Relatively Hyperbolic Groups with Semistable Peripheral Subgroups

We are interested in the asymptotic behavior of relatively hyperbolic groups. We consider a property of finitely presented groups that has been well studied for over 40 years called semistable fundamental group at $\infty$. A locally finite complex $Y$ has semistable fundamental group at $\infty$ if any two proper rays $r,s:[0,\infty)\to Y$ that converge to the same end of $Y$ are properly homotopic in $Y$. A finitely presented group $G$ has semistable fundamental group at $\infty$ if for some (equivalently any) finite complex $X$ with $\pi_{1}(X)=G$, the universal cover of $X$ has semistable fundamental group at $\infty$. (See section 3 for several equivalent notions of semistability.) It is unknown at this time, whether or not all finitely presented groups have semistable fundamental group at $\infty$, but in [Mih87] the problem is reduced to considering 1-ended groups. The finitely presented group $G$ satisfies a weaker geometric condition called semistable first homology at $\infty$ if and only if $H^{2}(G:\mathbb{Z}G)$ is free abelian (see [GM85]). The question of whether or not $H^{2}(G:\mathbb{Z}G)$ is free abelian for all finitely presented groups $G$ goes back to H. Hopf. Our main interest is in showing certain relatively hyperbolic groups have semistable fundamental group at $\infty$. The work of B. Bowditch [Bow99] and G. Swarup [Swa96] shows that if $G$ is a 1-ended word hyperbolic group then $\partial G$, the Gromov boundary of $G$, has no (global) cut point. M. Bestvina and G. Mess [BM91] (Propositions 3.2 and 3.3) show the absence of cut points in $\partial G$ implies $\partial G$ is locally connected. It was pointed out by R. Geoghegan that $G$ has semistable fundamental group at $\infty$ if and only if $\partial G$ has the shape of a locally connected continuum (see [DS78] for a proof of this fact). In particular, all 1-ended word hyperbolic groups have semistable fundamental group at $\infty$.

Relatively hyperbolic groups are a much studied generalization of hyperbolic groups. Semistability only makes sense for finitely generated groups. We only consider finitely presented groups $G$ in our main result. Later in this section and again in §5, we say what it means for a finitely generated group to be hyperbolic relative to a finite collection of finitely generated subgroups. If a finitely generated group $G$ is hyperbolic relative to a collection of finitely generated subgroups ${\bf P}$ the pair $(G,{\bf P})$ has a well-defined compact metric boundary (see §5), denoted $\partial(G,{\bf P})$. While all 1-ended hyperbolic groups have locally connected boundary without cut points, the space $\partial(G,{\bf P})$ may contain cut points. When $\partial(G,{\bf P})$ is connected, it is locally connected (see Theorem 2.3) and the Hahn-Mazurkiewicz Theorem (see Theorem 31.5 of [Wil70]) implies it is the continuous image of the interval $[0,1]$. This implies $\partial(G,{\bf P})$ is path connected and locally path connected, these facts are important in our approach to showing the relatively hyperbolic groups we consider have semistable fundamental group at $\infty$. The main theorem of [MSb] solves a semistability problem when $\partial(G,{\bf P})$ has no cut point. Note that there is no semistability hypothesis on the peripheral subgroups.

The primary semistability question for relatively hyperbolic groups following Theorem 1.1 is:

Some support for this conjecture appeared in the form of a result of C. Hruska and K. Ruane:

When $G$ is finitely presented, the homology version of the conjecture is resolved by the main theorem of [MSa].

While several results in [MSa] are useful to us, the techniques of that paper are insufficient to resolve the conjecture. A new idea of nearly geodesic homotopies in a cusped space is developed here and it is fundamental in proving our results. The main theorem of our paper resolves the conjecture when $G$ is finitely presented.

