##  1 Incremental computational scheme for an elastic-plastic simple laminate

In this appendix, we discuss the case of a simple laminate composed of two
elastic-plastic materials. As discussed below, the corresponding computational
model involves a nested iterative-subiterative Newton scheme. Its consistent
linearization is crucial so that the Newton method can be effectively used on
the structural level. In this work, this is achieved by using the automatic
differentiation (AD) technique that is available in _AceGen_ (Korelc2009;
KorelcWriggers2016), and below we use the compact AD-based notation introduced
in (Korelc2009).

The incremental constitutive equations of finite-strain plasticity are rather
standard (SimoHughes1998), and the details are omitted here. The specific AD-
based formulation of elastoplasticity which is adopted here follows that
developed in (Korelc2009), see also (KorelcStupkiewicz2014;
KorelcWriggers2016). On the other hand, the treatment of the laminated
microstructure is based on that developed in (Sadowski2017) for the
incremental Mori–Tanaka scheme (with due differences). The corresponding AD-
based formulation is provided below in the form of a pseudocode with only
short comments, while for the details the reader is referred to
(Sadowski2017).

The AD-based notation employed in the pseudocodes below uses a special
notation to denote the computational derivative, i.e., the derivative
evaluated by AD. The computational derivative is denoted by
$\hat{\delta}f/\hat{\delta}\brm{a}$, where $f$ is a function defined by an
algorithm (or computer program) in terms of independent variables collected in
vector $\brm{a}$. The actual dependencies present in the algorithm can be
overridden or modified by introducing the so-called AD exceptions that are
denoted by a vertical bar following the derivative with additional
specifications in the subscript. The details can be found in (Korelc2009;
KorelcWriggers2016).

Adopting the finite-strain framework, the elastic strain energy of phase $i$
is expressed as a function of the deformation gradient
$\bm{F}_{i}=\bm{F}_{i}^{n+1}$ and the vector $\brm{h}_{i}=\brm{h}_{i}^{n+1}$
of internal (history) variables at the current time step $t=t_{n+1}$,

$W_{i}=W_{i}(\bm{F}_{i},\brm{h}_{i}),$ (1)

and the Piola stress $\bm{P}_{i}=\bm{P}_{i}^{n+1}$ is thus given by

$\bm{P}_{i}=\frac{\partial
W_{i}(\bm{F}_{i},\brm{h}_{i})}{\partial\bm{F}_{i}}.$ (2)

Here and below, the superscript $n+1$ denoting the quantities at $t_{n+1}$ is
omitted to make the notation more compact. Time-discrete evolution of the
internal variables $\brm{h}_{i}$ is governed by a set of nonlinear equations
written symbolically in the residual form as

$\brm{Q}_{i}(\bm{F}_{i},\brm{h}_{i},\brm{h}_{i}^{n})=\brm{0},$ (3)

where $\brm{h}_{i}^{n}$ denotes the known internal variables at the previous
time step $t=t_{n}$. In computational plasticity, the incremental equations of
elastoplasticity are usually solved using the return-mapping algorithm, which
leads to the _state update algorithm_ that is outlined in Algorithm 1 using
the AD-based notation. In this algorithm, the local problem eq:local is solved
iteratively using the Newton method, and the derivative of the implicit
dependence of the solution $\brm{h}_{i}$ on $\bm{F}_{i}$ (denoted by
$\brm{G}_{i}$ in Algorithm 1) is computed in the standard manner
(Michaleris1994; Korelc2009),

$\frac{\partial\brm{h}_{i}}{\partial\bm{F}_{i}}=-\left(\frac{\partial\brm{Q}_{i}}{\partial\brm{h}_{i}}\right)^{-1}\frac{\partial\brm{Q}_{i}}{\partial\bm{F}_{i}}.$
(4)

StateUpdate[ ]: state update algorithm for phase $i$ 1.2 input: $\bm{F}_{i}$,
$\brm{h}_{i}^{n}$
$\phi_{i}^{\text{trial}}\leftarrow\phi_{i}(\bm{F}_{i},\brm{h}_{i}^{n})$
$\phi_{i}^{\text{trial}}<0$ $\brm{h}_{i}\leftarrow\brm{h}_{i}^{n}$
$\brm{G}_{i}\leftarrow\brm{0}$ $\brm{h}_{i}\leftarrow\brm{h}_{i}^{n}$
$\brm{A}_{i}\leftarrow\dfrac{\hat{\delta}\brm{Q}_{i}\left(\bm{F}_{i},\brm{h}_{i},\brm{h}_{i}^{n}\right)}{\hat{\delta}\brm{h}_{i}}$
tangent matrix,
$\brm{A}_{i}=\displaystyle\frac{\partial\brm{Q}_{i}}{\partial\brm{h}_{i}}$
$\Delta\brm{h}_{i}\leftarrow-\brm{A}_{i}^{-1}\brm{Q}_{i}$
$\brm{h}_{i}\leftarrow\brm{h}_{i}+\Delta\brm{h}_{i}$
$\|\Delta\brm{h}_{i}\|\leq tol$
$\brm{G}_{i}\leftarrow-\brm{A}_{i}^{-1}\left.\dfrac{\hat{\delta}\brm{Q}_{i}}{\hat{\delta}\bm{F}_{i}}\right|_{\brm{h}_{i}={\rm
const}}$
$\brm{G}_{i}=\displaystyle\frac{\partial\brm{h}_{i}}{\partial\bm{F}_{i}}$
$\brm{h}_{i}\leftarrow\brm{h}_{i}\Big{|}_{\gfrac{\RD\brm{h}_{i}}{\RD\bm{F}_{i}}=\brm{G}_{i}}$
introduce the implicit dependence of $\brm{h}_{i}$ on $\bm{F}_{i}$
$\bm{P}_{i}\leftarrow\left.\dfrac{\hat{\delta}W_{i}(\bm{F}_{i},\brm{h}_{i})}{\hat{\delta}\bm{F}_{i}}\right|_{\brm{h}_{i}={\rm
const}}$ AD exception ensures that $\bm{P}_{i}$ is computed correctly return:
$\brm{h}_{i}$, $\bm{P}_{i}$, $\brm{G}_{i}$

The specific constitutive functions of the finite-strain $J_{2}$ plasticity
model used in this work, see Section LABEL:sec:EPcomp, are summarized in Box
1, see the formulation I-C-C-b in Box 1 in (KorelcStupkiewicz2014). Box 1
provides also the corresponding definitions of the vector of internal
variables $\brm{h}_{i}$ and of the local residual $\brm{Q}_{i}$. For brevity,
the subscript $i$ is omitted in Box 1.

Constitutive equations of finite-strain $J_{2}$ plasticity with isotropic
hardening. Phase index $i$ is omitted for brevity.

Given: $\bm{F},\bm{C}_{{\rm p},n}^{-1},\gamma_{n}$ Find: $\bm{C}_{{\rm
p}}^{-1},\gamma$ $\bm{b}_{\rm e}=\bm{F}\bm{C}_{{\rm p}}^{-1}\bm{F}^{\rm T}$
$I_{1}=\tr{\bm{b}_{\rm e}},\quad I_{3}=\det{\bm{b}_{\rm e}}$
$W=\frac{1}{2}\mu\left(I_{1}-3-\log{I_{3}}\right)+\frac{1}{4}\lambda\left(I_{3}-1-\log{I_{3}}\right)$
$\bm{\tau}=2\bm{b}_{\rm e}\dfrac{\partial W}{\partial\bm{b}_{\rm e}}$
$\triangleright\;\text{automation:}\bm{\tau}\leftarrow 2\bm{b}_{\rm
e}\dfrac{\hat{\delta}W_{\rm e}}{\hat{\delta}\bm{b}_{\rm e}}\\!\\!$
$\bm{\tau}^{\prime}=\bm{\tau}-\frac{1}{3}\left(\tr{\bm{\tau}}\right)\bm{I}$
$\phi=\sqrt{\frac{3}{2}\bm{\tau}^{\prime}\cdot\bm{\tau}^{\prime}}-\sigma_{\rm
y}(\gamma)$ $\bm{n}=\dfrac{\partial\phi}{\partial\bm{\tau}}$
$\triangleright\;\text{automation:}\bm{n}\leftarrow\dfrac{\hat{\delta}\phi}{\hat{\delta}\bfs{\tau}}\\!\\!$
$\bm{\mathcal{Z}}=\bm{F}\bm{C}_{{\rm
p}}^{-1}-\exp{\left(-2\left(\gamma-\gamma_{n}\right)\bm{n}\right)}\bm{F}\bm{C}_{{\rm
p},n}^{-1}$ $\brm{h}=\left\\{{C}_{{\rm p},11}^{-1}-1,{C}_{{\rm
p},22}^{-1}-1,{C}_{{\rm p},33}^{-1}-1,{C}_{{\rm p},23}^{-1},{C}_{{\rm
p},13}^{-1},{C}_{{\rm p},12}^{-1},\gamma\right\\}$
$\brm{Q}=\left\\{\mathcal{Z}_{11},\mathcal{Z}_{22},\mathcal{Z}_{33},\mathcal{Z}_{23},\mathcal{Z}_{13},\mathcal{Z}_{12},\phi\right\\}$

Consider now a simple laminate in which both phases are governed by an
elastic-plastic material model. Expressing the local deformation gradients in
terms of $\bar{\bm{F}}$ and $\bm{c}$, as in Eq. eq:Fi, the macroscopic elastic
strain energy $\bar{W}=\langle W\rangle$ reads

$\bar{W}(\bar{\bm{F}},\bm{c},\brm{h}_{1},\brm{h}_{2})={(1-\eta)W_{1}(\bm{F}_{1},\brm{h}_{1})+\eta
W_{2}(\bm{F}_{2},\brm{h}_{2})},$ (5)

and the macroscopic stress is obtained as

$\bar{\bm{P}}=\frac{\partial\bar{W}(\bar{\bm{F}},\bm{c},\brm{h}_{1},\brm{h}_{2})}{\partial\bar{\bm{F}}}={(1-\eta)\frac{\partial
W_{1}}{\partial\bm{F}_{1}}\frac{\partial\bm{F}_{1}}{\partial\bar{\bm{F}}}+\eta\frac{\partial
W_{2}}{\partial\bm{F}_{2}}\frac{\partial\bm{F}_{2}}{\partial\bar{\bm{F}}}=(1-\eta)\bm{P}_{1}+\eta\bm{P}_{2}}.$
(6)

The unknown vector $\bm{c}$ is obtained by solving, using the Newton method,
the compatibility condition eq:comp:P written here in the residual form as

$\brm{R}(\bar{\bm{F}},\bm{c},\brm{h}_{1},\brm{h}_{2})=(\bm{P}_{2}-\bm{P}_{1})\bm{N}=\bm{0},$
(7)

where the internal variables $\brm{h}_{i}$ depend on $\bm{c}$ through
$\bm{F}_{i}$, thus $\brm{h}_{i}=\brm{h}_{i}(\bm{F}_{i}(\bar{\bm{F}},\bm{c}))$,
and this dependence must be taken into account when the residual $\brm{R}$ is
linearized. The complete computational scheme is summarized in Algorithm 1,
which includes consistent linearization of the nested iterative-subiterative
scheme. In particular, once the implicit dependencies are correctly identified
and introduced into the code, the consistent overall tangent (denoted as
$\bar{\mathbb{L}}^{\rm alg}$ in Algorithm 1) is obtained as the computational
derivative $\hat{\delta}\bar{\bm{P}}/\hat{\delta}\bar{\bm{F}}$.

AD-based formulation of the incremental scheme for an elasto-plastic two-phase
composite 1.2 input: $\bar{\bm{F}}$, $\bm{c}^{n}$, $\brm{h}_{1}^{n}$,
$\brm{h}_{2}^{n}$ $\bm{c}\leftarrow\bm{c}^{n}$
$\bm{F}_{1}\leftarrow\bar{\bm{F}}-\eta\bm{c}\otimes\bm{N}$
$\bm{F}_{2}\leftarrow\bar{\bm{F}}+(1-\eta)\bm{c}\otimes\bm{N}$
$\\{\brm{h}_{1},\bm{P}_{1},\brm{G}_{1}\\}\leftarrow\mathtt{StateUpdate}\left[\bm{F}_{1},\brm{h}_{1}^{n}\right]$
$\\{\brm{h}_{2},\bm{P}_{2},\brm{G}_{2}\\}\leftarrow\mathtt{StateUpdate}\left[\bm{F}_{2},\brm{h}_{2}^{n}\right]$
$\brm{R}\leftarrow\left(\bm{P}_{2}-\bm{P}_{1}\right)\bm{N}$
$\brm{B}\leftarrow\dfrac{\hat{\delta}\brm{R}}{\hat{\delta}\bm{c}}$ tangent
matrix, $\brm{B}=\dfrac{\partial\brm{R}}{\partial\bm{c}}$
$\Delta\bm{c}\leftarrow-\brm{B}^{-1}\brm{R}$
$\bm{c}\leftarrow\bm{c}+\Delta\bm{c}$ $\|\Delta\bm{c}\|\leq tol$
$\bm{c}\leftarrow\bm{c}\Big{|}_{\gfrac{\RD\bm{c}}{\RD\bar{\bm{F}}}=-\brm{B}^{-1}\gfrac{\hat{\delta}\brm{R}}{\hat{\delta}\bar{\bm{F}}}\big{|}_{\bm{c}={{\rm
const}}}}$ $\bm{F}_{1}\leftarrow\bar{\bm{F}}-\eta\bm{c}\otimes\bm{N}$
$\bm{F}_{2}\leftarrow\bar{\bm{F}}+(1-\eta)\bm{c}\otimes\bm{N}$
$\brm{h}_{1}\leftarrow\brm{h}_{1}\Big{|}_{\gfrac{\RD\brm{h}_{1}}{\RD\bm{F}_{1}}=\brm{G}_{1}}$
introduce the implicit dependence of $\brm{h}_{1}$ on $\bm{F}_{1}$
$\brm{h}_{2}\leftarrow\brm{h}_{2}\Big{|}_{\gfrac{\RD\brm{h}_{2}}{\RD\bm{F}_{2}}=\brm{G}_{2}}$
introduce the implicit dependence of $\brm{h}_{2}$ on $\bm{F}_{2}$
$\bar{W}\leftarrow\left(1-\eta\right)W_{1}(\bm{F}_{1},\brm{h}_{1})+\eta
W_{2}(\bm{F}_{2},\brm{h}_{2})$
$\bar{\bm{P}}\leftarrow\left.\dfrac{\hat{\delta}\bar{W}}{\hat{\delta}\bar{\bm{F}}}\right|_{\bm{c}=\rm{const},\,\brm{h}_{1}={\rm
const},\,\brm{h}_{2}={\rm const}}$ AD exception ensures that $\bar{\bm{P}}$ is
computed correctly $\bar{\mathbb{L}}^{\rm
alg}\leftarrow\dfrac{\hat{\delta}\bar{\bm{P}}}{\hat{\delta}\bar{\bm{F}}}$
return: $\bm{c}$, $\bar{\bm{P}}$, $\bar{\mathbb{L}}^{\rm alg}$, $\brm{h}_{1}$,
$\brm{h}_{2}$

In practice, the iterative Newton scheme in Algorithm 1 can be enhanced by a
line search technique which improves the robustness of the computational
scheme. The related details are omitted here.


