\Gamma_{A}$ is a
lattice in $SOL$ with the embedding given by $(x,y,z)\mapsto(Q(x,y),z)$ where
$QAQ^{-1}$ is a diagonal matrix. The quotient of $SOL/\Gamma_{A}$ is compact
so that $\Gamma_{A}$ is a uniform lattice in $SOL$.

####  2.1.3. Solvable Baumslag-Solitar groups

Solvable Baumslag-Solitar groups are groups given by the following
presentation.

$BS(1,n)=\langle t,a\mid tat^{-1}=a^{n}\rangle.$

Alternatively, we can also view $B(1,n)$ as the group of affine maps on
${\mathbb{Z}}[\frac{1}{n}]$, with $a$ acting as $i\rightarrow i+1$ and $t$ as
$i\rightarrow\frac{1}{n}i$, which determines the short exact sequence

$1\rightarrow{\mathbb{Z}}[\tfrac{1}{n}]\rightarrow
BS(1,n)\rightarrow\mathbb{Z}\rightarrow 1.$

Thus, we can also identify
$B(1,n)=\mathbb{Z}[\frac{1}{n}]\rtimes{\mathbb{Z}}$.

###  2.2. Quasi-isometries

######  Definition 2.1 (Quasi-isometry).

For $K\geq 1,C\geq 0$ and $(X,d_{X}),(Y,d_{y})$ metric spaces we say that
$f:(X,d_{X})\to(Y,d_{Y})$ is a $(K,C)$ _quasi-isometry_ if for all
$x,x^{\prime}\in X$

  1. (1)

$-C+\frac{1}{K}d_{X}(x,x^{\prime})\leq d_{Y}(f(x),f(x^{\prime}))\leq
Kd_{X}(x,x^{\prime})+C$

  2. (2)

for all $y\in Y$ there is $x\in X$ such that $d_{Y}(f(x),y)\leq C$.

If only the first condition is satisfied we say that $f$ is a $(K,C)$ _quasi-
isometric embedding_ and if only the second condition is satisfied we say that
$f$ is _coarsely surjective_. We say that two spaces are _quasi-isometrically
equivalent_ if for some $K\geq 1$ and $C\geq 0$ there is a $(K,C)$ quasi-
isometry between them and in most cases we don’t specify the constants $K$ and
$C$. A _coarse inverse_ of a quasi-isometry $f:X\to Y$ is a quasi-isometry
$\bar{f}:Y\to X$ such that both $d_{sup}(f\circ\bar{f},Id_{Y})<\infty$ and
$d_{sup}(\bar{f}\circ f,Id_{X})<\infty$ where $d_{sup}$ is the sup metric and
$Id_{X},Id_{Y}$ are the identify maps on $X$ and $Y$ respectively. More
generally we can define an equivalence class on self-quasi-isometries of a
metric space $X$ where for $f,g:X\to X$ we set $f\sim g$ if
$d_{sup}(f,g)<\infty$. The set of these equivalence classes is a group under
the composition of representatives which we call the _quasi-isometry group_
$QI(X)$.

In the special case when $C=0$ then we say that $f$ is a _biLipschitz_ map and
if in addition $K=1$ then $f$ is an _isometry_. We can also define the group
of self-biLipschitz maps of a metric space $Bilip(X)$ or self-isometries
$Isom(X)$ but for these we do not need to look at equivalence classes and in
general these groups do not embed into $QI(X)$.

####  2.2.1. Finitely generated groups as metric spaces.

A finitely generated group can be treated as a metric space by fixing a finite
generating set $S$ and declaring $d_{S}(g,h)=\|g^{-1}h\|_{S}$ where
$\|x\|_{S}$ counts the minimum number of generators of $S$ or their inverses
needed to write the word $x$. This distance depends on $S$ but all such
metrics are _quasi-isometric_ in that if $S$ and $S^{\prime}$ are both finite
generating sets for $G$ then the identity map
$Id:(G,d_{S})\to(G,d_{S^{\prime}})$ is a quasi-isometry.

####  2.2.2. Pattern preserving quasi-isometries

As outlined in the introduction if a quasi-isometry $\Psi:G\to G$ coarsely
permutes left cosets of a certain subgroup $H\leq G$ then we can call it
pattern preserving. In this paper we focus on the case of
$G=B\rtimes{\mathbb{Z}}$ and $H={\mathbb{Z}}$.

######  Definition 2.2.

If $G=B\rtimes{\mathbb{Z}}$ is a semi-direct product, we say that a quasi-
isometry $\Psi:G\to G$ is a pattern preserving quasi-isometry if it coarsely
permutes the left cosets of $H={\mathbb{Z}}$.

###  2.3. Model spaces and Horocyclic products.

######  Definition 2.3.

We say that a proper geodesic metric space $X$ is a _model space_ for a
finitely generated group $G$ if $G$ acts on $X$ properly discontinuously and
cocompactly by isometries.

By the Fundamental Lemma of geometric group theory if $X$ is a model space for
$G$ then $G$ with any word metric is quasi-isometric to $X$. Any Cayley graph
of $G$ with respect to a finite generating set is a model space for $G$ and if
$G$ is a cocompact lattice in a Lie group then this Lie group with any left
invariant Riemannian metric is also a model space for $G$.

For the three types of groups we are interested in (lamplighter groups,
solvable Baumslag-Solitar groups, and $\Gamma_{A}$ lattices in SOL), we can
construct model spaces as horocylic products.

######  Definition 2.4.

Given two metric spaces $X_{1},X_{2}$, each equipped with a surjective
continuous function $h_{i}:X_{i}\to{\mathbb{R}}$, called the height functions,
the horocyclic product is a subset of the direct product $X_{1}\times X_{2}$
given by

$X_{1}\times_{h}X_{2}:=\\{(x,y)\in X_{1}\times X_{2}|h_{1}(x)+h_{2}(y)=0\\}.$

In this paper we will consider at most the case when $X_{1},X_{2}$ are
$CAT(-1)$ spaces and $h_{i}$ are Busemann functions and most often the case
when the $X_{i}$ are either regular trees or a hyperbolic plane. We often draw
the two spaces $X_{1}$ and $X_{2}$ so that the height function for $X_{1}$
corresponds to moving upward on the page and the height function for $X_{2}$
corresponds to moving downward on the page. Using this visualization, points
in the horocylic product correspond to pairs of points at the same height on
the page (see Figure 1. We can then define a height function
$h:X_{1}\times_{h}X_{2}\to{\mathbb{R}}$ given by $h_{1}=-h_{2}$.

Figure 1. _Top:_ The Diestel-Leader graph as a horocyclic product two trees.
_Middle:_ The model space for $BS(1,2)$ as a horocyclic product of the
hyperbolic plane with a tree.
_Bottom:_ The model space of a lattice in SOL as a horocyclic product of two
hyperbolic planes. Note a part of this figure is taken from a figure in [FM98]

We let $T_{n+1}$ be the $n+1$ valent oriented tree with $n$ incoming edges and
one outgoing edge. This orientation allows us to define a height by fixing a
basepoint $v_{0}\in T_{n+1}$ and letting $h:T_{n+1}\to{\mathbb{R}}$ be the
unique map with $h(v_{0})=0$ that maps vertices to integers, edges to
intervals and preserves the orientation in that if $(v,w)$ is an edge oriented
from $v$ to $w$ then $h(w)=h(v)+1$.