All of our work is done in a “cusped” space $X$ for $(G,{\bf P})$ (see §5). When $X$ is Gromov hyperbolic then the pair $(G,{\bf P})$ is said to be relatively hyperbolic or that $G$ is hyperbolic relative to ${\bf P}$. This cusped space is a locally finite 2-complex on which $G$ acts by isometries, but not co-compactly (see §5). It follows from ([Bow12], §6 and §9) that the Bowditch boundary for a relatively hyperbolic pair $(G,{\bf P})$ agrees with the Gromov boundary of $X$. Throughout the paper this boundary is denoted $\partial(G,{\bf P})$ and is called the boundary of the relatively hyperbolic pair $(G,{\bf P})$.

The base space $Y$ in $X$ is a universal cover of a finite complex with fundamental group $G$. There are closed neighborhoods $X_{m}$ of $Y$ in $X$ which are also universal covers of finite complexes with fundamental group $G$ so $G$ has semistable fundamental group at $\infty$ if and only if some (equivalently any) $X_{m}$ has semistable fundamental group at $\infty$. The proof of Theorem 9.3 shows that for some $m$, the space $X_{m}$ (and hence $G$) has semistable fundamental group at $\infty$.

Any proper ray in $X_{m}$ is properly homotopic to a proper ray in $Y$. We show two nearly geodesic rays in $Y$ are properly homotopic in $X$ by nearly geodesic homotopies. Using Theorem 8.3, we cut out disks in the domains of our homotopies on which these homotopies stray out of $X_{m}$ (for some large fixed integer $m$). The geodesic nature of our homotopies allows us to show that the disks can only occur in a locally finite way (see Claims 1 and 2 of the proof of Theorem 9.1) and hence we can properly fill in our homotopies on these disks by homotopies with image in $X_{m}$. This is where we use the hypothesis that the peripheral subgroups are 1-ended and semistable. The resulting homotopies are then combined in a standard way to finish the proof of the theorem. Our nearly geodesic homotopies and the local finiteness arguments of the Claims are the key insights that drive our proofs.

The remainder of the paper is organized as follows. The first order of business is to reduce our problem to the case where $G$ is 1-ended and the peripheral subgroups are 1-ended and have semistable fundamental group at $\infty$. This is accomplished in Section 2. Once the reduction is accomplished we need to know boundaries of the resulting relatively hyperbolic groups are path connected and locally path connected. This is accomplished in Section 2. Finally, groups covered by our Main Theorem and not covered by earlier results are described is Section 2. We develop basic semistability background in Section 3. Section 4 is a short section on hyperbolicity and thin triangles. In Section 5 we review the construction of cusped spaces for a relatively hyperbolic group and discuss some of its properties. We develop the idea of a filter and a filter map in Section 6. Filters are graphs in $[0,1]\times[0,\infty)$ that are geodesically mapped into our cusped space and allow us to produce nearly geodesic homotopies (filter maps). Theorem 6.3 is the main result of this section and all of our initial homotopies are built using this result. Triangulations of our cusped space and $[0,1]\times[0,\infty)$ are developed in Section 7. Filters maps are turned into our first simplicial homotopies in this section. Section 8 contains several tracking results. For each vertex $v$ of $Y$ we construction a geodesic ray $r_{v}$ in $X$ that stays close to $Y$. If $s_{v}$ is a geodesic ray at the base point $\ast$ in $Y$ converging to the same boundary point as does $r_{v}$, then we show that each point of $r_{v}$ is within $\delta$ (the hyperbolicity constant for $X$) of $s_{v}$. The rays $r_{v}$ are important in the construction of filters and filter maps. Theorem 6.1 of [MSa] is introduced in order to cut out the parts of our simplicial homotopies that leave $X_{m}$. Finally, our Main Theorem is proved in Section 9.

We begin with a finitely presented group $G$ and a finite collection ${\bf P}$ of finitely generated subgroups of $G$ such that $G$ is hyperbolic relative to ${\bf P}$. The members of ${\bf P}$ are finitely presented by the following result (proved in [DG13]). For a more general result see [DGO17], Theorem 2.11.