# Near Ascending HNN-Extensions and a Combination Result for Semistability at
Infinity

Michael Mihalik

###### Abstract

Semistability at $\infty$ is an asymptotic property of finitely presented
groups that is needed in order to effectively define the fundamental group at
$\infty$ for a 1-ended group. It is an open problem whether or not all
finitely presented groups have semistable fundamental group at $\infty$. While
many classes of groups are known to contain only semistable at $\infty$
groups, there are only a few combination results for such groups. Our main
theorem is such a result.

Main Theorem. Suppose $G$ is the fundamental group of a connected reduced
graph of groups, where each edge group is infinite and finitely generated, and
each vertex group is finitely presented and either 1-ended and semistable at
$\infty$ or has an edge group of finite index. Then $G$ is 1-ended and
semistable at $\infty$.

An important part of the proof of this result is the semistability part of the
following:

Theorem. Suppose $H_{0}$ is an infinite finitely presented group, $H_{1}$ is a
subgroup of finite index in $H_{0}$, $\phi:H_{1}\to H_{0}$ is a monomorphism
and $G=H_{0}\ast_{\phi}$ is the resulting HNN extension. Then $G$ is 1-ended
and semistable at $\infty$. If additionally, $H_{0}$ is 1-ended, then $G$ is
simply connected at $\infty$.

##  1 Introduction

A connected locally finite complex $K$ has semistable fundamental group at
$\infty$ if any two proper rays converging to the same end of $K$ are properly
homotopic. If for any compact $C\subset K$ there is a compact $D\subset K$
such that loops in $K-D$ are homotopically trivial in $K-C$, then $K$ is
simply connected at $\infty$. If $G$ is a finitely presented group then $G$
has semistable (respectively simply connected) fundamental group at $\infty$
if for some (equivalently any) finite complex $X$ with $\pi_{1}(X)=G$, the
universal cover of $X$ has semistable (respectively simply connected)
fundamental group at $\infty$. Both semistability and simple connectivity at
$\infty$ are quasi-isometry invariants of finitely presented groups ([Geo08]
and [Bri93]). If a finitely presented group $G$ has semistable fundamental
group at $\infty$ (respectively is simply connected at $\infty$) then
$H^{2}(G,\mathbb{Z}G)$ is free abelian (respectively trivial) (See Theorems
16.5.1 and 16.5.2, [Geo08]). A open question (apparently due to H. Hopf) ask
if $H^{2}(G,\mathbb{Z}G)$ is free abelian for all finitely presented groups
$G$. There are many classes of groups, all of whose (finitely presented)
members are know to have semistable fundamental group at $\infty$. These
include word hyperbolic groups (combining work of B. Bowditch [Bow99], G.
Levitt [Lev98], G. Swarup [Swa96] and M. Bestvina and G. Mess [BM91]), CAT(0)
cube groups (S. Shepherd [She]), Artin and Coxeter groups (Theorems 1.1 and
1.4, [Mih96]), many relatively hyperbolic groups ((Theorem 1.1, [MS21]) and
(Theorem 1.5, [HM])) and groups containing a finitely generated infinite
subcommensurated subgroup of infinite index (Theorem 1.9, [Mih16]). There are
only a few semistability combination results. Perhaps the most significant one
states:

###### Theorem 1.1

([MT92]) (MTComb) If $G$ is the fundamental group of a finite graph of groups
where each vertex group is finitely presented with semistable fundamental
group at $\infty$, and each edge group is finitely generated, then $G$ has
semistable fundamental group at $\infty$.

Notice that finitely generated free groups trivially have semistable
fundamental group at $\infty$. Exotic 1-ended groups can be written as
$G_{1}\ast_{G_{3}}G_{2}$ for $G_{i}$ a finitely generated free group. These
groups have semistable fundamental group at $\infty$ by Theorem 1.1.

###### Example 1.2

(Examples3) Let $F_{i}$ be the free group of rank $i>0$. Three groups
$\Lambda_{1}$, $\Lambda_{2}$ and $\Lambda_{3}$ are constructed in [Rat07].
Each group $\Lambda_{i}$ can be decomposed in two ways as amalgamated products
$F_{9}\ast_{F_{81}}F_{9}$, such that $F_{81}$ has index 10 in both $F_{9}$
factors. The group $\Lambda_{1}$ is a finitely presented, torsion free simple
group. The groups $\Lambda_{2}$ and $\Lambda_{3}$ are not simple, but
$\Lambda_{2}$ is virtually simple and $\Lambda_{3}$ has no non-trivial finite
quotients.

Since the $F_{81}$ edge group has finite index in each vertex group, it is
commensurated in each $\Lambda_{i}$. This gives a second way of seeing the
$\Lambda_{i}$ have semistable fundamental group at $\infty$ (see Theorem 2.2).
Our main theorem generalizes this phenomenon.

###### Theorem 1.3

(SSDecomp) Suppose $G$ is the fundamental group of a connected reduced graph
of groups, where each edge group is infinite and finitely generated, and each
vertex group is finitely presented and either 1-ended and semistable at
$\infty$ or has an edge group of finite index. Then $G$ is 1-ended and
semistable at $\infty$.

First a clarification. If an edge $e$ is a loop at a vertex $v$ in our graph,
then the group $G_{v}$ has two subgroups $G_{1}$ and $G_{2}$, both of which
are isomorphic to $G_{e}$. There is an isomorphism $\phi:G_{1}\to G_{2}$
determining a subgroup of $G$ that is an HNN-extension of $G_{v}$. If either
$G_{1}$ or $G_{2}$ has finite index in $G_{v}$, we say $G_{e}$ has finite
index in $G_{v}$.

In the setting of this theorem, even when the graph is an edge path loop (with
at least 2 vertices) and each edge group has finite index in its initial
vertex, it seems that one cannot use commensurability ideas in order to prove
semistability. Instead, for such a loop, the intersection of the edge groups
will have finite index in the first vertex group producing an important near
ascending HNN-extension subgroup as considered in the following result.

###### Theorem 1.4

(MMFI) Suppose $H_{0}$ is an infinite finitely presented group, $H_{1}$ is a
subgroup of finite index in $H_{0}$, $\phi:H_{1}\to H_{0}$ is a monomorphism
and $G=H_{0}\ast_{\phi}$ is the resulting HNN extension. Then $G$ is 1-ended
and semistable at $\infty$. If additionally, $H_{0}$ is 1-ended, then $G$ is
simply connected at $\infty$.

If $H_{0}=H_{1}$ then $G$ is an ascending HNN-extension with base group
$H_{0}$. This case of Theorem 1.4 was proved as (Theorem 3.1, [Mih85]), but is
insufficient for our purposes. The proof of Theorem 1.4 is fundamentally more
difficult than that of (Theorem 3.1, [Mih85]). This theorem is a critical
piece of our proof of the Main Theorem.

Semistability preliminaries are covered in §2. We prove that a near ascending
HNN-extension of an infinite finitely generated group is 1-ended in §3. The
proof of Theorem 1.4 is presented in §4 and the proof of the Main Theorem is
concluded in §5.

##  2 Semistability and End Preliminaries (SS)

While semistability makes sense for multiple ended spaces, we are only
interested in 1-ended spaces in this article. Suppose $K$ is a locally finite
connected CW complex. A ray in $K$ is a continuous map $r:[0,\infty)\to K$. A
continuous map $f:X\to Y$ is proper if for each compact subset $C$ of $Y$,
$f^{-1}(C)$ is compact in $X$. Proper rays $r,s:[0,\infty)\to K$ converge to
the same end if for any compact set $C$ in $K$, there is an integer $k(C)$
such that $r([k,\infty))$ and $s([k,\infty))$ belong to the same component of
$K-C$. The space $K$ has semistable fundamental group at $\infty$ if any two
proper rays $r,s:[0,\infty)\to K$ that converge to the same end are properly
homotopic (there is a proper map $H:[0,1]\times[0,\infty)\to X$ such that
$H(0,t)=r(t)$ and $H(1,t)=s(t)$). Note that when $K$ is 1-ended, this means
that $K$ has semistable fundamental group at $\infty$ if any two proper rays
in $K$ are properly homotopic. Suppose $C_{0},C_{1},\ldots$ is a collection of
compact subsets of a locally finite 1-ended complex $K$ such that $C_{i}$ is a
subset of the interior of $C_{i+1}$ and $\cup_{i=0}^{\infty}C_{i}=K$, and
$r:[0,\infty)\to K$ is proper, then $\pi_{1}^{\infty}(K,r)$ is the inverse
limit of the inverse system of groups:

$\pi_{1}(K-C_{0},r)\leftarrow\pi_{1}(K-C_{1},r)\leftarrow\cdots$

This inverse system is pro-isomorphic to an inverse system of groups with
epimorphic bonding maps if and only if $K$ has semistable fundamental group at
$\infty$ (see Theorem 2.1 of [Mih83] or Theorem 16.1.2 of [Geo08]).

(Theorem 2.1, [Mih83]) and (Lemma 9, [Mih86a]) promote several equivalent
notions of semistability. We will use the second formulation in the next
result to prove Theorem 1.4 and the third formulation in the proof of Theorem
1.3.

###### Theorem 2.1

(ssequiv) Suppose $K$ is a 1-ended, locally finite and connected CW-complex.
Then the following are equivalent:

  1. 1.

$K$ has semistable fundamental group at $\infty$.

  2. 2.

For some (equivalently any) proper ray $r:[0,\infty)\to K$ and compact set
$C$, there is a compact set $D$ such that for any third compact set $E$ and
loop $\alpha$ based on $r$ and with image in $K-D$, $\alpha$ is homotopic
rel$\\{r\\}$ to a loop in $K-E$, by a homotopy with image in $K-C$.

  3. 3.

For any compact set $C$ there is a compact set $D$ such that if $r$ and $s$
are rays based at $v$ and with image in $K-D$, then $r$ and $s$ are properly
homotopic rel$\\{v\\}$, by a proper homotopy in $K-C$.

If in addition $K$ is simply connected, then:

  4. 4.

If $r$ and $s$ are rays based at $v$ then $r$ and $s$ are properly homotopic
$rel\\{v\\}$.

In a connected locally finite CW-complex $K$, every proper ray is properly
homotopic to a proper edge path ray and every loop at a vertex of $K$ is
homotopic (by a small homotopy) to an edge path loop.

A subgroup $H$ of a group $G$ is commensurated in $G$ if for each $g\in G$,
the subgroup $H\cap(g^{-1}Hg)$ has finite index in both $H$ and $g^{-1}Hg$.
The following two results will be used in our proof of Theorem 1.3. There is a
notion of semistability at $\infty$ for finitely generated groups that agrees
with our definitions when the group is finitely presented. We mention this
fact since it comes up in the following two results.

###### Theorem 2.2

([CM14]) (MainCM) If a finitely generated group $G$ has an infinite, finitely
generated, commensurated subgroup $Q$, and $Q$ has infinite index in $G$, then
$G$ is 1-ended and semistable at $\infty$.

###### Theorem 2.3

(Theorem 3, [Mih86b]) (MComb) Suppose $A$ and $B$ are finitely generated 1 or
2-ended subgroups of the finitely generated group $G$ and both $A$ and $B$ are
semistable at $\infty$. If the set $A\cup B$ generates $G$ and the group
$A\cap B$ contains a finitely generated infinite subgroup then $G$ has
semistable fundamental group at $\infty$.

The conditions of the following result are the most general we know in order
to define ends of a space. In this article we are only interested in locally
finite CW-complexes.

###### Theorem 2.4

(ES) Suppose $X$ is connected, locally compact, locally connected and
Hausdorff and $C$ is compact in $X$, then $C$ union all bounded components of
$(X-C)$ is compact, and $X-C$ has only finitely many unbounded components.
Here bounded means compact closure.

If $X$ satisfies hypotheses of Theorem 2.4, then the number of ends of $X$ is
the largest number of unbounded components one can obtain in the compliment of
a compact subset of $X$. If this number is unbounded, then $X$ has infinitely
many ends. If $K$ is a locally finite, connected CW complex, then $K$ has
1-end if and only if any two proper rays in $K$ converge to the same end. Note
that if two proper rays are properly homotopic, then they converge to the same
end.

##  3 Near Ascending HNN-Extensions are 1-ended (NHH1E)

If $A$ is a group with finite generating set $\mathcal{A}$ then let
$\Gamma(A,\mathcal{A})$ be the Cayley graph of $A$ with respect to
$\mathcal{A}$. This means that the vertex set of $\Gamma(A,\mathcal{A})$ is
$A$ and there is an edge (labeled $a$) between the vertex $v$ and $w$ if
$va=w$ for some $a\in\mathcal{A}$. The following result is somewhat standard.
We include it for completeness.

###### Theorem 3.1

(MMFI1) Suppose $H_{0}$ is an infinite finitely generated group, $H_{1}$ is a
subgroup of finite index in $H_{0}$, $\phi:H_{1}\to H_{0}$ is a monomorphism
and $G=H_{0}\ast_{\phi}$ is the resulting HNN extension. Then $G$ is 1-ended.