We use the log model of the hyperbolic plane $\mathbb{H}^{2}$ so that if
$(x,y)$ represent the usual upper half plane coordinates with metric
$ds^{2}=(dx^{2}+dy^{2})/y^{2}$ we use instead $(x,t)$ where $t=\ln{y}$ in
which case the metric is given by $ds^{2}=e^{-2t}dx^{2}+dt^{2}$. This allows
us to define the _height_ of point $h:\mathbb{H}^{2}\to{\mathbb{R}}$ as
$h(x,t)=t$.

With this notation we can create model spaces for our three families of groups
as follows.

  * •

lamplighter groups $L_{n}$: The Cayley graph given by the above presentation,
also called the Diestel-Leader graph, is the horocyclic product

$DL(n,n)=T_{n+1}\times_{h}T_{n+1}.$

In this graph, two vertices $(x_{1},y_{1}),\ (x_{2},y_{2})\in DL(n,n)$ are
joined by an edge whenever both $x_{1}$, $x_{2}$ and $y_{1}$, $y_{2}$ are
joined by an edge in their respective trees.

  * •

lattices in SOL, $\Gamma_{A}$: A model space can be constructed as a lattice
inside the horocyclic product

${\mathbb{H}}^{2}\times_{h}{\mathbb{H}}^{2}.$

This space is isometric to $SOL={\mathbb{R}}^{2}\rtimes{\mathbb{R}}$ with the
standard Riemannian metric

$ds^{2}=e^{-2t}dx^{2}+e^{2t}dy^{2}+dt^{2}.$

  * •

solvable Baumslag-Solitar groups, BS($1,n$): The model space constructed by
Farb and Mosher in [FM98] is a horocylic product.

$X_{n}={\mathbb{H}}^{2}\times_{h}T_{n+1}.$

The one caveat here is that we have to either rescale the edge lengths in the
tree to be $\ln{n}$ or rescale the curvature of the hyperbolic plane. See
[FM98].

####  2.3.1. Distance in horocyclic products.

In the Diestel-Leader graph $DL(n,n)$, distance is given by

$d((x_{1},x_{2}),(y_{1},y_{2}))=d_{T_{n+1}}(x_{1},y_{1})+d_{T_{n+1}}(x_{2},y_{2})-|h(x_{1})-h(y_{1})|$

where $d_{T_{n+1}}$ is distance in the tree $T_{n+1}$ and $h$ is the height
function on the horocyclic product [Ber01]. On more general horocyclic
products a similar formula coarsely holds, see Theorem A in [Fer23]. In the
case of $SOL$ any two points at the same height $t$ can be connected by a path
that lies at height $t$ and so we can compute induced distances along height
level sets. These are given by $d_{t}((p,t),(q,t))=e^{t}|\Delta
x|+e^{-t}|\Delta y|$ see for example [EFW12].

###  2.4. Vertical geodesics and boundaries

######  Definition 2.5.

Given a space $X$ with a height function $h:X\to{\mathbb{R}}$ we can define a
_vertical geodesic_ as a geodesic in $X$ that maps isometrically onto
${\mathbb{R}}$ via the height map $h$.

Let $\mathcal{L}$ denote the space of all vertical geodesics in the space $X$.
Given $\gamma,\gamma^{\prime}\in\mathcal{L}$, there are two natural
equivalence relations $\sim_{u}$ and $\sim_{\ell}$ on $\mathcal{L}$:

$\gamma\sim_{u}\gamma^{\prime}\text{ if and only if
}\lim_{t\to\infty}d(\gamma(t),\gamma^{\prime}(t))<\infty$

and

$\gamma\sim_{\ell}\gamma^{\prime}\text{ if and only if
}\lim_{t\to-\infty}d(\gamma(t),\gamma^{\prime}(t))<\infty.$

We then define the _lower boundary_ of $X$ to be

$\partial_{\ell}X:=\mathcal{L}/\sim_{\ell}$

and the _upper boundary_ to be

$\partial^{u}X:=\mathcal{L}/\sim_{u}.$

For example, on $\mathbb{H}^{2}$ with its standard height function we have
$\partial_{\ell}\mathbb{H}^{2}\simeq{\mathbb{R}}$ while
$\partial^{u}\mathbb{H}^{2}$ is a single point $\\{\infty\\}$. In general, for
a Gromov hyperbolic space $X$ with height $h$ given by a Busemann function,
you can identify $\partial_{\ell}X=\partial_{\infty}X\setminus\\{\infty\\}$
where $\partial_{\infty}X$ is the usual visual boundary of $X$ and
$\partial^{u}X=\\{\infty\\}$.

In the case of $T_{n+1}$ we have a nice identification of
$\partial_{\ell}T_{n+1}$ with power series $\sum_{i=L}^{\infty}a_{i}n^{i}$
where $L\in{\mathbb{Z}}$ and $a_{i}\in{\mathbb{Z}}_{n}$, that is, with
elements of ${\mathbb{Q}}_{n}$. To see this identification, first for any
vertex $v\in T_{n+1}$ label all $n$ edges coming to $v$ with labels
$0,1,\ldots,n-1$. Then a vertical geodesic determines a bi-infinite sequence
$\\{a_{i}\\}$. Since any vertical geodesic must eventually join the geodesic
labeled with all $0$’s we can identify this sequence with
$\sum_{i=L}^{\infty}a_{i}n^{i}$ for some $L\in{\mathbb{Z}}$.

######  Lemma 2.6.

If $X_{1}\times_{h}X_{2}$ is a horocyclic product of Gromov hyperbolic spaces
$X_{1},X_{2}$ with height functions given by Busemann functions then we have
the following boundary identifications

$\partial_{\ell}(X_{1}\times_{h}X_{2})=\partial_{\ell}X_{1}\textrm{ and
}\partial^{u}(X_{1}\times_{h}X_{2})=\partial_{\ell}X_{2}.$

###### Proof.

This follows easily from the fact that the height function in the second
factor is reversed. ∎

######  Corollary 2.7.

The lower and upper boundaries of $L_{n},BS(1,n)$ and $\Gamma_{A}$ are as
follows:

  * •

For lamplighter groups $L_{n}$:

$\partial_{\ell}(DL(n,n))=\partial_{\ell}T_{n+1}\simeq{\mathbb{Q}}_{n},\
\partial^{u}(DL(n,n))=\partial_{\ell}T_{n+1}\simeq{\mathbb{Q}}_{n}$

  * •

For solvable Baumslag-Solitar groups $BS(1,n)$:

$\partial_{\ell}(X_{n})=\partial_{\ell}\mathbb{H}^{2}\simeq{\mathbb{R}},\
\partial^{u}(X_{n})=\partial_{\ell}T_{n+1}\simeq{\mathbb{Q}}_{n}$

  * •

For $\Gamma_{A}$ lattices in $SOL$:

$\partial_{\ell}(SOL)=\partial_{\ell}\mathbb{H}^{2}\simeq{\mathbb{R}},\
\partial^{u}(SOL)=\partial_{\ell}\mathbb{H}^{2}\simeq{\mathbb{R}}$

We also need to endow these boundaries with a metric. When $X$ is $CAT(-1)$,
such as in our case, there is a well studied family of metrics on
$\partial_{\ell}X$: the _parabolic visual boundary metrics_. In the case of
$\mathbb{H}^{2}$ and $T_{n+1}$, the usual metrics on ${\mathbb{R}}$ and
${\mathbb{Q}}_{n}$, defined by the Euclidean and $n$-adic norms respectively,
are examples of parabolic visual metrics so we will just use those.
Geometrically we should think of these metrics as $d(x,x^{\prime})=a^{t_{0}}$
where $t_{0}$ is the height at which the vertical geodesics defined by
$x,x^{\prime}$ come close and $a>0$ is a parameter which in the case of
$T_{n+1}$ can be chosen to be $n$ and in the case of the curvature $-1$
hyperbolic plane $a=e$. The notion of coming close is also a little different
in the two settings. In the $T_{n+1}$ case coming close will mean intersecting
for the first time while in the hyperbolic plane we mean coming within some
fixed bounded distance of each other. This last notion also works for general
$CAT(-1)$ spaces so anytime we use these boundary metrics we assume that the
$X_{i}$ are $CAT(-1)$ spaces.