The reduction we want comes directly from:

In his thesis [Das20] A. Dasgupta proves that the only cut points in a connected boundary of a finitely generated relatively hyperbolic group are parabolic. Dasgupta combines this result with a result of Bowditch to prove:

As noted in the introduction, the Hahn-Mazurkiewicz Theorem combines with Theorem 2.3 to show:

Results of B. Bowditch (see Theorem 2.13 of [MSb]) determine that cut points appear in $\partial(G,{\bf P})$ precisely when $(G,{\bf P})$ admits a non-trivial graph of groups decomposition that is a ‘proper peripheral splitting’. Notice that in the following combination result of M. Mihalik and S. Tschantz, there is no restriction on the number of ends of any of the groups involved.

Combining Theorems 1.1 and 2.5 with the splitting result of Bowditch shows many relatively hyperbolic groups (with boundary cut points) have semistable fundamental group at $\infty$, but a broad collection of examples are described near the end of Section 2 of [MSb] that are covered by the Main Theorem of this paper and not by previous results. In particular, for any finitely generated (but not finitely presented) recursively presented group $Q$ and finitely presented group $P$ containing a subgroup isomorphic to $Q$, a finitely presented group $G=A\ast_{Q}P$ is described that is hyperbolic relative to $P$. Here $A$ is finitely generated but not finitely presented. If $P$ has semistable fundamental group at $\infty$ then our Main Theorem 1.5 implies $G$ has semistable fundamental group at $\infty$. The techniques of [MSb] break down for such groups.

The best reference for the notion of semistable fundamental group at $\infty$ is [Geo08] and we use this book as a general reference throughout this section. 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 set $C$ in $Y$, $f^{-1}(C)$ 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:

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]). It is an elementary exercise to see that semistable fundamental group at $\infty$ is an invariant of proper homotopy type and S. Brick [Bri93] proved that semistability is a quasi-isometry invariant. When $K$ is 1-ended with semistable fundamental group at $\infty$, $\pi_{1}^{\infty}(K,r)$ is independent of proper base ray $r$ (in direct analogy with the fundamental group of a path connected space being independent of base point). Theorem 2.1 of [Mih83] and Lemma 9 of [Mih86], provide several equivalent notions of semistability. Conditions 2 and 3 are the semistability criterion used in the proof of our main theorem.

If $G$ is a finitely presented group and $X$ is a finite connected complex with $\pi_{1}(X)=G$ then $G$ has semistable fundamental group at $\infty$ if the universal cover of $X$ has semistable fundamental group at $\infty$. This definition only depends on $G$ (see the proof of Theorem 3 of [LR75] or the opening paragraph of section 16.5 of [Geo08]) and it is unknown if all finitely presented groups have semistable fundamental group at $\infty$.

There are a number of equivalent forms of hyperbolicity for geodesic metric spaces. In this paper we use the following thin triangles definition.

Let $c_{x}=f(c_{x}^{\prime})$, $c_{y}=f(c_{y}^{\prime})$ and $c_{z}=f(c_{z}^{\prime})$. We call the points $c_{x},c_{y},c_{z}$ the internal points of $\triangle$. There is a unique continuous function $t_{\triangle}:\triangle^{\prime}\to T_{\triangle}$ of $\triangle^{\prime}$ onto a tripod $T_{\triangle}$, where $t_{\triangle}$ is an isometry on the edges of $\triangle^{\prime}$, and $T_{\triangle}$ is a tree with one vertex $w$ of degree 3, and vertices $x^{\prime\prime},y^{\prime\prime},z^{\prime\prime}$ each of degree one, such that $d(w,z^{\prime\prime})=d(z^{\prime},c_{y}^{\prime})=d(z^{\prime},c_{x}^{\prime})$ etc. (See Figure 1.)

Let $f_{\triangle}$ be the composite map $f_{\triangle}\equiv t_{\triangle}\circ f^{-1}:\triangle\to T_{\triangle}$. We say that $\triangle(x,y,z)$ is $\delta-thin$ if fibers of $f_{\triangle}$ have diameter at most $\delta$ in $X$. In other words, for all $p,q$ in $\triangle$,

The space $X$ is ($\delta$) hyperbolic if there is a constant $\delta$ such that all geodesic triangles in $X$ are $\delta$ thin.