Proof: Choose a finite generating set $\mathcal{H}_{0}$ for $H_{0}$ where
$\mathcal{H}_{1}\subset\mathcal{H}_{0}$ generates $H_{1}$. Then
$\mathcal{H}_{0}$ along with $t$ (the stable letter) generates $G$. Consider:

$\Gamma_{1}=\Gamma(H_{1},\mathcal{H}_{1})\subset\Gamma_{0}=\Gamma(H_{0},\mathcal{H}_{0})\subset\Gamma=\Gamma(G,\\{t\\}\cup\mathcal{H}_{0}).$

If $v$ (an element of $G$) is a vertex of $\Gamma$, then $v\Gamma_{0}$ and
$v\Gamma_{1}$ are translates of $\Gamma_{0}$ and $\Gamma_{1}$ respectively.
Note that if $w$ is a vertex of $v\Gamma_{i}$ ($i\in\\{0,1\\}$) then
$v\Gamma_{i}=w\Gamma_{i}$. Our metric on $\Gamma$ is the edge path metric. If
$v$ is a vertex of $\Gamma$ and $M>0$ an integer, then $B(v,M)$ is the ball of
radius $M$ about $v$. Let $\ast$ be the identity vertex of $\Gamma$. There is
a continuous map $P:\Gamma\to\mathbb{R}$ such that $P(\ast)=0$, each edge
labeled $t$ is mapped to an interval $[n,n+1]$ for some $n\in\mathbb{Z}$ and
each edge labeled by an element of $\mathcal{H}_{0}$ is mapped to an integer.
This map extends the homomorphism $G\to\mathbb{Z}$ that kills the normal
closure of $H_{0}$.

We show that for any integer $N$ there is an integer $M>N$ such that any two
vertices of $\Gamma-B(\ast,M)$ can be joined in $\Gamma-B(\ast,N)$. There are
only finitely many $v\Gamma_{0}$ that intersect $B(\ast,N)$ non-trivially.
Choose $M$ such that for each such $v\Gamma_{0}$, all bounded components of
$v\Gamma_{0}-B(\ast,N)$ belong to $B(\ast,M)$ (see Theorem 2.4). The next
lemma follow easily from the fact that $A_{1}$ has finite index in $A_{0}$.

###### Lemma 3.2

(FIclose) Suppose $\mathcal{A}_{1}\subset\mathcal{A}_{0}$ are finite
generating sets for the groups $A_{1}\subset A_{0}$ and that $A_{1}$ has
finite index in $A_{0}$. Then there is an integer
$N_{\ref{FIclose}}(A_{0},\mathcal{A}_{0},A_{1},\mathcal{A}_{1})$ such that if
$v$ and $w$ are verteices of $\Gamma(A_{0},\mathcal{A}_{0})$ then there is an
edge path in $\Gamma(A_{0},\mathcal{A}_{0})$ of length $\leq
N_{\ref{FIclose}}(G_{0},\mathcal{A}_{0},G_{1},\mathcal{A}_{1})$ from $v$ to a
vertex of $w\Gamma_{1}(A_{1},\mathcal{A}_{1})$.

From this point on we let
$N_{\ref{FIclose}}=N_{\ref{FIclose}}(H_{0},\mathcal{H}_{0},H_{1},\mathcal{H}_{1})$.
Suppose $e=(v,w)$ is an edge labeled $t$ and either $v$ or $w$ is in
$B(\ast,M)$. Let $\alpha_{v}$ be an edge path at $v$ such that each edge of
$\alpha$ is labeled by an element of $\mathcal{H}_{1}$ and such that the end
point $a$ of $\alpha_{v}$ is in $\Gamma-B(\ast,M+1)$. Let $e_{1}=(a,b)$ be the
edge at $a$ labeled $t$ and note that $e_{1}$ has image in $\Gamma-B(\ast,M)$.
For each $h\in\mathcal{H}_{1}$, we have $t^{-1}ht=\phi(h)\in H_{0}$ and so
there is an edge path $\beta_{w}$ in $w\Gamma_{0}$ from $w$ to $b$. Now
suppose $x$ and $y$ are vertices of $\Gamma-B(\ast,M)$. We want to find a path
in $\Gamma-B(\ast,N)$ connecting $x$ and $y$. Let $\tau$ be any edge path from
$x$ to $y$ in $\Gamma$. If $e=(v,w)$ is an edge of $\tau$ labeled $t$, such
that either $v$ or $w$ belongs to $B(\ast,M)$ then replace $e$ by the path
$(\alpha_{v},e_{1},\beta_{w}^{-1})$ (above). In this way we produce an edge
path $\tau_{1}$ from $x$ to $y$ such that each $t$ labeled edge of $\tau_{1}$
belongs to $\Gamma-B(\ast,M)$. Each maximal subpath of $\tau_{1}$ (none of
whose edges is labeled $t$) has both end points in $\Gamma-B(\ast,M)$. Hence
it suffices to show the end points of such paths can be joined by an edge path
in $\Gamma-B(\ast,N)$.

Figure 1: One Endedness

Recall that $P:\Gamma\to\mathbb{R}$ and $P(v\Gamma_{0})=P(v)$. We have:

$P(B(\ast,N))=[-N,N].$

Suppose $\gamma$ is an edge path with end points in $\Gamma-B(\ast,M)$, each
edge of $\gamma$ is labeled by an element of $\mathcal{H}_{0}$ and $\gamma$
intersects $B(\ast,N)$, then $-N\leq P(\gamma)\leq N$. The strategy is to
“slide” $\gamma$ upward to a path $\gamma_{k}$ in level $N+1$ (which
automatically avoids $B(\ast,N)$). Assume the initial point of $\gamma$ is $v$
and the terminal point is $w$ (see Figure 1). By the definition of $M$, the
vertex $v$ is in an unbounded component of $v\Gamma_{0}-B(\ast,N)$. Hence
there is an edge path $\tau_{1}$ in $v\Gamma_{0}-B(\ast,N)$ from $v$ to
$v_{1}\in v\Gamma_{0}-B(\ast,M+N_{\ref{FIclose}}+1)$. Similarly choose
$\tau^{\prime}_{1}$ in $v\Gamma_{0}-B(\ast,N)$ from $w$ to $w_{1}\in
v\Gamma_{0}-B(\ast,M+1)$. Note that:

$v\Gamma_{0}=v_{1}\Gamma_{0}=w\Gamma_{0}=w_{1}\Gamma_{0}.$

By the definition of $N_{\ref{FIclose}}$ we choose an edge path $\tau_{2}$ of
length $\leq N_{\ref{FIclose}}$ in $w_{1}\Gamma_{0}-B(\ast,M+1)$ from $v_{1}$
to $v_{2}\in w_{1}\Gamma_{1}-B(\ast,M+1)$. Let $e=(v_{2},v_{3})$ and
$e^{\prime}=(w_{1},w_{2})$ be edges labeled $t$. Let $\gamma^{\prime}$ be an
edge path in $w_{1}\Gamma_{1}$ from $v_{2}$ to $w_{1}$. The conjugation
relations $t^{-1}ht=\phi(h)$ for $h\in\mathcal{H}_{1}$ (applied to each edge
of $\gamma^{\prime}$) give an edge path $\gamma_{1}$ in $v_{3}\Gamma_{0}$ from
$v_{3}$ to $w_{2}$. Then:

$\\{v_{3},w_{2}\\}\subset v_{3}\Gamma_{0}-B(\ast,M).$

Note that the paths $(\tau_{1},\tau_{2},e)$ (from $v$ to $v_{3}$) and
$(\tau_{1}^{\prime},e^{\prime})$ (from $w$ to $w_{2}$) have image in
$\Gamma-B(\ast,N)$. Hence we need only show there is an edge path in
$\Gamma-B(\ast,N)$ from $v_{3}$ to $w_{2}$. Note that
$P(\gamma_{1})=P(\gamma)+1\geq-N+1$. Repeating this process at most $2N+1$
times produce a path from $v$ to $w$ in $\Gamma-B(\ast,N)$, and so $G$ is
1-ended. $\Box$

##  4 The Proof of Theorem 1.4 (PFMMFI)

This theorem generalizes both the Main Theorem and Theorem 3.1 of [Mih85]
(where $H_{0}=H_{1}$ and $G$ is an ascending HNN-extension of $H_{0}$). Our
proof of Theorem 1.4 has similarities to the one in [Mih85], but requires
several new ideas.

Proof: Theorem 3.1 implies that $G$ is 1-ended (even when $H_{0}$ is finitely
generated and not finitely presented). Again we consider the following finite
presentation of $G$:

$\mathcal{P}=\langle
t,\mathcal{H}_{0}:\mathcal{R},t^{-1}ht=\hat{\phi}(h)\hbox{ for all
}h\in\mathcal{H}_{1}\rangle$

where $\mathcal{P}_{0}=\langle\mathcal{H}_{0}:\mathcal{R}\rangle$ is a finite
presentation for $H_{0}$. We assume $\mathcal{H}_{1}\subset\mathcal{H}_{0}$
and $\mathcal{P}_{1}=\langle\mathcal{H}_{1}:\mathcal{R}_{1}\rangle$ (for some
$\mathcal{R}_{1}\subset\mathcal{R}_{0}$) is a finite presentation for $H_{1}$.
Here $\hat{\phi}:F(\mathcal{H}_{1})\to F(\mathcal{H}_{0})$ is a homomorphism
of free groups realizing $\phi$ and the relations $t^{-1}ht=\hat{\phi}(h)$ are
called conjugation relations. Let

$q:F(\mathcal{H}_{0}\cup\\{t\\})\to G$

be a quotient homomorphism with kernel equal to the normal closure of
$\mathcal{R}$ union the set of conjugation relations.

$\textstyle{F(\mathcal{H}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\phi}}$$\scriptstyle{q}$$\textstyle{F(\mathcal{H}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{H_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{H_{0}}$

Let $\Gamma_{1}\subset\Gamma_{0}\subset\Gamma=\Gamma(\mathcal{P})$ be the
Cayley 2-complexes for $\mathcal{P}_{1}$, $\mathcal{P}_{0}$ and $\mathcal{P}$
respectively. The 1-skeleton of $\Gamma$ is the Cayley graph of $G$ with
respect to the generating set $\mathcal{H}_{0}\cup\\{t\\}$. At each vertex
there is a 2-cell attached according to each relation of $\mathcal{P}$. If the
edges of an edge path in $\Gamma$ are labeled by $\mathcal{S}^{\pm 1}$ for
$\mathcal{S}\subset\mathcal{H}_{0}$, then we call it an $\mathcal{S}$ path. If
$v(\in G)$ is a vertex of $\Gamma$ then $v\Gamma_{1}\subset
v\Gamma_{0}\subset\Gamma$. Note that any edge path in $v\Gamma_{0}$ is an
$\mathcal{H}_{0}$ path. There is a continuous map $p:\Gamma\to\mathbb{R}$ such
that each vertex is mapped to an integer, a (directed) edge $e$ labeled $t$ is
mapped linearly by $p$ to an interval $[n,n+1]$ (so that $p(e(0))=n$ and
$p(e(1))=n+1$) and $p(v\Gamma_{0})=\\{p(v)\\}$. A conjugation 2-cell is mapped
to an interval $[n,n+1]$. We say a vertex, edge or 2-cell is in level $n$ if
$p$ maps it to $n$. If $X$ is a 2-complex, let $X^{(1)}$ be the 1-skeleton of
$X$, endowed with the edge path metric. Suppose $A$ is a subcomplex of $X$ and
$A^{(1)}$ is the 1-skeleton of $A$. Let $B(A^{(1)},n)$ be the ball of radius
$n$ in $X^{(1)}$ around $A^{(1)}$. For $n\geq 1$, let $St^{n}(A)$ be
$B(A^{(1)},n)$ union all 2-cells whose boundaries belong to $B(A^{(1)},n)$.

($\ast$) Let $N_{0}$ be such that $|\hat{\phi}(h)|+3\leq N_{0}$ for all
$h\in\mathcal{H}_{1}$. Then any (closed) 2-cell arising from a conjugation
relation belongs to $St^{N_{0}}(v)$ for any vertex of that cell.

All homotopies of paths will be relative to end points and we simply say the
paths are homotopic. Note that ($\ast$) implies:

###### Lemma 4.1

(oldslide) Suppose $\alpha$ is an $\mathcal{H}_{1}$ path from $v$ to $w$ in
$\Gamma$. Let $e$ and $d$ be the edges labeled $t$ at $v$ and $w$
respectively. Then the edge path $(e^{-1},\alpha,d)$ is homotopic to an
$\mathcal{H}_{0}$ path $\beta$ by a homotopy $H$ with image in
$St^{N_{0}}(im(\alpha))$. Furthermore $p(H)\subset[p(v),p(v+1)]$.

We apply the next lemma in both this section and in the proof of Lemma 5.1.

###### Lemma 4.2

(SSL1) Suppose $A_{0}$ and $A_{1}$ are finitely presented groups, and $A_{1}$
has finite index in $A_{0}$. Let $\mathcal{Q}=\langle\mathcal{A}_{0}:R\rangle$
be a finite presentation of $A_{0}$ such that
$\mathcal{A}_{1}\subset\mathcal{A}_{0}$ generates $A_{1}$ (so that the Cayley
graph $\Gamma(A_{1},\mathcal{A}_{1})$ is a subset of the Cayley 2-complex
$\Gamma(\mathcal{Q})$). There is an integer $N_{\ref{SSL1}}(\mathcal{Q})$ such
that if $\alpha$ is an edge path in $\Gamma(\mathcal{Q})$ with $\alpha(0)=v$
and end point in $v\Gamma(A_{1},\mathcal{A})$ then $\alpha$ is homotopic to an
edge path in $v\Gamma(A_{1},\mathcal{A}_{1})$ by a homotopy $H$ in
$\Gamma(\mathcal{Q})$ with image in
$St^{N_{\ref{SSL1}}(\mathcal{Q})}(im(\alpha))$.