###  2.5. $(\epsilon,M)$-quadrilaterals

Our next step is to understand how vertical geodesics in our horocyclic
product come together. We will be able to do this in terms of the boundary
metrics which we denote $d_{\ell}$ and $d_{u}$ for the lower and upper
boundaries respectively.

######  Definition 2.8.

Given vertical geodesics $p,q$ in a horocyclic product $X_{1}\times_{h}X_{2}$,
we define

$\delta(p,q):=d_{\ell}(p,q)d_{u}(p,q).$

In the lamplighter case $\log_{n}(\delta(p,q))$ has a very nice geometric
interpretation, specifically it measures the distance between the vertical
geodesics $p$ and $q$ inside the Diestel-Leader graph (see Figure 2 for an
explanation).

Figure 2. The distance between geodesics $p$ and $q$ is given by the length of
the segment from $B$ to $C$ (or equivalently $A$ to $D$). If $h$ denotes the
height function on $DL(n,n)$. Then $d_{\ell}(p,q)=n^{h(B)}$ and
$d_{u}(p,q)=n^{-h(C)}$ so that $\delta(p,q)=n^{h(B)-h(C)}$ and therefore
$\log_{n}(\delta(p,q))$ is exactly the distance between the geodesics $p$ and
$q$.

For more general horocyclic products a coarse version of this interpretation
can be deduced using Theorem B from [Fer23]. In particular the picture in
Figure 2 needs to be modified slightly so that the points $A,B,C,D$ are now
replaced by balls of a fixed radius. Theorem B ensures that the shortest path
between the vertical geodesics $p$ and $q$ stays bounded distance from
vertical geodesics that either start in $A$ and end in $D$ or start in $B$ and
end in $C$.

The height interval at which this geodesic between $p$ and $q$ lies is also
important. It coincides coarsely with the interval defined by
$\log{d_{\ell}(p,q)}$ and $-\log{d_{u}(p,q)}$ where the logarithm base is
determined by the base chosen for the boundary metrics.

######  Definition 2.9.

Suppose $X=X_{1}\times_{h}X_{2}$ is a horocyclic product of two $CAT(-1)$
spaces $X_{i}$ and $d_{\ell},d_{u}$ two parabolic visual metrics. For vertical
geodesics $p,q$ suppose $d_{\ell}(p,q)=a^{t_{\ell}}$ and
$d_{u}(p,q)=a^{-t_{u}}$ then we call any $t_{0}\in[t_{u},t_{\ell}]$ a coarse
height at which $p$ and $q$ come close.

Since we will only use this notion coarsely and when $\delta(p,q)$ is small it
will not matter which coarse height in the interval we chose. In particular it
suffices to look at one endpoint or the other.

In our three families of groups, $\delta$ also has nice algebraic
interpretations:

  * •

lamplighter groups $L_{n}$: For $p=(x_{i})$,
$q=(y_{i})\in\bigoplus{\mathbb{Z}}_{n}$, define $d_{\ell}(p,q)=n^{-l_{+}}$
where $l_{+}$ is the maximal integer such that $x_{i}=y_{i}$ for all
$i<l_{+}$. In other words, $l_{+}$ is the smallest integer such that
$x_{i}\neq y_{i}$. Similarily $d_{u}(p,q)=n^{l_{-}}$ where $l_{-}$ is the
minimal integer such that $x_{i}=y_{i}$ for all $i>l_{-}$. Outside this
interval of length $l_{-}-l_{+}$, $x_{i}=y_{i}$, i.e. this interval is the
support of $p-q$ and so we denote it by $supp(p-q)$ and its length by
$|supp(p-q)|$. Then $\delta(p,q):=n^{|supp(p-q)|}$.

  * •

lattices in SOL: If $p=(x_{1},x_{2}),\ q=(y_{1},y_{2})\in
B\subset{\mathbb{R}}^{2}$ then we have
$\delta(p,q):=|x_{1}-y_{1}|\cdot|x_{2}-y_{2}|=|\Delta x||\Delta y|$. Because
of how $B$ embeds in ${\mathbb{R}}^{2}$ we can have $|\Delta x|,|\Delta y|$
arbitrarily small but never zero unless $p=q$.

  * •

solvable Baumslag-Solitar groups: Here we have
$\delta(p,q):=|p-q|_{\mathbb{R}}|p-q|_{\mathbb{Q}_{n}}.$ Note that if
$p,q\in\mathbb{Z}[1/n]$ and if we write $p-q=rn^{k}$ where $r$ contains no
powers of $n$ then $\delta(p,q)=|r|n^{k}\cdot n^{-k}=|r|$.

######  Definition 2.10.

An _$(\epsilon,M)$ -quadrilateral_ is a collection of four distinct points
$p_{i}\in B$ for $i=1,2,3,4$ with $\delta(p_{i},p_{i+1})\leq\epsilon$ and
$\delta(p_{i},p_{i+2})\geq M$ where the indices are taken mod $4$. We
represent these points in a matrix

$P=\begin{bmatrix}p_{1}&p_{2}\\\ p_{4}&p_{3}\end{bmatrix}.$

In the case that $p_{1}+p_{3}=p_{2}+p_{4}$ we call $P$ an
$(\epsilon,M)$-parallelogram instead.

Figure 3. A schematic of an $(\epsilon,M)$ quadrilateral in the model space
for $BS(1,2)$, given by vertical geodesics $0,v,w,$ and $v+w$. Every dashed
grey segment has length 1. The coarse height $t_{v}$ is the height where the
geodesics 0 and $v$ are closest together. At $t_{v}$, we can see that the
distance between $0$ and $v$ is small and the distance between $w$ and $v+w$
is small but the distance between $0$ and $v+w$ is large. The distance between
$0$ and $w$ is realized at height $t_{w}$, which is also a small distance.

We will always choose $M\gg\epsilon$. In this case we have a nice geometric
interpretation: an $(\epsilon,M)$-quadrilateral is one for which the vertical
geodesics represented by $p_{i}$ come coarsely close cyclically but are far
across the diagonal. The only way this is possible is that they are arranged
as in Figure 3, that is the height at which the pairs $p_{1},p_{2}$ and
$p_{3},p_{4}$ come close is very far from the height at which $p_{2},p_{3}$
and $p_{4},p_{1}$ come together. Note that rotating the entries of an
$(\epsilon,M)$-parallelogram results in another $(\epsilon,M)$ parallelogram.

######  Lemma 2.11.

If $X=X_{1}\times_{h}X_{2}$ is a horocyclic product of $CAT(-1)$ spaces and
$P$ an $(\epsilon,M)$ quadrilateral for $M\gg\epsilon$ with entries $p_{i}$
for $i=1,2,3,4$ then if $t_{i}$ is a coarse height along which $p_{i}$ and
$p_{i+1}$ come close then up to relabeling $t_{1}\gg t_{2}$, $t_{3}\gg t_{4}$
while $t_{2}\ll t_{3}$, $t_{4}\ll t_{1}$ and $t_{i},t_{i+2}$ are close. In
other words the geodesics form a picture very similar to Figure 3.