In a hyperbolic geodesic metric space $X$ the boundary $\partial X$ can be defined in a number of ways. In Section III.H.3 of [BH99] $\partial X$ is defined as the set of equivalence classes $[r]$ of geodesic rays $r$, where $r$ and $s$ are equivalent if there is a number $K\geq 0$ such that $d(r(k),s(k))\leq K$ for all $k\geq 0$. We say $r$ converges to $[r]$. Note that if such a $K$ exists for $r,s$ based at $p$, then our thin triangle condition forces $d(r(k),s(k))\leq\delta$ for all $k\geq 0$. (Simply consider the geodesic triangle formed by $r([0,k+K])$, $s([0,k+K])$ and a geodesic (of length $\leq K$) connecting $r(k+K)$ to $s(k+K)$. The internal points on $r$ and $s$ are beyond $r(k)$ and $s(k)$ respectively.)

If $X$ is a $\delta$ hyperbolic geodesic metric space then there is a metric $d$ on $\partial X$ (induced from an inner product on $X$) such that $(\partial X,d)$ is compact (see Proposition 3.7 [BH99]). Intuitively, if $r(0)=s(0)$ then $[r]$ is ‘close’ to $[s]$ if $r$ and $s$ fellow travel for a ‘long’ time.

Given a finitely generated group $G$ and a collection of finitely generated subgroups ${\bf P}$ of $G$ there are a number of equivalent definitions of what it means for the pair $(G,{\bf P})$ to be relatively hyperbolic or $G$ to be relatively hyperbolic with respect to ${\bf P}$. Theorem 5.2 enables us to say the pair $(G,{\bf P})$ is relatively hyperbolic if a certain cusped space is Gromov hyperbolic, so we take this as our definition. The Gromov boundary of this cusped space is the boundary of the pair $(G,{\bf P})$ and is denoted $\partial(G,{\bf P})$. This boundary agrees with the Bowditch boundary of the pair $(G,{\bf P})$.

D. Groves and J. Manning [GM08] investigate a locally finite space $X$ derived from a finitely generated group $G$ and a collection ${\bf P}$ of finitely generated subgroups. The following definitions are directly from [GM08]

A) $\mathcal{H}^{(0)}=\Gamma(0)\times(\{0\}\cup\mathbb{N})$

B) $\mathcal{H}^{(1)}$ contains the following three types of edges. The first two types are called horizontal, and the last type is called vertical.

(B1) If $e$ is an edge of $\Gamma$ joining $v$ to $w$ then there is a corresponding edge $\bar{e}$ connecting $(v,0)$ to $(w,0)$.

(B2) If $k>0$ and $0<d_{\Gamma}(v,w)\leq 2^{k}$, then there is a single edge connecting $(v,k)$ to $(w,k)$.

(B3) If $k\geq 0$ and $v\in\Gamma^{(0)}$, there is an edge joining $(v,k)$ to $(v,k+1)$.

C) $\mathcal{H}^{(2)}$ contains three kinds of 2-cells:

(C1) If $\gamma\subset\mathcal{H}^{(1)}$ is a circuit composed of three horizontal edges, then there is a 2-cell (a horizontal triangle) attached along $\gamma$.

(C2) If $\gamma\subset\mathcal{H}^{(1)}$ is a circuit composed of two horizontal edges and two vertical edges, then there is a 2-cell (a vertical square) attached along $\gamma$.

(C3) If $\gamma\subset\mathcal{H}^{(1)}$ is a circuit composed of three horizontal edges and two vertical ones, then there is a 2-cell (a vertical pentagon) attached along $\gamma$, unless $\gamma$ is the boundary of the union of a vertical square and a horizontal triangle.

(1) $\mathcal{D}(x)=0$ if $x\in\Gamma$,

(2) $\mathcal{D}(x)=k$ if $x$ is a vertex $(v,k)$, and

(3) $\mathcal{D}$ restricts to an affine function on each 1-cell and on each 2-cell.

For each $i$, and each $t\in T_{i}$, let $\Gamma_{i,t}$ be the full subgraph of the Cayley graph $\Gamma(G,S)$ which contains $tP_{i}$. Each $\Gamma_{i,t}$ is isomorphic to the Cayley graph of $P_{i}$ with respect to the generators $P_{i}\cap S$. Then define

where the graphs $\Gamma_{i,t}\subset\Gamma(G,S)$ and $\Gamma_{i,t}\subset\mathcal{H}(\Gamma_{i,t})$ are identified in the obvious way.