Proof: Say $\alpha=(e_{1},\ldots,e_{n})$ with vertices $v=v_{0},\ldots,v_{n}$.
If $i\in\\{1,\ldots,n-1\\}$ then there is an edge path $\phi_{i}$ of length
$\leq\bar{N}_{\ref{FIclose}}=N_{\ref{FIclose}}(A_{0},\mathcal{A}_{0},A_{1},\mathcal{A}_{1})$
from $v_{i}$ to $w_{i}\in v\Gamma(A_{1},\mathcal{A}_{1})$. Define $\phi_{0}$
and $\phi_{n}$ to be trivial paths and define $w_{0}=v_{0}$ and $w_{n}=v_{n}$.
Choose an integer $M$ such that if two vertices of
$\Gamma(A_{1},\mathcal{A}_{1})$ are within $2\bar{N}_{\ref{FIclose}}+1$ in
$\Gamma(\mathcal{Q})$ then there is an edge path in
$\Gamma(A_{1},\mathcal{A}_{1})$ of length $\leq M$ connecting these vertices.
Choose $N_{\ref{SSL1}}(\mathcal{Q})$ so that any edge path loop in
$\Gamma(\mathcal{Q})$ of length $\leq 2\bar{N}_{\ref{FIclose}}+M+1$ is
homotopically trivial in
$St^{N_{\ref{SSL1}}(\mathcal{Q})}(a)\subset\Gamma(\mathcal{Q})$ for any vertex
$a$ of the loop. For $i\in\\{1,\ldots,n\\}$ let $\tau_{i}$ be an
$\mathcal{H}_{1}$ edge path (in $v\Gamma(A_{1},\mathcal{A}_{1})$) of length
$\leq M$ from $w_{i-1}$ to $w_{i}$. Each loop
$(\phi_{i},\tau_{i+1},\phi_{i+1}^{-1},e_{i+1}^{-1})$ is homotopically trivial
in $St^{N_{\ref{SSL1}}(\mathcal{Q})}(v_{i})\subset\Gamma(\mathcal{Q})$. This
implies that $\alpha$ is homotopic to $(\tau_{1},\ldots,\tau_{n})$, by a
homotopy in
$St^{N_{\ref{SSL1}(\mathcal{Q})}}(im(\alpha))\subset\Gamma(\mathcal{Q})$.
$\Box$

Next we recall some basic group theory for HNN-extensions. Suppose $w\in
F(\mathcal{H}_{0})$ and $q(w)=1\in G$. Then the number of letters of $w$
labeled $t$ is the same as the number of letters of $w$ labeled $t^{-1}$. If
some letter of $w$ is labeled $t$, then (cyclically), there is a subword
$t^{-1}w^{\prime}t$ of $w$ such that $q(w^{\prime})\in H_{1}$. Suppose
$\alpha$ is an edge path in $\Gamma$ with consecutive vertices
$v_{0},\ldots,v_{n}$. If the edge $(v_{i-1},v_{i})$ is labeled $a_{i}$ and is
directed from $v_{i}$ to $v_{i+1}$ then let $b_{i}=a_{i}$. If this edge is
directed $(v_{i},v_{i-1})$ then let $b_{i}=a^{-1}$. We say the word
$b_{1}\cdots b_{n}$ ($\in F(\mathcal{H}_{0}\cup\\{t\\}$) corresponds to
$\alpha$ and we write $\bar{\alpha}=b_{1}\cdots b_{n}$.

An edge path has backtracking if there are adjacent edges $e$ and $e^{-1}$.
The next lemma effectively allows us to move lowest level maximal subpaths of
certain loops upward. Lemmas 4.3 and 4.4 are used in the simply connected at
$\infty$ part of our result. The no backtracking hypothesis is not an issue
for simply connected at $\infty$ arguments, but when a base ray must be
respected in the semistability part of our theorem, we cannot assume there is
(cyclically) no backtracking in certain loops.

###### Lemma 4.3

(smallmove) Suppose $\gamma$ is an edge path loop in $\Gamma$ such that: at
least one edge of $\gamma$ is labeled $t$; $p(\gamma)=[A,B]$; and (cyclically)
$\gamma$ has no backtracking. Assume $\alpha$ is (cyclically) a maximal
subpath of $\gamma$ such that $p(\alpha)=A$. Let $e$ and $d$ be the edges of
$\gamma$ that (cyclically) immediately precede and follow $\alpha$, so that
$\beta=(e,\alpha,d)$ is (cyclically) a subpath of $\gamma$. Then
$q\bar{\alpha}\in H_{1}$, $\bar{\beta}=t^{-1}\bar{\alpha}t$ and $\beta$ is
homotopic to an $\mathcal{H}_{0}$ path $\tau$ by a homotopy $H$ with image in
$St^{N_{0}+N_{\ref{SSL1}}}(im(\alpha))$ and with $p(H)\subset[A,A+1]$.

Proof: The maximality of $\alpha$ implies $\bar{\beta}=t^{-1}\bar{\alpha}t$.
We next argue that $q(\bar{\alpha})\in H_{1}$. By the remarks preceding this
lemma, there is (cyclically) a subword of $\bar{\gamma}$ of the form
$t^{-1}\bar{\beta}t$ where $q(\bar{\beta})\in H_{1}$. If
$\bar{\beta}=\bar{\alpha}$ then indeed $q(\bar{\alpha})\in H_{1}$. Otherwise,
$\bar{\alpha}$ and $\bar{\beta}$ do not overlap. Furthermore (since $\gamma$
has no backtracking) $e$ and $d$ are not adjacent to $\beta$ (see Figure 2).
If $\bar{\beta}\neq\bar{\alpha}$ then replace $t^{-1}\bar{\beta}t$ in
$\bar{\gamma}$ by an $\mathcal{H}_{0}$-word. This produces a word with two
fewer $t$ letters, but the $t$ letters corresponding to $d$ and $e$ remain.
Continuing, we eventually conclude that $q(\bar{\alpha})\in H_{1}$. Let $v$ be
the initial vertex of $\alpha$, then the end point of $\alpha$ is in
$v\Gamma_{1}$.

Figure 2: Sliding $\alpha$ upward to $\tau$

By Lemma 4.2, $\alpha$ is homotopic to an edge path $\alpha_{1}$ in
$v\Gamma_{1}$ by a homotopy $\tilde{H}$ (in $v\Gamma_{0}$) with image in
$St^{N_{\ref{SSL1}}}(im(\alpha))$. In particular, the image of $p\tilde{H}$ is
$p(v)=A$. Now we can slide $\alpha_{1}$ up one level (between $t$ labeled
edges) to a path $\tau$, by a homotopy $\hat{H}$ which only uses 2-cells
arising from conjugation relations on the letters of $\alpha_{1}$ (see Figure
2). Note that the image of this homotopy lies between levels $A$ and $A+1$ and
$\hat{H}$ has image in $St^{N_{0}}(im(\alpha_{1}))$. Combine $\tilde{H}$ and
$\hat{H}$ to form the homotopy $H$ and finish the proof. $\Box$

###### Lemma 4.4

(slideupward) Suppose $C$ is a compact subcomplex of $\Gamma$,
$p(C)\subset[A,B]$ and $\gamma$ is an edge path loop (without backtracking) in
$\Gamma-C$. Then either $\gamma$ is homotopically trivial in $\Gamma-C$ or
there is a homotopy in $\Gamma-C$ of $\gamma$ to an edge path loop
$\gamma_{1}$ such that $p(\gamma_{1})$ has image in $[A-1,\infty)$, the
homotopy fixes the edges of $\gamma$ in levels $A-1$ and above and the only
edges of $\gamma_{1}$ above level $A-1$ (including $t$-labeled edges
connecting levels $A-1$ and $A$) are edges of $\gamma$.

Proof: Let $\bar{\gamma}\ (\in F(\mathcal{H}_{0}\cup\\{t\\}))$ be the
(reduced) word labeling $\gamma$. If $\gamma$ has image in a single level,
then either $\gamma$ lies in a level at or above level $A-1$ (and we are
finished) or $\gamma$ lies in a level below level $A-1$. In the later case,
$\gamma$ is homotopically trivial in that level (by a homotopy avoiding $C$).
Instead, we assume some letter of $\bar{\gamma}$ is $t$.

Suppose $\alpha$ is (cyclically) a maximal subpath of $\gamma$ in the lowest
level of $\Gamma$ traversed by $\gamma$. If $p(\alpha)\geq A-1$ we are
finished. If $e$ and $d$ are the edges preceding and following $\alpha$, then
Lemma 4.3 implies that $(e,\alpha,d)$ is homotopic to an $\mathcal{H}_{0}$
path $\tau$ (with $p(\tau)=p(\alpha)+1$) by a homotopy $H$ with
$p(H)\subset(-\infty,A-1]$. In particular, $H$ avoids $C$. Replace
$(e,\alpha,d)$ in $\gamma$ by $\tau$, eliminate backtracking and continue
applying Lemma 4.3 to (cyclically) maximal subpaths in a lowest level until
all edges of the resulting path are either in a single level below level $A-1$
(in which case $\gamma$ is homotopically trivial in $\Gamma-C$) or in levels
$A-1$ and above. $\Box$

Now assume that $H_{0}$ is 1-ended. In this case, our goal is to show that $G$
is simply connected at $\infty$. Let $C$ be a finite subcomplex of $\Gamma$
and assume that $p(C)\subset[A,B]$. Let $D$ be a finite subcomplex of $\Gamma$
containing $St^{(B-A+2)(N_{\ref{SSL1}}+N_{0})}(C)$. Also assume that if $g\in
G$ then $g\Gamma_{0}-D$ is a single unbounded component (since $H_{0}$ is one
ended). Let $\gamma$ be an edge path loop in $\Gamma-D$. It suffices to show
that $\gamma$ is homotopically trivial in $\Gamma-C$. The idea is to move
$\gamma$ to a loop in a single level above level $B$ by a homotopy that avoids
$C$. Once there, it is homotopically trivial in that level and hence that
homotopy avoids $C$. Suppose $p(\gamma)=[A_{1},B_{1}]$.

Inductively we show that for $0\leq k\leq B-A+2$, the loop $\gamma$ is either
homotopically trivial in $\Gamma-C$ (as desired) or homotopic to an edge path
loop $\mu_{k}$ by a homotopy in $\Gamma-C$ such that $\mu_{k}$ has image in
$\Gamma-St^{(B-A+2-k)(N_{\ref{SSL1}}+N_{0})}(C)$ and
$p(\mu_{k})\subset[A-1+k,\infty)$. Note that for $k=B-A+2$, we have
$p(\mu_{k})\subset[B+1,\infty)$ (so $\mu_{k}$ is above $C$).

Figure 3: Sliding $\gamma$ Up

If $A_{1}\geq A-1$, then let $\mu_{0}=\gamma$. If $A_{1}<A-1$ then Lemma 4.4
implies that either $\gamma$ is homotopically trivial in $\Gamma-C$ (and we
are finished) or there is a homotopy in $\Gamma-C$ of $\gamma$ to an edge path
loop $\gamma_{1}^{\prime}$ such that
$p(\gamma_{1}^{\prime})\subset[A-1,\infty)$, the homotopy fixes the edges of
$\gamma$ in levels $A-1$ and above, and the only edges of
$\gamma_{1}^{\prime}$ above level $A-1$ (including $t$-labeled edges
connecting levels $A-1$ and $A$) are edges of $\gamma$. If
$\gamma_{1}^{\prime}$ has image in level $A-1$ then it is homotopically
trivial in that level (hence missing $C$) and again we are finished. Let
$\gamma_{1}$ be obtained from $\gamma_{1}^{\prime}$ by eliminating
backtracking (which can only occur in level $A-1$). If $\alpha_{1}$ is a
maximal subpath of $\gamma_{1}$ in level $A-1$ and $e$ and $d$ are the edges
of $\gamma_{1}$ that (cyclically) immediately precede and follow $\alpha_{1}$,
then $e$ and $d$ are edges of $\gamma$ labeled $t$. In particular if $v$ and
$w$ are the initial and end points of $\alpha_{1}$ then $v$ and $w$ are in the
single unbounded component of $v\Gamma_{0}-D$. Hence there is an
$\mathcal{H}_{0}$ path $\alpha_{2}$ (without backtracking) in $v\Gamma_{0}-D$
from $v$ to $w$. Now $\alpha_{1}$ and $\alpha_{2}$ are homotopic by a homotopy
in $v\Gamma_{0}$ (in level $A-1$) and hence in $\Gamma-C$ (see Figure 3). Let
$\mu_{0}$ be obtained from $\gamma_{1}$ by replacing each maximal
$\mathcal{H}_{0}$ subpath $\alpha_{1}$ of $\gamma_{1}$ with a corresponding
$\alpha_{2}$. Note that there is no backtracking in $\mu_{0}$. We have that
$\gamma$ is homotopic to $\mu_{0}$ by a homotopy in $\Gamma-C$, the image of
$\mu_{0}$ is in $\Gamma-D$ and $p(\mu_{0})\subset[A-1,\infty)$.