###### Proof.

Similar lemmas have appeared in other places such as [EFW12] but we give a
sketch here because our definition is slightly different. Since we are working
coarsely, we write $\gtrsim$ instead of $\geq$ to mean there is some bounded
error either additive or multiplicative. First we note that $t_{i}$ and
$t_{i+1}$ must lie at sufficiently different heights if
$\delta(p_{i},p_{i+2})\geq M$. This is because there is always a path of
length at most $|t_{i}-t_{i+1}|+4\log\epsilon$ joining $p_{i}$ and $p_{i+2}$.
Now without loss of generality suppose $t_{1}\gg t_{2}$. For contradiction
suppose $t_{2}\gg t_{3}$. Note by the triangle inequality, we have

$d_{\ell}(p_{1},p_{4})\geq
d_{\ell}(p_{1},p_{2})-d_{\ell}(p_{2},p_{3})-d_{\ell}(p_{3},p_{4})\gtrsim
a^{t_{1}}-a^{t_{2}}-a^{t_{3}}\gtrsim a^{t_{1}}$

while

$d_{u}(p_{1},p_{4})\gtrsim a^{-t_{3}}-a^{-t_{2}}-a^{-t_{1}}\gtrsim
a^{-t_{3}}.$

But then $\delta(p_{1},p_{4})\gtrsim a^{t_{1}-t_{3}}\gg 0$ which contradicts
that $\delta(p_{1},p_{4})\leq\epsilon$. ∎

Next we verify this geometric picture for $(\epsilon,M)$-parallelograms in our
three cases.

######  Proposition 2.12.

Let $G=B\rtimes{\mathbb{R}}$ be either a lamplighter group $L_{n}$, a solvable
Baumslag-Solitar group $BS(1,n)$ or $\Gamma_{A}$ a lattice in $SOL$. Let

$P=\begin{bmatrix}0&w\\\ v&w+v\end{bmatrix}.$

be an $(\epsilon,M)$ parallelogram in $B$ with $\epsilon\ll M$. Let $t_{w}$ be
the coarse height at which the vertical geodesics defined by $0$ and $w$ come
close and similarly let $t_{v}$ be the coarse height at which the vertical
geodesics defined by $0$ and $v$ come close. Then for $M\gg\epsilon$ we have
that $||t_{w}|-|t_{v}||\gg 0$ and hence geometrically our parallelogram has
the shape denoted in Figure 3.

###### Proof.

As defined above $t_{w}$ can be any point in the interval defined by
$\log{d_{\ell}(0,w)}$ and $-\log{d_{u}(0,w)}$ and similarly for $t_{v}$. Since
we are only interested in the coarse picture the precise values of $t_{w}$ and
$t_{v}$ are unnecessary. Again in the lamplighter group the picture is
clearest.

  * •

lamplighter groups $L_{n}:$ Consider $w=(x_{i})$,$v=(y_{i})$ in
$\bigoplus{\mathbb{Z}}_{n}$ with $supp(w)=[i_{w},j_{w}]$ and
$supp(v)=[i_{v},j_{v}]$. These intervals are the heights at which the two
geodesics come close to the geodesic defined by $0$. The condition that
$\delta(0,w),\delta(0,v)$ are small here means that both $|i_{w}-j_{w}|$ and
$|i_{v}-j_{v}|$ are small, while $\delta(w,v)$ large means that the size of
the support of $w-v$ is large. More specifically for $\epsilon\ll M$, we can
assume that up to exchanging $w$ and $v$ we have that
$supp(v-w)=[i_{w},j_{v}]$. This means that $|i_{w}-j_{v}|$ is large so that
the vertical geodesics defined by $w$ and $v$ come close at very different
heights. By translation this means that $w,w+v$ and $v,w+v$ also come together
at very different heights and form the geometric picture from Figure 3.

  * •

solvable Baumslag-Solitar groups:  Let $w,v\in{\mathbb{Z}}[\frac{1}{n}]$ where
$w=r_{w}n^{k_{w}}$ and $v=r_{v}n^{k_{v}}$. Then assume that
$\delta(0,w)=|r_{w}|$ and $\delta(0,v)=|r_{v}|$ are small. By relabeling we
may assume $k_{v}\geq k_{w}$. Then $w-v=(r_{w}-r_{v}n^{k_{v}-k_{w}})n^{k_{w}}$
and $\delta(w,v)=(r_{w}-r_{v}n^{k_{v}-k_{w}})$. If $\delta(w,v)$ is large then
$k_{v}\gg k_{w}$ since $r_{w}$ and $r_{v}$ are small. Now $k_{w},k_{v}$ are
heights at which the vertical geodesics defined by $w,v$ come close to the
geodesic defined by $0$. In fact they are exactly the heights at which these
geodesics branch in the tree direction from the vertical geodesic defined by
zero. Again we have a quadrilateral like Figure 3.

  * •

lattices in SOL:  If $w=(x_{w},y_{w}),v=(x_{v},y_{v})\in
B\subset{\mathbb{R}}^{2}$ then $\delta(0,w)=|x_{w}||y_{w}|$ and
$\delta(0,v)=|x_{v}||y_{v}|$ while $\delta(w,v)=|x_{w}-x_{v}||y_{w}-y_{v}|$.
Without loss of generality we can assume that $|x_{w}-x_{v}|$ is large. Then
the difference between $t_{w}=\log{|x_{w}|}$ and $t_{v}=\log{|x_{v}|}$ is
large so the two geodesics defined by $w$ and $v$ come close at very different
heights.

∎

###  2.6. Quasi-isometries of horocyclic products

As mentioned in the introduction quasi-isometries of our three families of
groups $B\rtimes{\mathbb{Z}}$ are well understood. Up to composition with an
isometry and up to bounded distance they fix level sets of ${\mathbb{Z}}$.
This was first proved in the $BS(1,n)$ case by [FM98] and in the other two
cases by Eskin-Fisher-Whyte [EFW12, EFW13]. They also show that such maps must
also coarsely preserve vertical geodesics and induce biLipschitz maps of the
lower and upper boundaries. The biLipschitz constants of these boundary maps
only depend on the quasi-isometry constants in the following sense: There
exists $N,K$ such that

$N/Kd_{\ell}(p,q)\leq d_{\ell}(\psi(p),\psi(q))\leq NKd_{\ell}(p,q)$

and

$1/(NK)d_{u}(p,q)\leq d_{u}(\psi(p),\psi(q))\leq K/Nd_{u}(p,q)$

where $K$ only depends on the quasi-isometry constants. By choosing an
appropriate isometry to compose with we can also assume that $N=1$. In
addition any two biLipschitz maps of the two boundaries define a quasi-
isometry of our group. This allows us to identify the quasi-isometry groups of
our three groups as

  * •

$QI(\Gamma_{A})=(Bilip({\mathbb{R}})\times
Bilip({\mathbb{R}})\rtimes){\mathbb{Z}}_{2}$

  * •

$QI(BS(1,n))=Bilip({\mathbb{R}})\times Bilip({\mathbb{Q}}_{n})$

  * •

$QI(DL(n,n))=(Bilip({\mathbb{Q}}_{n})\times
Bilip({\mathbb{Q}}_{n}))\rtimes{\mathbb{Z}}_{2}$

where $Bilip({\mathbb{R}})$ and $Bilip({\mathbb{Q}}_{n})$ denote the groups of
biLipschitz self maps of ${\mathbb{R}}$ and ${\mathbb{Q}}_{n}$ respectively.