The space $X(G,{\bf P},S)$ is called the cusped space for $G$, ${\bf P}$ and $S$. If $G$ and the $P_{i}$ have finite presentations, let $\mathcal{A}=\langle S;R\rangle$ be such a presentation that includes sub-presentations of the $P_{i}$. We add 2-cells to $\Gamma(G,S)$ to form the Cayley 2-complex of this presentation. The resulting expansion of $X(G,{\bf P},S)$ is called the cusped space for $G$, ${\bf P}$ and $\mathcal{A}$ and is denoted $X(G,{\bf P},\mathcal{A})$. The next result shows cusped spaces are fundamentally important spaces. We prove our results in cusped spaces.

Assume $G$ is finitely presented and hyperbolic with respect to the subgroups ${\bf P}=\{P_{1},\ldots,P_{n}\}$ and $S$ is a finite generating set for $G$ containing generating sets for the $P_{i}$. The $P_{i}$ and their conjugates are called peripheral subgroups of $G$. For a finite presentation $\mathcal{A}$ of $G$ with respect to $S$, let $Y(\mathcal{A})$ be the Cayley 2-complex for $\mathcal{A}$. So $Y$ is simply connected with 1-skeleton $\Gamma(G,S)$, and the quotient space $G/Y$ has fundamental group $G$. The cusped space $X(G,{\bf P},S)$ is quasi-isometric to the cusped space $X(G,{\bf P},\mathcal{A})$ and so one is hyperbolic if and only if the other is hyperbolic, and these two spaces have the same boundary. For $g\in G$ and $i\in\{1,\ldots,n\}$ we call $gP_{i}$ a peripheral coset in a cusped space. The depth functions on the horoballs over the peripheral cosets extend to $X(G,{\bf P},\mathcal{A})$. So that

where $\mathcal{D}^{-1}(0)=Y$ and for each horoball $H$ (over a peripheral coset) we have $H\cap\mathcal{D}^{-1}(m)=H(m)$, $H\cap\mathcal{D}^{-1}[0,m]=H_{m}$ and $H\cap\mathcal{D}^{-1}[m,\infty)=H^{m}$. We call each $H^{m}$ an $m$-horoball.

Given two points $x$ and $y$ in a horoball $H$, there is a shortest path in $H$ from $x$ to $y$ of the form $(\alpha,\tau,\beta)$ where $\alpha$ and $\beta$ are vertical and $\tau$ is horizontal of length $\leq 3$. Note that if $\alpha$ is non-trivial and ascending and $\beta$ is non-trivial and descending, then $\tau$ has length either 2 or 3.

If $Y(\mathcal{A})$ is the Cayley 2-complex for the finite presentation $\mathcal{A}$ of the group $G$, then the isometric action of $G$ on $Y$ extends to an isometric action of $G$ on $X(G,{\bf P},\mathcal{A})$. This action is height preserving. In the following lemmas, $X=X(G,{\bf P},\mathcal{A})$.