If $A_{1}>A-1$, let $\mu_{1}=\gamma$. Otherwise let $\alpha_{2}$ be a maximal
subpath of $\mu_{0}$ in level $A-1$. Let $e$ and $d$ be the edges of $\mu_{0}$
that (cyclically) immediately precede and follow $\alpha_{2}$ (see Figure 3).
The path $\beta=(e,\alpha_{2},d)$ is such that
$\bar{\beta}=t^{-1}\bar{\alpha}_{2}t$. Lemma 4.3 implies that $\beta$ is
homotopic to an $\mathcal{H}_{0}$ path $\tau$ by a homotopy $H$ with image in
$St^{N_{0}+N_{\ref{SSL1}}}(im(\alpha_{2}))$ and with $p(H)\subset[A-1,A]$. In
particular $H$ (and $\tau$) avoids $St^{(B-A+1)(N_{\ref{SSL1}}+N_{0})}(C)$.
For all such maximal subpaths $\alpha_{2}$ of $\mu_{0}$ in level $A-1$ replace
$(e,\alpha_{2},d)$ by the corresponding $\tau$. Eliminate backtracking (all in
level $A$) and call the resulting path $\mu_{1}$. The result is that $\mu_{0}$
(and $\gamma$) is homotopic to an edge path loop $\mu_{1}$ with image in
$\Gamma-St^{(B-A+1)(N_{\ref{SSL1}}+N_{0})}(C)$ by a homotopy in $\Gamma-C$ and
$p(\mu_{1})\subset[A,\infty)$. If $A_{1}>A$, let $\mu_{2}=\gamma$. Otherwise,
we can obtain $\mu_{2}$ by apply Lemma 4.3 to each maximal subpath of
$\mu_{1}$ in level $A$, as long as $\mu_{1}$ does not have image in level $A$.
If $\mu_{1}$ has image in level $A$, then $\mu_{1}$ is an $\mathcal{H}_{0}$
loop at say the vertex $v$. Lemma 4.2 implies that $\mu_{1}$ is homotopic to
an $\mathcal{H}_{1}$ edge path loop $\mu_{1}^{\prime}$ at $v$ by a homotopy in
$St^{N_{\ref{SSL1}}}(im(\mu_{1}))$. If $e$ is the edge labeled $t$ at $v$ then
Lemma 4.1 implies that $(e^{-1},\mu_{1}^{\prime},e)$ is homotopic to an
$\mathcal{H}_{0}$ path by a homotopy $H$ with image in
$St^{N_{0}}(im(\mu_{1}^{\prime}))$ and such that $p(H)\subset[p(v),p(v)+1]$.
This means that $(e^{-1},\mu_{1},e)$ is homotopic to an $\mathcal{H}_{0}$ path
in level $A+1$ by a homotopy in $\Gamma-St^{(B-A)(N_{\ref{SSL1}}+{N_{0}}}(C)$.
If $\mu_{1}$ is not in level $A$ we apply Lemma 4.3 to maximal subpaths of
$\mu_{1}$ in level $A$, so that in either case we have that $\mu_{1}$ is
homotopic to an edge path loop $\mu_{2}$ (without backtracking) with image
avoiding $St^{(B-A)(N_{\ref{SSL1}}+{N_{0}}}(C)$, by a homotopy in $\Gamma-C$
and $p(\mu_{2})\subset[A+1,\infty)$.

Repeating this procedure $k=B-A+2$ times (starting with $\mu_{0}$) we obtain
an edge path loop $\mu_{k}$ (without backtracking) with $p(\mu_{k})$ either
equal to $B+1$ (if $B_{1}\leq B+1$) or a subset of $[B+1,B_{1}]$ (if
$B_{1}>B+1$). We have $\gamma$ is homotopic to $\mu_{k}$ by a homotopy in
$\Gamma-C$. In the latter case, use Lemma 4.3 to move lowest level maximal
$\mathcal{H}_{0}$ subpaths of $\mu_{k}$ (and of the resulting loops) up one
level by homotopies with image above level $B$ and (hence avoiding $C$) until
a loop in a single level is obtained. This loop is homotopically trivial in
that level. This completes the proof that if $H_{0}$ is 1-ended then $G$ is
simply connected at $\infty$.

Now we turn to the semistability part of our theorem. The next lemma allows us
to replace a path with labeling $(t^{-1},\bar{\alpha},t)$ (where
$\bar{\alpha}$ is an $\mathcal{H}_{0}$ word) with a path comprised of subpaths
either in the level above the level of $\alpha$ or in the compliment of an
arbitrary compact set $E$. In Figure 4 the path $(e^{-1},\alpha,d)$ is
homotopic to $(\gamma_{1}^{\prime},c^{-1},\tau_{2},b,\gamma_{2}^{\prime-1})$
where $p(\gamma_{i}^{\prime})=p(\alpha)+1$ and the path $(c^{-1},\tau_{2},b)$
has image in $\Gamma-E$.

###### Lemma 4.5

(UpOut) Suppose:

  * $A.$

$D$ is a compact subcomplex of $\Gamma$ such that for each vertex $v\in\Gamma$
the space $v\Gamma_{0}-D$ is a union of unbounded path components, and

  * $B.$

$\beta=(e^{-1},\alpha,d)$ is an edge path of $\Gamma$ such that
$\bar{\beta}=t^{-1}\bar{\alpha}t$, $e$ and $d$ are edges of $\Gamma-D$,
$\alpha$ is an $\mathcal{H}_{0}$ path and the end points of $\alpha$ can be
joined by a path $\alpha^{\prime}$ in $\Gamma-D$ such that
$p(\alpha^{\prime}(t))\leq p(\alpha)$ for all $t$.

Then, for any compact set $E\subset\Gamma$ such that $D\subset E$, there is a
homotopy $M$ of $\beta$ to a path
$(\gamma_{1}^{\prime},c^{-1},\tau_{2},b,\gamma_{2}^{\prime-1})$ such that:

  1. 1.

$\tau_{2}$, $\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ are
$\mathcal{H}_{0}$ paths,

  2. 2.

$b$ and $c$ are $t$ edges,

  3. 3.

$(c^{-1},\tau_{2},b)$ has image in $\Gamma-E$,

  4. 4.

for $i\in\\{1,2\\}$, the end points of $\gamma_{i}^{\prime}$ can be connected
by an edge path $\gamma_{i}^{\prime\prime}$ with image in $\Gamma-D$ such that
$p(\gamma_{i}^{\prime\prime}(t))\leq p(\gamma_{i}^{\prime})$ for all $t$, and

  5. 5.

$p(M)$ has image in $[p(\alpha),p(\alpha)+1]$.

Proof: Suppose $\rho_{1}$ and $\rho_{2}$ are paths with the same end points.
We use the notation $\rho_{1}\buildrel K\over{\sim}\rho_{2}$ when $K$ is a
homotopy of $\rho_{1}$ to $\rho_{2}$ relative to end points. Say the initial
(terminal) vertex of $\alpha$ is $v$ ($w$). Since $v\Gamma_{1}\subset
v\Gamma_{0}$ and $w\in v\Gamma_{0}-D$, hypothesis (A.) implies that there is
an $\mathcal{H}_{0}$ path $\tau_{1}$ at $w$ with image in $v\Gamma_{0}-D$ such
that the end point $z$ of $\tau_{1}$ belongs to
$v\Gamma_{1}-St^{N_{\ref{FIclose}}+1}(E)$. (See Figure 4.) Let $\gamma_{1}$ be
an $\mathcal{H}_{1}$ path from $v$ to $z$ and $K_{1}$ a homotopy in
$v\Gamma_{0}$ realizing:

$(\alpha,\tau_{1})\buildrel K_{1}\over{\sim}\gamma_{1}.$

Lemma 3.2 implies there is an $\mathcal{H}_{0}$ edge path $\tau_{2}$ of length
$\leq N_{\ref{FIclose}}$ from $z$ to $y\in w\Gamma_{1}$. Let $\gamma_{2}$ be
an $\mathcal{H}_{1}$ path from $w$ to $y$ and $K_{2}$ a homotopy in
$w\Gamma_{0}$ realizing:

$(\tau_{1},\tau_{2})\buildrel K_{2}\over{\sim}\gamma_{2}.$

Let $c$ and $b$ be the $t$ edges at $z$ and $y$ respectively. Since
$z\in\Gamma-St^{N_{\ref{FIclose}}+1}(E)$ we have that $(c^{-1},\tau_{2},b)$
has image in $\Gamma-E$. At this point conclusions (2.) and (3.) are
satisfied. By Lemma 4.1, there are $\mathcal{H}_{0}$ paths
$\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ such that:

$(e^{-1},\gamma_{1},c)\buildrel L_{1}\over{\sim}\gamma_{1}^{\prime}\hbox{ and
}(d^{-1},\gamma_{2},b)\buildrel L_{2}\over{\sim}\gamma_{2}^{\prime}$

with $p(L_{i})\subset[p(\alpha),p(\alpha)+1]$. Conclusion (1.) is satisfied.
Hypothesis (B.) and the definitions of $\tau_{1}$ and $\tau_{2}$ imply that
$\gamma_{1}^{\prime\prime}=(e^{-1},\alpha^{\prime},\tau_{1},c)$ is a path in
$\Gamma-D$ connecting the end points of $\gamma_{1}^{\prime}$, and
$\gamma_{2}^{\prime\prime}=(d^{-1},\tau_{1},\tau_{2},b)$ is a path in
$\Gamma-D$ connecting the end points of $\gamma_{2}^{\prime}$. Conclusion (4.)
is satisfied. Combining $K_{i}$ and $L_{i}$ to form the homotopy $M_{i}$ we
have:

$(e^{-1},\alpha,\tau_{1},c)\buildrel M_{1}\over{\sim}\gamma_{1}^{\prime}\hbox{
and }(d^{-1},\tau_{1},\tau_{2},b)\buildrel
M_{2}\over{\sim}\gamma_{2}^{\prime}$

with $p(M_{i})=[p(\alpha),p(\alpha)+1]$.

-1

Figure 4: Sliding $\beta=(e^{-1},\alpha,d)$ up and out

The homotopy $M_{2}$ implies that:

$\tau_{1}\buildrel
M_{2}^{\prime}\over{\sim}(d,\gamma_{2}^{\prime},b^{-1},\tau_{2}^{-1}).$

Substituting for $\tau_{1}$ in $M_{1}$ we have:

$(e^{-1},\alpha,d,\gamma_{2}^{\prime},b^{-1},\tau_{2}^{-1},c)\sim\gamma_{1}^{\prime}$

and so:

$(e^{-1},\alpha,d)\buildrel
M\over{\sim}(\gamma_{1}^{\prime},c^{-1},\tau_{2},b,\gamma_{2}^{\prime-1})$

where $p(M)=[p(\alpha),p(\alpha)+1]$, so that (5.) is satisfied. $\Box$

Use of the next lemma is one of the primary difference between the proof of
(Theorem 3.1, [Mih85]) and the current argument. It is one of the main
ingredients in the proof of Theorem 1.4. If one thinks of $t$-labeled edges of
$\Gamma$ as vertical and $\mathcal{H}_{0}$ labeled edges as horizontal, then
Lemma 4.6 explains how to push a vertical edge $e$ that is “far enough” from a
compact set $C\subset\Gamma$, to a vertical edge $e_{1}$ that is “arbitrarily
far” from $C$, by a homotopy that misses $C$.

###### Lemma 4.6

(push) Given a compact subcomplex $C$ of $\Gamma$ there is a compact
subcomplex $D_{\ref{push}}(C)$ of $\Gamma$ containing
$St^{N_{0}+N_{\ref{SSL1}}}(C)$ such that for any third compact set $E$ and any
edge $e$ labeled $t$ in $\Gamma-D_{\ref{push}}(C)$, there is an edge path
$(\alpha,e_{1},\beta)$ from $e(0)$ to $e(1)$ such that $\alpha$ and $\beta$
are $\mathcal{H}_{0}$ paths, the edge $e_{1}$ has image in $\Gamma-E$ and
label $t$, and $e$ is homotopic to $(\alpha,e_{1},\beta)$ by a homotopy with
image in $\Gamma-C$.

Proof: Note that if $w$ and $v$ are vertices of $\Gamma$ and $w\in
v\Gamma_{0}$ then $w\Gamma_{0}=v\Gamma_{0}$. For $K$ a subcomplex of $\Gamma$,
we take $St(K)$ in $\Gamma$. (See $(\ast)$ for the definition of $N_{0}$.)
There are only finitely many $v\Gamma_{0}$ that intersect
$St^{N_{\ref{SSL1}}+N_{0}}(C)$. Choose $D$ a finite subcomplex of $\Gamma$
such that if $v\Gamma_{0}\cap St^{N_{0}+N_{\ref{SSL1}}}(C)\neq\emptyset$, then
$St^{N_{0}+N_{\ref{SSL1}}}(C)$ union all bounded components of
$v\Gamma_{0}-St^{N_{0}+N_{\ref{SSL1}}}(C)$ belong to $D$ (see Theorem 2.4).
Let $v=e(0)$. If $v\in\Gamma-D$ then by the definition of $D$, there are edge
paths in $v\Gamma_{0}-St^{N_{0}+N_{\ref{SSL1}}}(C)$ from $v$ to vertices
arbitrarily far from $C$. In particular there is an edge path $\alpha_{1}$ in
$v\Gamma_{0}-St^{N_{0}+N_{\ref{SSL1}}}(C)$ from $v$ to $w_{1}\in
v\Gamma_{0}-St^{N_{\ref{FIclose}}+1}(E\cup D)$. By Lemma 3.2 there is an edge
path $\alpha_{2}$ (of length $\leq N_{\ref{FIclose}}$) from $w_{1}$ to $w\in
v\Gamma_{1}-St(E\cup D)$. In particular, $\alpha_{2}$ avoids $D$. Then
$\alpha=(\alpha_{1},\alpha_{2})$ has image in
$v\Gamma_{0}-St^{N_{0}+N_{\ref{SSL1}}}(C)$. By Lemma 4.2, $\alpha$ is
homotopic to an edge path $\alpha^{\prime}$ (with image in $v\Gamma_{1}$) by a
homotopy in $v\Gamma_{0}-St^{N_{0}}(C)$. Let $e_{1}$ be the edge labeled $t$
at $w$. Since $w\in v\Gamma_{0}-St(E)$, $e_{1}$ avoids $E$. Combining
conjugation 2-cells (one for each edge of $\alpha^{\prime}$), the path
$\alpha^{\prime}$ is homotopic to an edge path $(e,\hat{\beta},e_{1}^{-1})$ by
a homotopy in $St^{N_{0}}(im(\alpha^{\prime}))\subset\Gamma-C$. Let
$\beta=\hat{\beta}^{-1}$. $\Box$

Lemma 4.6 allows us to focus on edge paths in $v\Gamma_{0}$ ($v\in G$) with
end points outside of a compact set $E$. Theorem 2.4 implies that for any
compact set $E\subset\Gamma$ and $v\in G$, $E$ union all bounded components of
$v\Gamma_{0}-E$ is compact. Since $E$ only intersects finitely many
$v\Gamma_{0}$ non-trivially, $E$ is contained in a compact set $E^{\prime}$
such that for any $v\in G$ the set $v\Gamma_{0}-E$ is a union of unbounded
components.