The same is true for many other examples of horocyclic products of hyperbolic
spaces with Busemann height functions $X_{1}\times_{h}X_{2}$. Any time we can
show that height level sets are coarsely preserved we can conclude that

$QI(X_{1}\times_{h}X_{2})\simeq Bilip(\partial_{\ell}X_{1})\times
Bilip(\partial_{\ell}X_{2})$

with an additional ${\mathbb{Z}}_{2}$ factor in case $X_{1}=X_{2}$. For the
latest work on this see [Fer23].

The following lemmas are immediate from the definition of $\delta$ and
$(\epsilon,M)$-parallelograms.

######  Lemma 2.13.

If $\Psi$ is a coarsely height preserving quasi-isometry of
$X_{1}\times_{h}X_{2}$ inducing a biLipschitz map $\psi$ on the boundaries,
and $p,q$ are vertical geodesics then

$1/K^{2}\delta(p,q)\leq\delta(\psi(p),\psi(q))\leq K^{2}\delta(p,q),$

where $K$ is the uniform biLipschitz boundary constant and depends only on the
quasi-isometry constants of $\Psi$.

######  Lemma 2.14.

If $P$ is an $(\epsilon,M)$-quadrilateral in $X_{1}\times_{h}X_{2}$ and $\Psi$
is a coarsely height preserving quasi-isometry inducing a biLipschitz map
$\psi$ on the boundary, then $\psi(P)$ is an
$(\epsilon^{\prime},M^{\prime})$-quadrilateral where $\epsilon^{\prime}\leq
K^{2}\epsilon$ and $M^{\prime}\geq M/K^{2}$ where $K$ depends only on the
quasi-isometry constants of $\Psi$.

##  3\. Comparative Study of Results in Baumlaug Solitar groups and SOL

In this section, we review the rigidity results of pattern-preserving quasi-
isometries for lattices in SOL and solvable Baumslag-Solitar groups. We
present the proofs of these rigidity results in [Tab00, Sch96] from the
geometric perspective in terms of the horocyclic product as a comparative
study to the lamplighter group.

###  3.1. Pattern rigidity for lattices in SOL

Recall that a matrix $A\in SL(2,{\mathbb{Z}})$ which is diagonalizable with no
eigenvalues on the unit circle defines a lattice
$\Gamma_{A}=\mathbb{Z}^{2}\rtimes{\mathbb{Z}}$ in
SOL$={\mathbb{R}}^{2}\rtimes{\mathbb{R}}$. We equip $SOL$ with the standard
Riemannian metric $ds^{2}=e^{-2t}dx^{2}+e^{2t}dy^{2}+dt^{2}$. Recall that the
lattice embeds in $SOL$ using the diagonalizing matrix and hence identifies
the two eigenspaces of $A$ with the two coordinate spaces $x$ and $y$. The
cosets of the vertical subgroup in $\Gamma_{A}$ are identified by their
${\mathbb{R}}^{2}$ coordinates.

In this section, we recall Schwartz’s proof [Sch96] in the horocyclic product
model of lattices in SOL and prove that if a quasi-isometry preserves cosets
of the vertical subgroup $\langle(0,0,1)\rangle<\Gamma_{Z}$ then the map is
affine. Careful readers may notice that the notation here is different from
those in [Sch96]. We focus more on the geometry of the horocyclic product and
use $(\epsilon,M)$-quadrilateral to do the argument, which is coarsely the
same as the quadrilaterals used in Schwartz’s proof.

######  Proposition 3.1.

[Sch96] Let $\Psi$ be a quasi-isometry that coarsely permutes the left cosets
of the vertical subgroup of $\Gamma_{A}$. Then for any $\epsilon>0$ there is
an $M\gg\epsilon$ that depends on $\epsilon$ and the quasi-isometry constants
of $\Psi$ such that if $P$ is an $(\epsilon,M)$-parallelogram the $\Psi(P)$ is
also a parallelogram.

###### Proof.

For simplicity, assume that

$\Psi(P)=\begin{bmatrix}0&b\\\ c&d\end{bmatrix}.$

This is always possible by post composing by an affine map. It suffices to
prove that $d=b+c$.

\begin{overpic}[abs,unit=1mm,scale={.2}]{hyperbolas.png} \put(31.0,23.0){$A$}
\put(32.0,50.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$B$}
\put(62.0,22.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$C$}
\put(56.0,47.0){$d=b+c$} \par\end{overpic} Figure 4. The shaded region, $A$,
bounded by the black hyperbolas centered around the origin consists of points
$p$ with $\delta(0,p)<\epsilon^{\prime}$. While regions $B$(red and dashed)
and $C$ (blue and cross-hatched) are those points $p$ with
$\delta(b,p)<\epsilon^{\prime}$ and $\delta(c,p)<\epsilon^{\prime}$
respectively. If $M$ is large enough then the intersection of $B$ and $C$ has
two regions, and each of these regions contains only one lattice point.

Recall by Lemma 2.14 that an $(\epsilon,M)$-quadrilateral gets mapped to an
$(\epsilon^{\prime},M^{\prime})$-quadrilateral where $\epsilon^{\prime}\leq
K^{2}\epsilon$ and $M^{\prime}\geq M/K^{2}$.

Observe that both $p=d$ and $p=b+c$ satisfy the following two inequalities:

$\delta(p,b)<\epsilon^{\prime},\quad\delta(p,c)<\epsilon^{\prime}$

These inequalities define two regions of the plane $B$,$C$ bounded by
hyperbolas (see Figure 4). The solution of these two inequalities lies in the
intersection of the regions. For fixed $\epsilon^{\prime}$ we can increase
$M^{\prime}$ so that if $\delta(b,c)>M^{\prime}$ then the intersection of two
regions consists of two disjoint regions. One region contains $0$ and the
other contains both $d$ and $b+c$. By increasing $M$ and hence $M^{\prime}$ we
can shrink these regions of intersection and ensure that both regions only
contain a single lattice point, in other words that $d=b+c$. Hence, for every
$\epsilon$ there exists a sufficiently large $M$ such that when $P$ is an
$(\epsilon,M)$-parallelogram then $\Psi(P)$ is an
$(\epsilon^{\prime},M^{\prime})$-quadrilateral and hence also a parallelogram.
∎

In order to show that all parallelograms are mapped to parallelograms,
Schwartz shows that you can “generate” all parallelograms with ‘large’ ones.
In our notation, this means that for any $(\epsilon,M)$, we can find a
generating set $\Sigma\subset B$ so that for any $v,w\in\Sigma$ we have that

$\begin{bmatrix}0&v\\\ w&v+w\end{bmatrix}$

is an $(\epsilon,M)$-parallelogram.