Our reductions imply the group $G$ is 1-ended as is each $P_{i}\in{\bf P}$. Let $\mathcal{A}$ be a finite presentation for $G$ that contains finite sub-presentations for the $P_{i}$. Let $X$ be the cusped space $X(G,{\bf P},\mathcal{A})$. The compact metric space $\partial(G,{\bf P})$ is path connected and locally path connected (Theorem 2.4). The space $Y\subset X$ is the Cayley 2-complex of $\mathcal{A}$. For a peripheral coset $gP_{i}$, let $\Gamma(gP_{i})$ be the copy of the Cayley 2-complex of $P_{i}$ in $Y$ containing $g$. We use $\hat{d}$ for our metric on $\partial X=\partial(G,\bf{P})$. Any proper ray in $X$ is properly homotopic to a proper edge path ray in the 1-skeleton of $X$. Hence when we show a space has semistable fundamental group at $\infty$ it suffices to show all proper edge path rays are properly homotopic. Since $X$ is quasi-isomorphic to the 1-skeleton of $X$, one is hyperbolic if and only if the other is hyperbolic. Let $d$ be the edge path metric on $X^{(1)}$, the 1-skeleton of $X$. If $A$ is a subcomplex of $X^{(1)}$ let $B_{n}(A)$ be the neighborhood of radius $n$ about $A$. For any subcomplex $A$ of $X$, define $St(A)$ to be $A$, union all vertices connected by an edge to a vertex of $A$, union all 2-cells of $X$ all of whose vertices belong to $St(A)$. Define $St_{n}(A)$ inductively as $St(St_{n-1}(A))$. Note that if $A^{(1)}$ is the 1-skeleton of $A$, then $B_{n}(A^{(1)})$ is the 1-skeleton of $St_{n}(A)$. In particular, if for $n\geq 1$, the 1-skeleton of $St_{n}(v)$ is $B_{n}(v)$ for all vertices $v$ of $X$.

(1) Each vertex is of the form $(t,n)$ for some integer $n\geq 0$ and some $t\in[0,1]$. The points $(0,0)$ and $(0,1)$ are vertices of $F$.

(2) Each edge of $F$ is either vertical or horizontal. A vertical edge is the convex hull of vertices $(t,n)$ and $(t,n+1)$. If $(t,n)$ is a vertex of $F$, then $(t,n)$ and $(t,n+1)$ are the vertices of a vertical edge (so every vertex is connected by an edge to exactly one vertex directly above it).

(3) A horizontal edge is the convex hull of the vertices $(t,n)$ and $(s,n)$ for some integer $n\geq 0$ and numbers $0\leq t<s\leq 1$. The horizontal edges at height $n$ form an edge path from $(0,n)$ to $(1,n)$ with consecutive vertices $(0,n),(t_{1},n),(t_{2},n),\ldots,(1,n)$ where $t_{i}<t_{i+1}$ for all $i$. (Note that the first coordinates of vertices at height $n$ are a subset of the first coordinates of vertices at height $n+1$.)

Note that each component of $[0,1]\times[0,\infty)-F$ is a rectangle that is bounded by an edge path loop with exactly two vertical edges, one horizontal edge at height $n$ and all other edges horizontal at height $n+1$.

The idea is to build filters and proper homotopies that map any vertical edge path in the filter to a geodesic edge path in the 1-skeleton of $X$. Infinitely many of these homotopies will then be combined in a proper way to show that every proper ray in $Y$ is properly homotopic to a certain (nearly geodesic) ray in $Y$ by a proper homotopy in $X_{M}$ for some fixed integer $M$.

Let $\delta\geq 1$ be the hyperbolicity constant for $X$. Given $\epsilon>0$ there is $N(\epsilon)>0$ such that if $x,y\in\partial X$ and $r_{x},r_{y}$ are geodesic edge path rays at $\ast$ converging to $x$ and $y$ respectively with $d(r_{x}(N(\epsilon)),r_{y}(N(\epsilon)))\leq 2\delta+1$ then $\hat{d}(x,y)\leq\epsilon$. Given $N>0$ there is $\epsilon_{1}(N)$ such that if $x,y\in\partial X$ and $\hat{d}(x,y)<\epsilon_{1}(N)$ then for any geodesics $r_{x}$ and $r_{y}$ at $\ast$ converging to $x$ and $y$ respectively, $d(r(N),s(N))\leq 2\delta+1$.

Since $\partial X$ is compact, connected and locally path connected we have: Given $\epsilon>0$ there is $\rho(\epsilon)>0$ such that if $x,y\in\partial X$ and $\hat{d}(x,y)\leq\rho(\epsilon)$ then there is a path connecting $x$ and $y$ in $\partial X$ of diameter $\leq\epsilon$. Combining these results we have:

For technical reasons, we need the following result to apply to edge path rays $s_{1}^{\prime}$ and $s_{2}^{\prime}$ that are only “nearly” geodesic. In applications $s_{i}^{\prime}$ will be the concatenation of a finite edge path and a geodesic edge path ray. The edge path ray $s_{i}^{\prime}$ will synchronously track a geodesic edge path ray.