The point of the next lemma is to show the path $\gamma$ is homotopic to a
path comprised of several $H_{0}$ subpaths in $\Gamma-E$ and a path
$\gamma_{1}$ “one level higher” than the path $\gamma$. It will be inductively
applied in Lemma 4.9.

###### Remark 4.7

(Rslideup) Lemma 4.8 is valid when $F=\emptyset$. In this case,
$St^{k}(F)=\emptyset$ for all $k\geq 0$. We will applied this special case to
paths $\gamma$ in a level above the highest level of a certain compact set
$C\subset\Gamma$. In this case the homotopy $H$ of Lemma 4.8 avoids $C$ since
$p(H)$ has image above the highest level of $C$.

###### Lemma 4.8

(slideup) Suppose $E$ and $F$ are finite subcomplexes of $\Gamma$, such that
$E$ contains $St^{N_{\ref{SSL1}}+N_{0}}(F)$, and $g\Gamma_{0}-E$ is a union of
unbounded components for any $g\in G$. If $\gamma$ is an edge path in
$v\Gamma_{0}-St^{N_{\ref{SSL1}}+N_{0}}(F)$ with end points in $\Gamma-E$, then
there is a homotopy $H$ with image in $\Gamma-F$, of $\gamma$ to an edge path
$(\tau,e,\gamma_{1},e^{\prime-1},\tau^{\prime-1})$ where $e$ and $e^{\prime}$
are edges labeled $t$; $\tau$ and $\tau^{\prime}$ are $\mathcal{H}_{0}$ paths
in $\Gamma-E$; and $\gamma_{1}$ is an $\mathcal{H}_{0}$ path in $\Gamma-F$ (so
that $p(\gamma)+1=p(\gamma_{1})$) with end points in $\Gamma-E$. Furthermore
$p(H)$ has image in $[p(v),p(v)+1]$.

Proof: In Figure 5, the path $(\tau_{1},\tau_{2})$ will be our path $\tau$ and
combining the two homotopies $K_{1}$ and $K_{2}$ will give the homotopy $H$.
Let $v$ be the initial and $w$ the terminal vertex of $\gamma$. Since
$w\Gamma_{0}-E$ is a union of unbounded components, there is an
$\mathcal{H}_{0}$ path $\tau^{\prime}$ in $w\Gamma_{0}-E$ from $w$ to a vertex
$w_{1}\in w\Gamma_{0}-St(E)$. See Figure 5. There is an $\mathcal{H}_{0}$ path
$\tau_{1}$ in $v\Gamma_{0}-E$ from $v$ to $v_{1}\in
v\Gamma_{0}-St^{N_{\ref{FIclose}}+1}(E)$. Note that:

$v\Gamma_{0}=w\Gamma_{0}=v_{1}\Gamma_{0}=w_{1}\Gamma_{0}.$

-1

Figure 5: Sliding $\gamma$ upward to $\gamma_{1}$

By Lemma 3.2 there is an $\mathcal{H}_{0}$ path $\tau_{2}$ of length $\leq
N_{\ref{FIclose}}$ from $v_{1}$ to a vertex $v_{2}$ of $w_{1}\Gamma_{1}$. Note
that $im(\tau_{2})\subset v_{1}\Gamma-St(E)$ (and so $v_{2}\in
w_{1}\Gamma_{1}-St(E)$). By Lemma 4.2 the path
$\nu=(\tau_{2}^{-1},\tau_{1}^{-1},\gamma,\tau^{\prime}_{1})$ is homotopic to
an $\mathcal{H}_{1}$ path $\gamma^{\prime}$, by a homotopy $K_{1}$ with image
in $St^{N_{\ref{SSL1}}}(im(\nu))$. Since $E$ contains
$St^{N_{\ref{SSL1}}+N_{0}}(F)$ and $\gamma$ avoids
$St^{N_{\ref{SSL1}}+N_{0}}(F)$, the path $\nu$ avoids
$St^{N_{\ref{SSL1}}+N_{0}}(F)$. Hence $K_{1}$ avoids $St^{N_{0}}(F)$. In
particular, $\gamma^{\prime}$ avoids $St^{N_{0}}(F)$.

Let $e$ be the edge labeled $t$ at $v_{2}$ (with end point $v_{3}$) and let
$e^{\prime}$ be the edge labeled $t$ at $w_{1}$ (with end point $w_{2}$). By
Lemma 4.1, the path $\gamma^{\prime}$ is homotopic to
$(e,\gamma_{1},e^{\prime-1})$ (where $\gamma_{1}$ is an $\mathcal{H}_{0}$
path) by a homotopy $K_{2}$ with image in $St^{N_{0}}(im(\gamma^{\prime}))$.
Since $\gamma^{\prime}$ avoids $St^{N_{0}}(F)$, $K_{2}$ (and $\gamma_{1}$)
avoids $F$. Since $v_{2},w_{1}\in\Gamma-St(E)$, we have
$v_{3},w_{2}\in\Gamma-E$. Now combine $K_{1}$ and $K_{2}$ to produce $H$ and
let $\tau=(\tau_{1},\tau_{2})$ to finish the proof. $\Box$

###### Lemma 4.9

(slideC) Suppose $C$ and $E$ are finite subcomplexes of $\Gamma$ such that
$p(C)\subset[A,B]$ and $E$ contains $St^{(B-A+2)(N_{\ref{SSL1}}+N_{0})}(C)$.
Also assume that if $g\in G$ then $g\Gamma_{0}-E$ is a union of unbounded
components. If $\gamma$ is an edge path in
$v\Gamma_{0}-St^{(B-A+2)(N_{\ref{SSL1}}+N_{0})}(C)$, with end points in
$\Gamma-E$ and $p(v)\in[A-1,B]$ then there is a homotopy of $\gamma$ (with
image in $\Gamma-C$) to a path in $\Gamma-E$.

Proof: Apply Lemma 4.8 to $\gamma$ with
$F=St^{(B-A+1)(N_{\ref{SSL1}}+N_{0})}(C)$. This gives a homotopy
$H_{1}^{\prime}$ with image in $\Gamma-F$ of $\gamma$ to
$(\beta_{1},\gamma_{1},\beta_{1}^{\prime})$ (here $\beta_{1}=(\tau,e)$ and
$\beta_{1}^{\prime}=(e^{\prime-1},\tau^{\prime-1})$) where $\beta_{1}$ and
$\beta_{1}^{\prime}$ have image in $\Gamma-E$ and $p(\gamma)+1=p(\gamma_{1})$.
Next, apply Lemma 4.8 to $\gamma_{1}$ with
$F=St^{(B-A)(N_{\ref{SSL1}}+N_{0})}(C)$ to produce
$(\beta_{2},\gamma_{2},\beta_{2}^{\prime})$. So after applying Lemma 4.8 at
most $B-A+2$ times, we have a homotopy $H^{\prime}$ with image in $\Gamma-C$
of $\gamma$ to $(\beta,\tilde{\gamma},\beta^{\prime})$ (the paths $\beta$ and
$\beta^{\prime}$ are the concatenations of the $\beta_{i}$ and
$\beta_{i}^{\prime}$ respectively) where $\beta$ and $\beta^{\prime}$ have
image in $\Gamma-E$ and $p(\tilde{\gamma})=B+1$. Of course the image of
$\tilde{\gamma}$ may intersect $E$, but $\tilde{\gamma}$ has image in level
$B+1$ (above $C$) and we may apply Lemma 4.8 to $\tilde{\gamma}$ with
$F=\emptyset$ (see Remark 4.7), to obtain a homotopy $\tilde{H}$ of
$\tilde{\gamma}$ to $(\alpha_{1},\tilde{\gamma}_{1},\alpha_{1}^{\prime})$
where $\alpha_{1}$ and $\alpha_{1}^{\prime}$ are $\mathcal{H}_{0}$ paths in
$\Gamma-E$, $p(\tilde{\gamma}_{1})=B+2$ and $\tilde{H}$ has image in
$[B+1,B+2]$ (and hence $\tilde{H}$ avoids $C$). Continue applying Lemma 4.8
(with $F=\emptyset$) to obtain $\alpha_{2},\ldots,\alpha_{n}$ and
$\alpha_{2}^{\prime},\ldots,\alpha_{n}^{\prime}$ all with image in $\Gamma-E$
and path $\tilde{\gamma}_{n}$ with $p(\tilde{\gamma}_{n})=B+n+1$ larger than
the largest integer in $p(E)$. Let $\alpha=(\alpha_{1},\ldots,\alpha_{n})$,
$\alpha^{\prime}=(\alpha_{1}^{\prime},\ldots,\alpha_{n}^{\prime})$. We have
$\gamma$ is homotopic by a homotopy with image in $\Gamma-C$ to
$(\beta,\alpha,\tilde{\gamma}_{n},\alpha^{\prime},\beta^{\prime})$, a path
with image in $\Gamma-E$. $\Box$

Let $C_{1},C_{2},\ldots$ be a collection of compact subcomplexes of $\Gamma$
such that $C_{i}$ is a subset of the interior of $C_{i+1}$ and
$\cup_{i=1}^{\infty}C_{i}=\Gamma$. If $a$ and $b$ are vertices of
$v\Gamma_{0}$ and $p(v)=n$, then define the degree of $(a,b)$ (written
$deg(a,b)$) to be the largest integer $m$ such that there is an edge path
$\alpha$ in $\Gamma-C_{m}$ from $a$ to $b$ such that
$p(\alpha)\subset(-\infty,n]$.

The next result is in analogy with Lemma 4.4 of [Mih85]. The proof is
elementary, but it is one of the main ideas of the overall argument. The other
lemmas used in the proof of Theorem 1.4 explain how to carefully slide $H_{0}$
paths upward and $t$-edges horizontally. Lemma 4.10 will allow us to slide
certain $H_{0}$ paths downward. It is in the proof of Lemma 4.10 where we need
$H_{0}$ to be finitely presented (as opposed to only finitely generated).

###### Lemma 4.10

(Like4.4) Suppose $D$ is a compact subcomplex of $\Gamma$ and $W$ and $L$ are
unbounded path components of $v\Gamma_{0}-D$, (we allow $W=L$ as a
possibility) then one of the following two statements holds:

(i) The set $W\times L$ contains a collection of pairs of vertices
$(w_{1},l_{1})$, $(w_{2},l_{2}),\ldots$ such that all $w_{j}$ are distinct,
all $l_{j}$ are distinct and $deg(w_{j},l_{j})\geq j$, or

(ii) There are finite sets of vertices $S\subset W$ and $T\subset L$ such that
all pairs of vertices of $(W-S)\times(L-T)$ have degree less than some fixed
positive integer $N_{1}(W,L)$.

Proof: Begin selecting pairs of vertices $(w_{j},l_{j})$ satisfying the
hypothesis of $(i)$. If for some $j\geq 1$, $(w_{j},l_{j})$ cannot be selected
to satisfy hypothesis $(i)$, then all pairs of vertices of
$(W-\\{w_{1},w_{2},\ldots,w_{j-1})\times(L-\\{l_{1},l_{2},\ldots,l_{j-1})$
have degree less than $j$ (so that $(ii)$ is satisfied). $\Box$

We are ready to combine our lemmas to prove the semistability part of Theorem
1.4. Let $C$ be a finite subcomplex of $\Gamma$ and $r$ be the proper ray at
$\ast$, each of whose edges is labeled $t$. Choose integers $A$ and $B$ such
that

$p(C)\subset[A,B].$

Figure 6: The General Setup

Let $D_{1}$ be a finite subcomplex of $\Gamma$ containing
$St^{(B-A+2)(N_{\ref{SSL1}}+N_{0})}(C)$. Furthermore, if $v\Gamma_{0}$
intersects $D_{1}$ non-trivially and $p(v)=A-1$ (there are only finitely many
of these), then we require that all components of $v\Gamma_{0}-D_{1}$ are
unbounded. Next we define a finite subcomplex $D_{2}$ of $\Gamma$. Our first
requirement is that $D_{2}$ contains $D_{\ref{push}}(D_{1})$ (see Figure 6).
Suppose $v\Gamma_{0}$ intersects $D_{1}$ non-trivially, $p(v)=A-1$ and that
$W$ and $L$ are unbounded components of $v\Gamma_{0}-D_{1}$ satisfying Lemma
4.10 (ii) (where $D_{1}$ plays the role of $D$). Then we require that $D_{2}$
contains the finite sets $S\subset W$ and $T\subset L$ and the set
$C_{N_{1}(W,L)}$ given by Lemma 4.10 (recall that the $C_{i}$ are a cofinal
sequence of finite subcomplexes of $\Gamma$). We also require that for each
vertex $v\in\Gamma$ the space $v\Gamma_{0}-D_{2}$ is a union of unbounded
components. An important fact follows immediately from Lemma 4.10:

$(\ast)$ Suppose $v\Gamma_{0}$ intersects $D_{1}$ non-trivially, $p(v)=A-1$,
and $\alpha$ is an edge path in $\Gamma-D_{2}$ that begins and ends in
$v\Gamma_{0}$ such that $p(\alpha)\subset(-\infty,A-1]$. If the initial vertex
of $\alpha$ belongs to the (unbounded) component $W$ of $v\Gamma_{0}-D_{1}$
and the terminal vertex of $\alpha$ belongs to the (unbounded) component $L$
of $v\Gamma_{0}-D_{1}$, then $W\times L$ satisfies condition $(i)$ of Lemma
4.10.