Then for any parallelogram

$P=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}$

first write $c-a=d-b=v_{1}+\cdots v_{k}$ where $v_{i}\in\Sigma$ and define

$P_{j}=\begin{bmatrix}a+v_{1}+\cdots+v_{j-1}&b+v_{1}+\cdots+v_{j-1}\\\
a+v_{1}+\cdots+v_{j-1}+v_{j}&b+v_{1}+\cdots+v_{j-1}+v_{j}\end{bmatrix}.$

If $\Psi(P_{j})$ is a parallelogram for all $j=1,\ldots k$ then

$\psi(a)+\psi(b+v_{1})=\psi(b)+\psi(a+v_{1})$
$\psi(a+v_{1})+\psi(b+v_{1}+v_{2})=\psi(b+v_{1})+\psi(a+v_{1}+v_{2})$ $\vdots$
$\psi(a+v_{1}+\cdots v_{k-1})+\psi(b+d-b)=\psi(b+v_{1}+\cdots
v_{k-1})+\psi(a+c-a).$

By adding all equations we get $\psi(a)+\psi(d)=\psi(b)+\psi(c)$ and so
$\Psi(P)$ is also a parallelogram. To show that $\Psi(P_{j})$ is a
parallelogram, rotate the entries by $\pi/4$ and repeat the above argument for
each $P_{j}$. For example $P_{1}$ becomes

$P^{\prime}_{1}=\begin{bmatrix}a+v_{1}&a\\\ b+v_{1}&b\\\
\end{bmatrix}=\begin{bmatrix}b+(a-b)+v_{1}&b+(a-b)\\\ b+v_{1}&b\\\
\end{bmatrix}$

and if we write $a+v_{1}-(b+v_{1})=a-b=v_{11}+\cdots+v_{1k_{1}}$ as a sum of
generators in $\Sigma$ then

$P_{11}^{\prime}=\begin{bmatrix}b+v_{1}&b\\\
b+v_{1}+v_{11}&b+v_{11}\end{bmatrix},P_{12}^{\prime}=\begin{bmatrix}b+v_{1}+v_{11}&b+v_{11}\\\
b+v_{1}+v_{11}+v_{12}&b+v_{11}+v_{12}\end{bmatrix},\cdots$

all of which are $(\epsilon,M)$-parallelograms and so the result follows.

###  3.2. Rigidity of solvable Baumslag-Solitar groups

Next we rewrite and simplify Taback’s proof from [Tab00] using
$(\epsilon,M)$-quadrilaterals.

######  Proposition 3.2.

[Tab00] Let $\Psi$ be a quasi-isometry that induces a biLipschitz map $\psi$
that coarsely permutes the cosets of the vertical subgroup of $BS(1,n)$. Then
for any $\epsilon>0$ there is an $M\gg\epsilon$ that depends on $\epsilon$ and
the biLipschitz constants of $\psi$ such that if $P$ is an
$(\epsilon,M)$-parallelogram in $B={\mathbb{Z}}[\frac{1}{n}]$ then $\psi(P)$
is also a parallelogram.

###### Proof.

Suppose that $P$ is an $(\epsilon,M)$-parallelogram in
$B={\mathbb{Z}}[\frac{1}{n}]$. Then by Lemma 2.14 we know that $\psi(P)$ is an
$(\epsilon^{\prime},M^{\prime})$-quadrilateral where $\epsilon^{\prime}\leq
K^{2}\epsilon$ and $M^{\prime}\geq M/K^{2}$. Let

$\psi(P)=\begin{bmatrix}p_{1}&p_{2}\\\ p_{4}&p_{3}\end{bmatrix}$

and for $i=1,2,3,4$ write $p_{i+1}-p_{i}=r_{i}n^{k_{i}}$ where $r_{i}$ is an
integer not divisible by $n$. Then we know that the
$\delta(p_{i},p_{i+1})=|r_{i}|\leq\epsilon^{\prime}$ but $|k_{i}-k_{i+1}|$ is
large for all $i$ (mod $4$) by Lemma 2.11. Without loss of generality we may
assume that $k_{1}>k_{2}$ and $k_{3}>k_{4}$. Using this information we note
that

$\displaystyle p_{3}-p_{1}$ $\displaystyle=$
$\displaystyle(p_{3}-p_{2})+(p_{2}-p_{1})$ $\displaystyle=$ $\displaystyle
r_{2}n^{k_{2}}+r_{1}n^{k_{1}}=n^{k_{2}}\left(r_{2}+r_{1}n^{k_{1}-k_{2}}\right)$

while

$\displaystyle p_{1}-p_{3}$ $\displaystyle=$
$\displaystyle(p_{1}-p_{4})+(p_{4}-p_{3})$ $\displaystyle=$ $\displaystyle
r_{4}n^{k_{4}}+r_{3}n^{k_{3}}=n^{k_{4}}\left(r_{4}+r_{3}n^{k_{3}-k_{4}}\right)$

Now since $k_{1}>k_{2}$ and $k_{3}>k_{4}$ we know that
$r_{2}+r_{1}n^{k_{1}-k_{2}}$ and $r_{4}+r_{3}n^{k_{3}-k_{4}}$ are integers not
divisible by $n$ and so comparing the above two expressions we must have
$k_{2}=k_{4}$ and $r_{2}+r_{1}n^{k_{1}-k_{2}}=-r_{4}-r_{3}n^{k_{3}-k_{4}}$. By
symmetry we also have that $k_{1}=k_{3}$. This allows us to conclude that
$r_{2}=-r_{4}$ and $r_{1}=-r_{3}$. and so

$p_{2}-p_{1}=r_{1}n^{k_{1}}=-r_{3}n^{k_{3}}=p_{3}-p_{4}$

so that $p_{1}+p_{3}=p_{2}+p_{4}$ as desired.

∎

To show that $\psi$ sends all parallelograms to parallelogram, a similar
argument to the one for SOL is used to show that any parallelogram can be
written as a sum of ‘large’ parallelograms. This is possible essentially
because we can write $n^{k+1}$ as a sum of $n$ copies $n^{k}$ so that a
vertical geodesic that branches from the $0$ geodesic at height $k+1$ can be
written as a sum of vertical geodesics that branch at height $k$. In the
lamplighter case this is not possible and so it is not always possible to
generate all parallelograms with $(\epsilon,M)$-parallelograms. For more
details see Section 4.

##  4\. Rigidity and Non-rigidity in Lamplighter groups

In this section, we discuss pattern preserving quasi-isometries in Lamplighter
groups. In Section 4.1, we first show that, up to swapping tree factors, a
pattern preserving isometry of $DL(2,2)$ is induced by an generalized affine
map of $B=\bigoplus{\mathbb{Z}}_{2}$. The analogous fact about pattern
preserving isometries of the standard model spaces for $BS(1,n)$ and lattices
in SOL is immediate as these model spaces are more rigid then $DL(2,2)$. In
Section 4.2 we show that pattern preserving quasi-isometries of $DL(2,2)$ are
“affine on the large scale” similar to Propositions 3.1 and 3.2 for lattices
in SOL and $BS(1,n)$. However, unlike in $BS(1,n)$ and lattices in SOL, this
large scale behavior cannot be extended to all parallelograms. In Section 4.3,
we construct a family of pattern preserving quasi-isometries and give a
counterexample to show that the pattern preserving condition no longer implies
the same rigidity that we saw in the SOL and Baumslag-Solitar cases.

###  4.1. Pattern Preserving Isometries in Lamplighter Groups

In this section, we study pattern preserving isometries of the Diestel-Leader
graph, focusing on the case $DL(2,2)$ which is the model space for the
lamplighter group $L_{2}$. All isometries of $DL(2,2)$ can be written as a
composition of the following three types isometries: an isometry that swaps
the tree factors (we call any such isometry an “orientation reversing
isometry”), an isometry that translates up (or down) in the first tree and
down (or up) by the same amount in the second tree and an isometry that fixes
height and orientation. Specifically we have

$Isom(DL(2,2))=((Isom({\mathbb{Q}}_{2})\times
Isom({\mathbb{Q}}_{2}))\rtimes{\mathbb{Z}})\rtimes{\mathbb{Z}}_{2}.$

Thinking of $DL(2,2)$ as a Cayley graph, the first type of isometry cannot be
induced by left multiplication by a group element while the second type can be
chosen to be induced by left multiplication by elements of the vertical
subgroup $\left<t\right>$. The last type contains both isometries that are
induced by left multiplication by $B=\bigoplus{\mathbb{Z}}_{n}$ and also many
others that are not induced by left multiplication.