We construct the filter $F(s_{1}^{\prime},s_{2}^{\prime},\gamma,K)$. Choose an integer $N_{0}\geq 0$ as large as possible such that for any two points $w_{1}$ and $w_{2}$ in the image of $\gamma$ and any geodesic edge paths $q_{1}$ and $q_{2}$ at $\ast$ converging to $w_{1}$ and $w_{2}$ respectively, $d(q_{1}(N),q_{2}(N))\leq\delta$ for all integers $0\leq N\leq N_{0}$. (Note that $N_{0}\geq{\delta\over 2}$.) For $j$ an integer between $0$ and $N_{0}-1$, the only vertices of $[0,1]\times[0,\infty)$ are $(0,j)$ and $(1,j)$. The vertical edges are between $(0,j-1)$ and $(0,j)$, and $(1,j-1)$ and $(1,j)$. There is a horizontal edge between $(0,j)$ and $(1,j)$.

Next we define $f$ on $[0,1]\times[0,N_{0}]$ (and on every vertical line above a vertex of $[0,1]\times\{N_{0}\}$). This process is iterated to define $f$ and the filter $F$.

(i) $f(t,0)=\ast$ for all $t\in[0,1]$.

(ii) $f(0,t)=s_{1}^{\prime}(t)$ and $f(1,t)=s_{2}^{\prime}(t)$ for all $t\in[0,\infty)$.

(iii) For $n$ an integer in $\{1,2,\ldots,N_{0}-1\}$ let $f$ restricted to the edge $[0,1]\times\{n\}$ be an edge path of length $\leq 2K+\delta$ from $s_{1}^{\prime}(n)$ to $s_{2}^{\prime}(n)$. (Such a path exists since there is an edge path of length $\leq\delta$ from $s_{1}(n)$ to $s_{2}(n)$ and for $i\in\{1,2\}$, edge paths of length $\leq K$ from $s_{i}^{\prime}(n)$ to $s_{i}(n)$).

(iv) For $n\in\{1,2,\ldots,N_{0}-1\}$, let $f$ restricted to the rectangle $[0,1]\times[n-1,n]$ be a homotopy in $St_{A}(s_{1}^{\prime}(n))$ (given by Lemma 5.6) that kills the loop determined by $f$ restricted to the boundary of the rectangle.

(v) For $k\geq 0$, let $N_{k}=N_{0}+k\delta$. Choose points $0=t^{0}_{0}<t^{0}_{1}<\cdots<t^{0}_{k(0)}=1$ such that for any $i$ and two points $u_{1}$ and $u_{2}$ in $[t^{0}_{i},t^{0}_{i+1}]$ and any $\ast$ based geodesic edge paths $q_{1}\in\gamma(u_{1})$ and $q_{2}\in\gamma(u_{2})$, we have $d(q_{1}(n),q_{2}(n))\leq\delta$ for all $n\in[0,N_{1}]$. There are $k(0)+1$ vertices $(t^{0}_{0},N_{0}),(t^{0}_{1},N_{0}),\ldots(t^{0}_{k(0)},N_{0})$ at level $N_{0}$ in $F$ and a horizontal edge between $(t^{0}_{j},N_{0})$ and $(t^{0}_{j+1},N_{0})$ for each $j$. For each $n\in\{1,\ldots,k_{0}-1\}$ add a vertical edge path ray $\{t_{n}^{0}\}\times[n_{0},\infty)$ to $F$ (with vertices $(t_{n}^{0},n)$ for each integer $n\geq n_{0}$.) Let $r_{n}^{0}$ be a geodesic edge path at $\ast$ converging to $\gamma(t^{0}_{n})$. Let $r_{0}^{0}=s_{1}^{\prime}$ and $r_{k(0)}^{0}=s_{2}^{\prime}$. For $n\in\{0,\dots,k(0)\}$ and $a\in[N_{0},\infty)$ define $f(t^{0}_{n},a)=r_{n}^{0}(a)$. (This agrees with our earlier definition of $f$ on $\{0,1\}\times[N_{0},\infty)$.