Assume

$p(D_{2})\subset[A^{\prime},B^{\prime}]\hbox{ and }r([0,B^{\prime}])\subset
D_{2}.$

Let $E$ be an arbitrary finite subcomplex of $\Gamma$ (containing $D_{2}$).
Enlarge $E$ so that for any vertex $v\in\Gamma$, $v\Gamma_{0}-E$ is a union of
unbounded path components. If $\gamma$ is an edge path loop in $\Gamma-D_{2}$
and based on $r$, our goal is to show that $\gamma$ is homotopic $rel\\{r\\}$
to an edge path loop in $\Gamma-E$ by a homotopy in $\Gamma-C$ (see Theorem
2.1(2)). Say $\gamma(0)=r(K)$ for some $K>B^{\prime}$. If $\tau$ is an edge
path ending at $r(K)$, each of whose edges is labeled $t^{-1}$, then $\gamma$
is homotopic $rel\\{r\\}$ by a homotopy in $\Gamma-D_{2}$ to
$(\tau,\gamma,\tau^{-1})$ (see Figure 6). In particular, we may assume:

$p(E)\subset[A^{\prime\prime},B^{\prime\prime}])\hbox{ and
}\gamma(0)=r(K)\hbox{ for }K>B^{\prime\prime}.$

We cannot assume that $\gamma$ is cyclically without backtracking. It may be
that $\gamma$ has the form $(\tau,\gamma^{\prime},\tau^{\prime})$ where
cyclically $\tau$ and $\tau^{\prime}$ backtrack. If we eliminating $\tau$ and
$\tau^{\prime}$ our path $\gamma^{\prime}$ may not be based on $r$.
Unfortunately, we cannot apply Lemmas 4.3 or 4.4. Instead, we will apply Lemma
4.5 to $\gamma$, $D_{2}$ and $E$. Repeated applications of this lemma allow us
to move the maximal $\mathcal{H}_{0}$ subpaths of $\gamma$ that lie below
level $A-1$ to level $A-1$ (modulo some paths in $\Gamma-E$). Notice that if
$\alpha$ is a maximal $H_{0}$ subpath of $\gamma$ in a lowest level $L<A-1$ of
$\gamma$, then $\alpha$ has image in $\Gamma-D_{2}$ so that hypotheses (A.)
and (B.) of Lemma 4.5 are satisfied. Apply Lemma 4.5 to all maximal
$\mathcal{H}_{0}$ subpaths of $\gamma$ in level $L$. If $M$ is one of these
homotopies, then $p(M)\subset[L,L+1]$ and so avoids $C$. Say $\hat{\gamma}$ is
the resulting path. Then $\gamma$ is homotopic to $\hat{\gamma}$ by a homotopy
that avoids $C$. The only subpaths of $\hat{\gamma}$ in level $L$ have image
in $\Gamma-St(E)$. If $\alpha$ is a maximal $H_{0}$ subpath of $\hat{\gamma}$
in level $L+1$ then $\alpha$ is a concatenation of $H_{0}$ subpaths of
$\gamma$ (which have image in $\Gamma-D_{2}$) and paths of the form
$\gamma_{i}^{\prime}$ arising from Lemma 4.5. As the end points of these
$\gamma_{i}^{\prime}$ can be connected by edge paths in $\Gamma-D_{2}$ in
levels $\leq L+1$, we conclude that if $\hat{\alpha}$ is a maximal
$\mathcal{H}_{0}$ subpath of $\hat{\gamma}$ in level $L+1$ , then the end
points of $\hat{\alpha}$ can be joined by a path $\beta$ in $\Gamma-D_{2}$
such that $p(\beta(t))\leq L+1$ for all $t$. If $e$ and $d$ are the $t$ edges
at the initial and terminal vertex of $\hat{\alpha}$ then $e$ ($d$) is either
an edge of $\gamma$ or is in $\Gamma-E$. In either case, $e$ and $d$ avoid
$D_{2}$. Hence Lemma 4.5 can be applied to $(e^{-1},\hat{\alpha},d)$. Repeated
applications of Lemma 4.5 shows:

Fact 1. There is a homotopy $\gamma{\buildrel M_{1}\over{\sim}}\gamma_{1}$
with $p(M_{1})\subset\Gamma-C$ such that for any vertex $v$ of $\gamma_{1}$
either $p(v)\in[A-1,\infty)$ or $v\in\Gamma-E$, the homotopy $M_{1}$ fixes the
edges of $\gamma$ in levels $A-1$ and above, and the only edges of
$\gamma_{1}$ in levels $A$ and above are edges of $\gamma$ (and hence these
edges avoid $D_{2}$). Any $t$-edge of $\gamma_{1}$ not in $\Gamma-E$ is also a
$t$-edge of $\gamma$ and so is in $\Gamma-D_{2}$. Furthermore:

$(\ast\ast)$ Suppose $\tau$ is a maximal $\mathcal{H}_{0}$ subpath of
$\gamma_{1}$ with $p(\tau)=A-1$. Let $v_{0}$ and $v_{1}$ be the end points
$\tau$. Then Lemma 4.5 implies there is an edge path $\beta$ with
$im(\beta)\subset\Gamma-D_{2}$ connecting $v_{0}$ to $v_{1}$ with
$p(\beta)\subset(-\infty,A-1]$. If $W$ ($L$) is the component of
$v\Gamma_{0}-D_{1}$ containing $v_{0}$ ($v_{1}$) then $(\ast)$ implies
$W\times L$ satisfies condition $(i)$ of Lemma 4.10.

Figure 7: Moving $t$-Edges Horizontally

The next step is to move all $t$ edges of $\gamma_{1}$ outside of $E$. Apply
Lemma 4.6 (with $C=D_{1}$ and $D=D_{2}$) to each $t$-edge of $\gamma_{1}$ not
already in $\Gamma-E$, to obtain a homotopy $\gamma_{1}{\buildrel
M_{2}\over{\sim}}\gamma_{2}$ in $\Gamma-D_{1}$ where each $t$-edge of
$\gamma_{2}$ has image in $\Gamma-E$. It is important to observe that if $e$
is a $t$ edge of $\gamma_{1}$ connecting levels $A-1$ and $A$, then the
homotopy $M_{2}$ uses Lemma 4.6 to replace $e$ by a path
$(\alpha,e_{1},\beta)$ in $\Gamma-D_{1}$ connecting the end points of $e$ such
that $p(\alpha)=A-1$, $p(\beta)=A$ and $e_{1}$ (in $\Gamma-E$) has label $t$.
Since $\alpha$ avoids $D_{1}$, there is an analogue of $(\ast\ast)$ for
$\gamma_{2}$:

$(\overline{\ast\ast})$ If $\tau$ is a maximal subpath of $\gamma_{2}$ in
level $A-1$, then $\tau=(\alpha_{0}^{-1},\tau^{\prime},\alpha_{1})$ where
$\tau^{\prime}$ is a maximal subpath of $\gamma_{1}$ in level $A-1$ and each
of the paths $\alpha_{0}$ and $\alpha_{1}$ is either trivial (if the
corresponding end point of $\tau^{\prime}$ is already in $\Gamma-E$) or comes
from our replacement of edges labeled $t$ at the end points of $\tau^{\prime}$
(see Figure 7). So $\alpha_{0}$ and $\alpha_{1}$ are in level $A-1$, avoid
$D_{1}$ and both end in $\Gamma-E$. By $(\ast\ast)$ the end points of
$\tau^{\prime}$ are in unbounded components $W$ and $L$ of $v\Gamma_{0}-D_{1}$
and $W\times L$ satisfies condition $(i)$ of Lemma 4.10. As $\alpha_{0}$ and
$\alpha_{1}$ avoid $D_{1}$, the end points of $\tau$ are in $W$ and $L$ as
well.

Figure 8: Moving $\tau$ Down and Out

Again let $\tau$ (with end points $v_{0}$ and $v_{1}$) be a maximal subpath of
$\gamma_{2}$ in level $A-1$. Then $v_{0},v_{1}\in v_{0}\Gamma_{0}-E$. If
$W_{i}$ is the component of $v_{0}\Gamma_{0}-D_{1}$ containing $v_{i}$ (for
$i\in\\{0,1\\}$) then ($\overline{\ast\ast}$) implies that $W_{0}\times W_{1}$
satisfies condition $(i)$ of Lemma 4.10. Choose $N>1$ such that $E\subset
C_{N}$ and vertex pairs $(w_{0},w_{1})\in W_{0}\times W_{1}$ such that
$deg(w_{0},w_{1})\geq N$. For $i\in\\{0,1\\}$ let $\psi_{i}$ be an edge path
in $W_{i}$ from $v_{i}$ to $w_{i}$. Let $\mu$ be an edge path from $w_{0}$ to
$w_{1}$ in $\Gamma-C_{M}$ ($\subset\Gamma-E$) such that
$p(\mu)\subset(-\infty,A-1]$ (see Figure 8). Consider the edge path loop
$\rho=(\tau,\psi_{1},\mu^{-1},\psi_{0}^{-1})$ (in levels $\leq A-1$). The loop
$\rho$ is homotopic (in the image of $\rho$) to a loop $\rho^{\prime}$ without
backtracking. Lemma 4.3 can then be applied to $\rho^{\prime}$. After a
sequence of backtracking elimination and applications of Lemma 4.3 the
resulting loop has image in $v_{0}\Gamma_{0}$ and so is homotopically trivial
in level $A-1$. In particular $\tau$ is homotopic to
$(\psi_{0},\mu,\psi_{1}^{-1})$ by a homotopy in levels $\leq A-1$ and hence in
$\Gamma-C$. Since $\mu$ has image in $\Gamma-E$ and (for $i\in\\{1,2\\}$)
$\psi_{i}$ is an edge path in level $A-1$ with end points in $\Gamma-E$ and
image in $\Gamma-D_{1}$, we have shown:

$(\dagger)$ The path $\gamma$ is homotopic $rel\\{r\\}$ to an edge path loop
$\gamma_{3}$ in $\Gamma-D_{1}$ by a homotopy in $\Gamma-C$ such that each
$t$-edge of $\gamma_{3}$ has image in $\Gamma-E$ and each edge of $\gamma_{3}$
in any level $<A-1$ is in $\Gamma-E$.

Next suppose $\alpha$ is a maximal subpath of $\gamma_{3}$ in level
$L\in[A-1,B]$. The end points of $\alpha$ are in $\Gamma-E$ and $\alpha$ has
image in $\Gamma-D_{1}$. By Lemma 4.9, the path $\alpha$ is homotopic
$rel\\{0,1\\}$ to an edge path in $\Gamma-E$ by a homotopy in $\Gamma-C$.
Apply Lemma 4.9 to all such $\alpha$. The resulting path, call it
$\gamma_{4}$, is homotopic $rel\\{0,1\\}$ to $\gamma_{3}$ by a homotopy in
$\Gamma-C$. Notice that each edge of $\gamma_{4}$ in a level $\leq B$ is in
$\Gamma-E$. At this point we need only be concerned about maximal subpaths of
$\gamma_{4}$ which are in a level above level $B$ (and hence above the compact
set $C$). Repeated applications of Lemma 4.8 to such paths with $F=\emptyset$
(see Remark 4.7) move such paths out of $E$ by homotopies in $\Gamma-C$. This
finishes the proof that $G$ has semistable fundamental group at $\infty$.
$\Box$

##  5 The Proof of the Main Theorem

Before proving Theorem 1.3 we prove Lemma 5.1 (by an argument that parallels
the proof of Theorem 3 of [Mih86b]).

###### Lemma 5.1

(SS2) Suppose $G=A\ast_{C}B$ or $G=A\ast_{C}$, where $A$ and $B$ are finitely
presented, $C$ is infinite, finitely generated and in the first case, of
finite index in $B$. If $A$ is 1-ended and semistable at $\infty$, then $G$ is
1-ended and semistable at $\infty$.

Proof: We consider the case $G=A\ast_{C}B$ first. If $B=C$ or $A=C$, then $A$
has finite index in $G$ and so $G$ is 1-ended and semistable at $\infty$
(since semistability and the number of ends of a group are quasi-isometry
invariants). We assume $A\neq C\neq B$. Let $\mathcal{A}\cup\mathcal{B}$ be a
finite generating set for $G$, where $\mathcal{A}$ generates $A$,
$\mathcal{B}$ generates $B$ and $\mathcal{A}\cap\mathcal{B}=\mathcal{C}$
generates $C$. Let $\mathcal{P}$ be a finite presentation for $G$ with
generators $\mathcal{A}\cup\mathcal{B}$. Let $\Gamma$ be the Cayley 2-complex
for $\mathcal{P}$. We show $\Gamma$ has semistable fundamental group at
$\infty$. Let $\Gamma_{H}\subset\Gamma$ be the corresponding Cayley graph of
$H=A$, $B$ or $C$ with respect to $\mathcal{A}$, $\mathcal{B}$, and
$\mathcal{C}$ respectively. Choose a geodesic edge path line
$\ell=(\ldots,c_{-1},c_{0},c_{1},\ldots)$ in $\Gamma_{C}$ (so that each edge
$c_{i}$ has label $\bar{c}_{i}\in\mathcal{C}^{\pm 1}$. Assume the initial
vertex of $c_{0}$ is $\ast$, the identity vertex of $\Gamma_{C}$). Let
$r^{+}=(c_{0},c_{1},\ldots)$ and $r^{-}=(c_{-1}^{-1},c_{-2}^{-1},\ldots)$ be
the two (opposite direction) geodesic rays (beginning at $\ast$) of $\ell$.
Recall that the vertices of $\Gamma$ are the elements of $G$. Given any
compact set $D\subset\Gamma$ there are only finitely many vertices
$v\in\Gamma$ such that both $vr^{+}$ and $vr^{-}$ intersect $D$.

Let $s=(s_{0},s_{1},\ldots)$ be a proper edge path ray in $\Gamma$ with
initial point $\ast$. Then the label of $s_{i}$ is
$\bar{s}_{i}\in\mathcal{A}^{\pm 1}\cup\mathcal{B}^{\pm 1}$. Let
$\ast=w_{0},w_{1},\ldots$ be the consecutive vertices of $s$. For any vertex
$v\in\Gamma$ let $B(v,N)$ be the ball of radius $N$ about $v$ in the
1-skeleton of $\Gamma$. Select $r_{0}=r^{+}$, and for $i>0$, set $r_{i}$ equal
to $w_{i}r^{+}$ or $w_{i}r^{-}$ so that for any integer $N$ only finitely many
$r_{i}$ intersect $B(\ast,N)$. (See Figure 9.)