We can now prove Theorem 1.1. Namely that if $\Psi:DL(2,2)\to DL(2,2)$ is an
orientation preserving pattern preserving isometry then there is some $g\in
L_{2}$ such that $\Psi$ is bounded distance from left multiplication by $g$.
We will give a geometric proof of this proposition here but the result also
follows from Proposition 4.2 whose proof relies on understanding $\delta$.

###### Proof.

_(Of Theorem 1.1)_ Recall that orientation preserving means that the two tree
factors are not interchanged. This assumption is necessary since no group
element can interchange these factors. Any orientation preserving isometry
that preserves the set of cosets can be composed with the action of a group
element to obtain an isometry that sends the identity $\langle t\rangle$ coset
to itself. Similarly we can compose such an isometry with a group element
acting by translating up or down to obtain an isometry that preserves height.
Thus it is sufficient to assume that $\Psi$ is an isometry that preserves
orientation and height, and that the induced map $\psi$ permutes the cosets,
and fixes the identity coset and to show that $\Psi$ must be the identity map.

Figure 5. A height one portion of $DL(2,2)$ on the right and its tree factors
on the left. Vertex labels illustrate the proof of Theorem Theorem 1.1. The
red (dotted) line represents the vertical geodesic corresponding to the
identity coset in the right-hand factor.

Now, suppose that $\Psi$ is not the identity, and consider the action of
$\Psi$ on the tree to the left, in the horocylic product. There exists some
point $a$ which is not fixed, but is the child of some fixed point $c$.
Because $\Psi$ acts as a height preserving isometry, this means that $a$ and
$b$ (the two children of $c$) are permuted (See Figure 5).

We are also assuming that the identity coset $\langle t\rangle$ is preserved
by $\psi$. The identity coset is represented by a vertical geodesic in
$DL(2,2)$. This geodesic appears as a pair of vertical geodesics one in each
tree. In the right-hand tree, there is some segment of this geodesic at the
same height as the edge from $c$ to $a$. Label the vertices of this segment
$d$ and $f$ and the other child of $f$ as $e$ (See fig. 5). Note that $f$ and
$d$ are fixed point-wise by $\Psi$ because the identity coset is fixed by
$\psi$ and $\Psi$ is height preserving. Hence, $e$ is also fixed because
$\Psi$ is an isometry.

We have now produced a quadrilateral $(c,d),(b,f),(c,e),(a,f)$ in $DL(2,2)$ of
height one, where $\Psi$ permutes $(a,f)$ and $(b,f)$, but leaves $(c,d)$ and
$(c,e)$ fixed. However, because cosets of $\langle t\rangle$ are vertical
geodesics that partition the vertices of $DL(2,2)$, there are exactly two
cosets, passing through any height one quadrilateral. Any pattern preserving
isometry must either fix both cosets or permute them. If both cosets are fixed
then all four vertices are fixed, but in our constructed quadrilateral
$(a,f)$, and $(b,f)$ are not fixed so this is not the case for our
quadrilateral. If the cosets are permuted, then opposite edges of the
quadrilateral are permuted. In particular this means that the edges
$(c,d)-(a,f)$ and $(c,e)-(b,f)$ are permuted and the vertices that $(c,d)$ and
$(c,e)$ are permuted. This again contradicts our construction of this
quadrilateral. Thus, under our assumptions about $\Psi$, such a quadrilateral
is impossible, providing a contradiction.

∎

This proposition shows that our pattern preserving assumption is a strong
hypothesis in the case where the quasi-isometry is an isometry. We now
consider quasi-isometries that are not isometries, and try to imitate the
arguments made in the case of lattices in $SOL$ and $BS(1,n)$. We will see
similar behavior carry over for $(\epsilon,M)$ parallelograms for
$M\gg\epsilon$, but in general quasi-isometries of $DL(2,2)$ are not affine,
in contrast to the behavior we saw for the other two groups.

###  4.2. Large Parallelograms

Recall that in the lamplighter case $log_{2}(\delta(p,q))$ has a very nice
interpretation. Namely if $p=(x_{i}),q=(y_{i})\in\bigoplus{\mathbb{Z}}_{2}$
and $x_{i}=y_{i}$ for all $i>\ell_{-}$ and all $i<\ell_{+}$. Then
$log_{2}(\delta(p,q))=\ell_{-}-\ell_{+}$, that is, it measures the length of
the smallest interval such that $x_{i}=y_{i}$ outside this interval. This
interval is denoted $supp(p-q)$.

######  Proposition 4.1.

Let $\Psi$ be a quasi-isometry and $\psi$ the induced biLipschitz map that
coarsely permutes the cosets of the vertical subgroup of $L_{2}$. Then for any
$\epsilon>0$ there is an $M\gg\epsilon$ that depends on $\epsilon$ and the
quasi-isometry constants of $\Psi$ such that if $P$ is an
$(\epsilon,M)$-parallelogram in $B=\bigoplus{\mathbb{Z}}_{2}$ then $\psi(P)$
is also a parallelogram.

###### Proof.

By Lemma 2.14 we know that if $P$ is an $(\epsilon,M)$-quadrilateral then
$\psi(P)$ is an $(\epsilon^{\prime},M^{\prime})$-quadrilateral where
$\epsilon^{\prime}\leq K^{2}\epsilon$ and $M^{\prime}\geq M/K^{2}$. We need to
show that for any $\epsilon$ there is an $M$ such that if $P$ is an
$(\epsilon^{\prime},M^{\prime})$ quadrilateral then $P$ is a parallelogram. In
fact it is enough to choose the constants so that
$\log{M^{\prime}}>2\log{\epsilon^{\prime}}$.

######  Claim.

Let $P$ be an $(2^{S},2^{2S})$ quadrilateral then $P$ is a parallelogram.

This claim proves Lemma 1.3 from the introduction and follows nicely from the
interpretation of $\log_{2}(\delta(p,q))=supp(p-q)$. Suppose

$P=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}.$

Then

$max(supp(a-b),supp(a-c))\leq S$ $min(supp(a-d),supp(b-c))\geq 2S$

Next we will show that $supp(a-b)\cap supp(a-c)=\emptyset$. Consider the
smallest integer $N_{+}$ such that $n$ is outside $supp(a-b)\cup supp(a-c)$
for all $n\geq N_{+}$. We must have $a_{n}=b_{n}=c_{n}$ and so $n$ must not be
in $supp(b-c)$. Similarly, if $N_{-}$ is the largest integer such that for all
$n\leq N_{-}$, $n$ is outside $supp(a-b)\cup supp(a-c)$, then $n$ cannot be in
$supp(b-c)$. This means the largest integer in $supp(b-c)$ and the smallest
integer in $supp(b-c)$ are both in $supp(a-b)\cup supp(a-c)$. See Figure 6.
Hence, $supp(a-b)\cap supp(a-c)=\emptyset$, as if these subsets of integers
overlapped then $supp(b-c)$ would be too small, which contradicts the
assumption that the support is at least $2S$.

Note that by symmetry we also have that $supp(b-a)\cap
supp(b-d)=\emptyset,supp(d-b)\cap supp(d-c)=\emptyset$ and $supp(c-a)\cap
supp(c-d)=\emptyset$.