Figure 9: Stacking homotopies

If $\bar{s}_{i}\in\mathcal{A}^{\pm 1}$ then the rays $r_{i}$ and
$(s_{i},r_{i+1})$ are both $\mathcal{A}$-rays and so are properly homotopic
$rel\\{w_{i}\\}$ by a homotopy $K_{i}$ in $w_{i}\Gamma_{A}$ (see Theorem
2.1(4)). There may be infinitely many $j$ such that
$w_{j}\Gamma_{A}=w_{i}\Gamma_{A}$. Let $J_{i}=\\{j\geq
0:w_{j}\Gamma_{A}=w_{i}\Gamma_{A}\hbox{ and }\bar{s}_{j}\in\mathcal{A}^{\pm
1}\\}$. Theorem 2.1(3) implies that the set of all $K_{j}$ such that $j\in
J_{i}$, can be selected so that only finitely many have image which intersects
any given compact subset of $\Gamma$. Since only finitely many translates of
$\Gamma_{A}$ intersect a given compact set, we may assume that: If
$\bar{s}_{i}\in\mathcal{A}^{\pm 1}$ then $K_{i}$ is defined and for any
compact set $D$, only finitely many such $K_{i}$ have image that intersects
$D$.

If $\bar{s}_{k}\in\mathcal{B}^{\pm 1}$ then the ray $(s_{k}^{-1},r_{k})$ is a
$\mathcal{B}$ ray. Since $C$ has finite index in $B$, there is a proper
$\mathcal{C}$ ray $q_{k+1}$ at $w_{k+1}$ that is properly homotopic
$rel\\{w_{k+1}\\}$ to $(s_{k}^{-1},r_{k})$ by a homotopy $K_{k}^{\prime}$
whose image is within $N_{\ref{SSL1}}+N_{\ref{FIclose}}$ of the image of
$r_{k}$. (Let $\beta_{0}$ be the trivial path at $w_{k+1}$. If $e_{i}$ is the
$i^{th}$ edge of $(s_{k}^{-1},r_{k})$ then let $\beta_{i}$ be an edge path of
length $\leq N_{\ref{FIclose}}$ from the terminal point of $e_{i}$ to a vertex
of $w_{k+1}\Gamma_{C}$ . Apply Lemma 4.2 to the paths
$(\beta_{i-1}^{-1},e_{i},\beta_{i})$ and combine the resulting homotopies.)

Now $q_{k+1}$ and $r_{k+1}$ are both $\mathcal{A}$ (in fact, $\mathcal{C}$)
rays at $w_{k+1}$ and so are properly homotopic by a homotopy
$K_{k}^{\prime\prime}$ in $w_{k+1}\Gamma_{A}$. Again, Theorem 2.1(3) allows us
to choose the $K_{k}^{\prime\prime}$ such that for any compact set $D$, only
finitely many such $K_{k}^{\prime\prime}$ have image that intersect $D$.

Combine $K_{k}^{\prime}$ and $K_{k}^{\prime\prime}$ to form $K_{k}$, a proper
homotopy of $r_{k}$ to $(s_{k},r_{k+1})$. Combining the $K_{i}$ for $i\geq 0$,
provides a proper homotopy from $r^{+}$ to $s$ (see Figure 9). Hence every
proper edge path ray at $\ast$ is properly homotopic to $r^{+}$ and $G$ is
semistable at $\infty$ (see Theorem 2.1(4)). As any two proper rays in
$\Gamma$ are properly homotopic, the space $\Gamma$ (and hence the group $G$)
is 1-ended.

In the second case semistability follows directly from Theorem 1.1. It remains
to show that $A\ast_{C}$ is 1-ended. Let $\mathcal{C}\subset\mathcal{A}$ be
finite generating sets for $C$ and $A$ respectively. Let $t$ be the stable
letter and for each $c\in\mathcal{C}$, let $w_{c}$ be an $\mathcal{A}$ word
such that $t^{-1}ct=w_{c}$ is a conjugation relation for $G$. Let $\Gamma$ be
the Cayley graph of $G$ with respect to $\mathcal{A}\cup\\{t\\}$ and let
$\Gamma_{C}\subset\Gamma$ be the Cayley graph of $C$ with respect to
$\mathcal{C}$. Again choose a geodesic edge path line
$\ell=(\ldots,c_{-1},c_{0},c_{1},\ldots)$ in $\Gamma_{C}$, so that each edge
$c_{i}$ has label $\bar{c}_{i}\in\mathcal{C}^{\pm 1}$. Assume the initial
vertex of $c_{0}$ is $\ast$, the identity vertex of $\Gamma_{C}$. Let
$r^{+}=(c_{0},c_{1},\ldots)$ and $r^{-}=(c_{-1}^{-1},c_{-2}^{-1},\ldots)$ be
the two (opposite direction) geodesic rays (beginning at $\ast$) of $\ell$.
Given any compact set $D\subset\Gamma$ there are only finitely many vertices
$v\in\Gamma$ such that both $vr^{+}$ and $vr^{-}$ intersect $D$. Choose $E(D)$
a finite subcomplex of $\Gamma$ such that if $v$ is a vertex of $\Gamma-E$
then either $vr^{+}$ or $vr^{-}$ avoids $D$. It is enough to show that any two
vertices $v$ and $w$ in $\Gamma-E$ can be joined by an edge path in
$\Gamma-D$. Let $r_{v}$ be $vr^{+}$ or $vr^{-}$ avoiding $D$. Similarly select
$r_{w}$ avoiding $D$. It is enough to show that $r_{v}$ and $r_{w}$ converge
to the same end of $\Gamma$. Let $\alpha=(a_{0},a_{1},\ldots,a_{n})$ be an
edge path from $v$ to $w$. Assume $a_{i}$ has label
$\bar{a}_{i}\in\mathcal{A}^{\pm 1}\cup\\{t^{\pm 1}\\}$). For
$i\in\\{1,\ldots,n\\}$ let $v_{i}$ be the initial vertex of $a_{i}$ and let
$r_{i}$ be $v_{i}r^{+}$. Let $v_{0}=v$, $v_{n+1}=w$, $r_{0}=r_{v}$ and
$r_{n+1}=r_{w}$. It is enough to show that $r_{i}$ and $r_{i+1}$ converges to
the same end of $\Gamma$ for $i\in\\{0,\ldots,n\\}$. Choose
$i\in\\{0,1,\ldots,n\\}$. If $\bar{a}_{i}\in\mathcal{A}^{\pm 1}$ then the
proper rays $r_{i}$ and $(a_{i},r_{i+1})$ form a proper line in the 1-ended
space $v_{i}\Gamma_{1}$. Hence $r_{i}$ and $(a_{i},r_{i+1})$ converge to the
same end of $v_{i}\Gamma_{A}$. In particular, $r_{i}$ and $r_{i+1}$ converge
to the same end of $\Gamma$. If $\bar{a}_{i}=t$ then write
$r_{i}=(e_{1},e_{2},\ldots)$. Let $t_{k}$ be the edge labeled $t$ at the
initial vertex of $e_{k}$. There is an $\mathcal{A}$ path $z_{j}$ (labeled by
one of the conjugation words $w_{k}^{\pm 1}$), connecting the end points of
$(t_{k}^{-1},e_{k},t_{k+1})$. As $(z_{1},z_{2},\ldots)$ tracks $r_{i}$, it is
proper and converge to the same end of $\Gamma$ as does $r_{i}$. Also
$(z_{1},z_{2},\ldots)$ and $r_{i+1}$ are in the 1-ended space
$v_{i+1}\Gamma_{A}$, so they converge to the same end of $\Gamma$. Hence
$r_{i}$ and $r_{i+1}$ converge to the same end of $\Gamma$. If
$\bar{a}_{i}=t^{-1}$ then consider the $t$ edges at the vertices of $r_{i+1}$
and proceed as before. $\Box$

Proof: (of Theorem 1.3) Suppose $\mathcal{G}$ is our graph of groups
decomposition of $G$. First a reduction. Suppose there is a vertex $v$ of
$\mathcal{G}$ such that $G_{v}$ is 1-ended and semistable at $\infty$. If
there is an edge (loop) $e(v,v)$ of $\mathcal{G}$, then the HNN-extension of
$G_{v}$ determined by $e(v,v)$ is 1-ended and semistable at $\infty$ by Lemma
5.1. In this situation, replace $v$ and $e(v,v)$ by a single vertex
$v^{\prime}$ with vertex group the HNN-extension of $G_{v}$. The resulting
graph of groups is still a decomposition of $G$ and satisfies the hypothesis
of our theorem. We may assume:

(0) If $v$ is a vertex of $\mathcal{G}$ and $G_{v}$ is 1-ended and semistable
at $\infty$, then no edge at $v$ is a loop.

Next, suppose there is a vertex $v$ of $\mathcal{G}$ such that $G_{v}$ is not
both 1-ended and semistable at $\infty$ and an edge (loop) $e(v,v)$ (giving an
HNN-extension). Then there are subgroups $G_{1}$ and $G_{2}$ of $G_{v}$ (both
isomorphic to $G_{e}$) and an isomorphism $\phi:G_{1}\to G_{2}$ determining a
subgroup of $G$ that is an HNN-extension of $G_{v}$. If either $G_{1}$ or
$G_{2}$ has finite index in $G_{v}$, then Theorem 1.4 implies that the
resulting HNN-extension (call it $G_{v^{\prime}}$) of $G_{v}$ is 1-ended and
semistable at $\infty$. In this case $v$ and the loop $e(v,v)$ are replaced by
$v^{\prime}$. If $\mathcal{G}$ has only 1-vertex $v$, then our hypotheses and
reductions imply that $\mathcal{G}$ cannot have an edge so that $G=G_{v}$. Our
hypotheses imply that $G$ is 1-ended and semistable at $\infty$. Assume the
result is true for $\mathcal{G}$ with $N\geq 1$ vertices. Now assume
$\mathcal{G}$ has $N+1$ vertices. Suppose $e(v,w)$ is an edge of $\Gamma$, and
the vertex group $G_{v}$ (of $\mathcal{G}$) is 1-ended and semistable at
$\infty$. Our reduction (0) implies that $v\neq w$. If $G_{e}$ has finite
index in $G_{w}$, then Lemma 5.1 implies that the subgroup of $G$ generated by
$G_{v}$ and $G_{w}$ is 1-ended and semistable at $\infty$. If $G_{w}$ is
1-ended and semistable at $\infty$ then Theorem 2.3 implies that the subgroup
of $G$ generated by $G_{v}$ and $G_{w}$ is 1-ended and semistable at $\infty$.
In either case, reducing $\mathcal{G}$ by combining $e$ into a single vertex
(with group generated by $G_{v}\cup G_{w}$) gives a graph of groups
decomposition of $G$ with fewer vertices and satisfying the hypothesis of our
theorem. Inductively, $G$ is 1-ended and semistable at $\infty$. So we may
assume:

(1) If $v$ is a vertex of $\mathcal{G}$ and $G_{v}$ is 1-ended and semistable
at $\infty$, then each edge $e(v,w)$ of $\mathcal{G}$ is such that $v\neq w$,
$G_{e}$ has infinite index in $G_{w}$, and $G_{w}$ is not 1-ended and
semistable at $\infty$.

Similarly, Theorems 1.4 and 2.2 imply that we may assume:

(2) No edge (loop) $e(v,v)$ of $\mathcal{G}$ is such that $G_{e}$ has finite
index in $G_{v}$ and no edge $e(v,w)$ with $v\neq w$ is such that $G_{e}$ has
finite index in both $G_{v}$ and $G_{w}$.

By (1), there is a vertex $v_{0}\in\mathcal{G}$ such that $G_{v_{0}}$ is not
1-ended and semistable at $\infty$. By our hypotheses there is an edge
$e_{0}=e(v_{0},v_{1})$ of $\mathcal{G}$ such that $G_{e_{0}}$ has finite index
in $G_{v_{0}}$. By (2), $v_{0}\neq v_{1}$ and $G_{e_{0}}$ has infinite index
in $G_{v_{1}}$. By (1), $G_{v_{1}}$ is not 1-ended and semistable at $\infty$.
By our hypothesis there is an edge $e_{1}=e(v_{1},v_{2})$ of $\mathcal{G}$
such that $G_{e_{1}}$ has finite index in $G_{v_{1}}$ (so $e_{1}\neq e_{0}$)
and by (2), $v_{1}\neq v_{2}$ and $G_{e_{1}}$ has infinite index in
$G_{v_{2}}$. By (1), $G_{v_{2}}$ is not 1-ended and semistable at $\infty$.
Continuing, we arrive at a simple (non-crossing) edge path loop $\mathcal{L}$
in $\mathcal{G}$ (relabeling if necessary) $e_{0},e_{1},\ldots,e_{n}$ where
$e_{i}=e(v_{i},v_{i+1})$, $v_{i}\neq v_{i+1}$, $v_{n+1}=v_{0}$, no $G_{v_{i}}$
is 1-ended and semistable at $\infty$, and $G_{e_{i}}$ has finite index in
$G_{v_{i}}$ and infinite index in $G_{v_{i+1}}$. It is enough to show the
fundamental group $L$ of this loop of groups is 1-ended and semistable at
$\infty$ (since then we could collapse the loop to a single vertex and the
resulting graph of groups decomposition of $G$ would have fewer vertices and
still satisfy the hypothesis of our theorem).

Let $A$ be the fundamental group of the graph of groups obtained by removing
$e_{n}$ from $\mathcal{L}$. We consider $L$ to be an HNN-extension of $A$,
with stable letter $t$ and $t^{-1}ct\in G_{v_{0}}$ for $c\in
G_{e_{n}}<G_{v_{n}}$. The group $C=G_{e_{0}}\cap G_{e_{1}}\cap\cdots\cap
G_{e_{n}}$ has finite index in $G_{v_{0}}$. The subgroup of $L$ generated by
$G_{v_{0}}$ and $t$ is an HNN extension with base $G_{0}$ and associated
subgroup $C$ (of finite index in $G_{0}$). Theorem 1.4 implies this subgroup
is finitely presented, 1-ended and semistable at $\infty$. Similarly each of
the vertex groups of $\mathcal{L}$ belong to a subgroup of $L$ that is
finitely presented, 1-ended and semistable at $\infty$. Repeated applications
of Theorem 2.3 imply that $L$ is 1-ended and semistable at $\infty$. $\Box$

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