Now, fix a given $i$. We would like to show that for every $i$, the equation
$a_{i}+d_{i}=c_{i}+b_{i}$ holds. If $a_{i}=b_{i}=c_{i}=d_{i}$ then we are
done, so assume without loss of generality that $a_{i}\neq b_{i}$. That means
$i\in supp(a-b)$, and by the above claim this implies that $i\notin supp(a-c)$
and $i\notin supp(b-d)$ hence that $a_{i}=c_{i}$ and $b_{i}=d_{i}$. This shows
that $a_{i}+d_{i}=c_{i}+b_{i}$ as desired. ∎

Figure 6.  In an $(2^{S},2^{2S})$-parallelogram formed by $a,b,c,d$, we know
that $supp(a-b)$ and $supp(a-c)$ both have length at most $S$, while
$supp(b-c)$ has length at least $2S$. This implies that $supp(a-c)\cap
supp(a-b)=\emptyset$.

We have now shown that the map $\psi$ is parallelogram preserving when
restricted to $(\epsilon,M)$-parallelograms for $M\gg\epsilon$. This mirrors
the SOL and Baumslag Solitar cases. However, in the SOL and Baumslag-Solitar
case, the result generalizes to all parallelograms and not only ones that are
‘large’ enough. This is done by finding a generating set $\Sigma$ for $B$ such
that all parallelograms based on elements in $\Sigma$ are $(\epsilon,M)$
parallelograms with $M\gg\epsilon$.

In the lamplighter case, this argument only works when $(\epsilon,M)$ is very
small. Namely a generating set for $B$ must contain a generator for
${\mathbb{Z}}_{2}$ at each index $i$. Therefore the best one can hope for is
that quadrilaterals based on $\Sigma$ are $(\epsilon,M)$ quadrilaterals for
$\epsilon=2^{1}$ and $M=2^{2}$. In addition, such quadrilaterals are useful
only in the case that $\Psi$ is a $(1,0)$-quasi-isometry, i.e. an isometry.
This gives us a new proof of Theorem 1.1.

######  Proposition 4.2.

If $\Psi:L_{2}\to L_{2}$ is a $(1,0)$ pattern preserving quasi-isometry, then
$\psi$ must be parallelogram preserving.

###### Proof.

Consider the set $\Sigma$ of elements of $\bigoplus{\mathbb{Z}}_{2}$ where the
size of the support is equal to 1. Given two elements $w,v$ in $\Sigma$, we
can consider the parallelogram

$\begin{bmatrix}a&a+w\\\ a+v&a+w+v\end{bmatrix}.$

If $w=v$ then (because $n=2$), we get $\psi(w+v+a)=\psi(2w+a)=\psi(a)$ so
$\psi(a+w+v)+\psi(a)=2\psi(a)=0=\psi(a+w)+\psi(a+v)$. If $w\neq v$ then this
is a $(\epsilon,M)$-parallelogram for $\epsilon=2^{1}$ and $M=2^{2}$ and
$\psi$ is parallelogram preserving with respect to this parallelogram. Every
element of $\bigoplus{\mathbb{Z}}_{2}$ can be written as a sum of elements of
$S$, so the parallelograms of the above form generate all possible
parallelograms. ∎

Notice that as quasi-isometry constants get worse we need $M\gg\epsilon$ in
order for the image quadrilateral to be a $(2^{S},2^{2S})$ quadrilateral. In
SOL and $BS(1,n)$ this does not matter as much, since we can find a generating
set $\Sigma$ such that parallelograms based on this set are all
$(\epsilon,M)$-parallelograms. However, in the lamplighter case the situation
is different. Pick $\epsilon,M$ such that $\log_{2}M>2\log_{2}\epsilon+1$.
Suppose for contradiction that there is a generating set $\Sigma$ for
$\bigoplus{\mathbb{Z}}_{2}$ such that any two elements $v,w\in\Sigma$ generate
an $(\epsilon,M)$-parallelogram

$P=\begin{bmatrix}0&w\\\ v&w+v\end{bmatrix}.$

Since $supp(v+w)$ is an interval that is larger than the sum of the sizes of
$supp(v)$ and $supp(w)$ there must be some index $i_{0}$ not in $supp(v)\cup
supp(w)$ so there must be another generator $z\in\Sigma$ with $i_{0}\in
supp(z)$ but then without loss of generality $z$ and $v$ will not generate an
$(\epsilon,M)$-parallelogram.

In fact, in the next section we show that we can generate counterexamples
indicating that for larger quasi-isometry constants, the map $\psi$ may not be
parallelogram preserving for all parallelograms.

###  4.3. A Family of Quasi-isometries and a Counterexample

In this section we prove Theorem 1.2. Here is a way to create many examples of
maps $\psi:\bigoplus{\mathbb{Z}}_{2}\to\bigoplus{\mathbb{Z}}_{2}$ that are
bijective and biLipschitz on both boundaries. Such a map will induce a quasi-
isometry $\Psi:DL(2,2)\to DL(2,2)$.

Suppose that $S_{m}$ is the set of sequences of 0’s and 1’s of length $m$, and
let $\pi$ be an element of the symmetric group on $2^{m}$ elements that
permutes the elements of $S_{m}$. This induces a map $\psi$,
$\psi:\bigoplus{\mathbb{Z}}_{2}\to\bigoplus{\mathbb{Z}}_{2}$ by applying $\pi$
to the subsequence with indicies between 0 and $m-1$ and the identity map at
all other heights.

######  Claim.

Any such map $\psi:\bigoplus{\mathbb{Z}}_{2}\to\bigoplus{\mathbb{Z}}_{2}$
induced by $\pi\in S_{2^{m}}$ will be biLipschitz in both the upper and lower
boundaries, and the biLipschitz constants of the boundary maps will both be
less than or equal to $2^{m}$.

We will show this map is biLipschitz in the upper boundary with metric
$d_{u}$. The argument for the lower boundary component is similar. Consider
two arbitrary elements $p$ and $q$ in $\bigoplus{\mathbb{Z}}_{2}$. Let
$\ell_{+}$ be the maximal integer where $x_{i}=y_{i}$ for all $i<\ell_{+}$. If
$\ell_{+}$ is either less than $0$ or greater than $m-1$, then
$d_{u}(p,q)=d_{u}(\psi(p),\psi(q))$ and the claim follows. If
$0\leq\ell_{+}<m$, then $p$ and $q$ do not match at heights between $0$ and
$m$ and the permutation $\pi$ applied to the subsequence of $p$ and the
subsequence of $q$ has different results. Thus there is some $0\leq j<m$ such
that $\psi(x)_{j}\neq\psi(y)_{j}$. This means that the distance in $d_{u}$
between $p$ and $q$ has changed by at most a factor of $2^{m}$ under $\psi$.
This implies that $\psi$ is biLipschitz in the upper boundary with biLipschitz
constant less than or equal to $2^{m}$.

These examples give us many possible maps on $\bigoplus{\mathbb{Z}}_{2}$, each
of which must induce a quasi-isometry on the Lamplighter group. Among these
examples are examples where the map is not parallelogram preserving.

For example consider the map $\pi$ where $m=3$, which transposes $100$ to
$111$ and acts as the identity otherwise. In this map we see that

$\pi(100)+\pi(001)=111+001=110$

while

$\pi(100+001)=\pi(101)=101.$

So this $\pi$ is not affine and hence induces a map $\psi$ on
$\bigoplus{\mathbb{Z}}_{2}$ which is not affine.

This family of examples of biLipschitz maps leads us to the following
question.

######  Question 4.3.

Can all bijective biLipschitz maps be written as compositions of maps
generated by permutations in $S_{m}$ as in the examples above?

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