Acylindrical Actions on Trees and the Farrell-Jones Conjecture

TL;DR

This paper proves the Farrell-Jones Conjecture for groups acting acylindrically on trees, providing new insights into algebraic K- and L-theory, and generalizing previous results on Nil-groups for amalgamated free products.

Contribution

It establishes the Farrell-Jones Conjecture for acylindrical actions on trees and identifies associated Nil-groups with those of virtually cyclic groups, extending prior work.

Findings

Farrell-Jones Conjecture holds for acylindrical actions on trees.

Nil-groups associated with amalgamated free products can be explicitly identified.

Nil-groups vanish for regular rings and strictly acylindrical actions.

Abstract

We show that for groups acting acylindrically on simplicial trees the $K$- and $L$-theoretic Farrell-Jones Conjecture relative to the family of subgroups consisting of virtually cyclic subgroups and all subconjugates of vertex stabilisers holds. As an application, for amalgamated free products acting acylindrically on their Bass-Serre trees we obtain an identification of the associated Waldhausen Nil-groups with a direct sum of Nil-groups associated to certain virtually cyclic groups. This identification generalizes a result by Lafont and Ortiz. For a regular ring and a strictly acylindrical action these Nil-groups vanish. In particular, all our results apply to amalgamated free products over malnormal subgroups.

Tables

$E^k\left(T\right)$	Denote by $E^k\left(T\right)$ the set of all geodesic segments of length $k$ in $T$ whose endpoints are vertices.
$E^k\left(T\right)\cap S$	For any subtree $S\subset T$ we write $\sigma \in E^k\left(T\right)\cap S$ if $\sigma \subset S$ and $\sigma \in E^k\left(T\right)$.
$d_{E^k}$	The set $E^k\left(T\right)$ is countable and the obvious action of $G$ on $E^k\left(T\right)$ has finite point stabilisers, since the action $G\curvearrowright T$ is $k$-acylindrical. We fix a metric $d_{E^k}$ on $E^k\left(T\right)$ according to Lemma 7.1.
$\sigma$ lies on $\gamma$	We will often think of an element $\sigma$ in $E^k\left(T\right)$ as an ‘elongated point’, which may or may not lie on a given geodesic $\gamma$. Thus, we will frequently use phrases like $\sigma$ lies on $\gamma$, meaning that $\sigma$ is a subset of the image of $\gamma$.
$o\left(\gamma \right),t\left(\gamma \right)$	For a finite geodesic $\gamma :[a,b]\to T$ we denote by $o\left(\gamma \right):=\gamma \left(a\right)$ and $t\left(\gamma \right):=\gamma \left(b\right)$ its start- and endpoint, respectively. We also extend this notation to (bi-infinite) geodesic rays in the canonical way.
$m\left(\gamma \right),m\left(\sigma \right)$	For a finite geodesic $\gamma :[a,b]\to T$ we denote by $m\left(\gamma \right):=\gamma \left(\frac{a+b}{2}\right)$ its midpoint. Analogously, for an element $\sigma \in E^k\left(T\right)$ we denote by $m\left(\sigma \right)$ its midpoint. Note that, if $k$ is even, the midpoint of $\sigma$ is a vertex of $T$.
${\sigma }_v\left(\gamma \right)$	For a geodesic $\gamma$ and any vertex $v$ on $\gamma$ we denote (if it exists) by ${\sigma }_v\left(\gamma \right)$ the geodesic segment of length $k$ on $[o\left(\gamma \right),v]$ that ends in $v$.
$v+i,v-i$	For a geodesic $\gamma$, any $i\in \mathbb{N}$ and a vertex $v$ on $\gamma$ we denote (if it exists) by $v+i$ (resp. $v-i$) the unique vertex on $[v,t\left(\gamma \right)]$ (resp. $[o\left(\gamma \right),v]$) that has distance $i$ to $v$ (with respect to $d_T$). If we want to specify along which geodesic we take this $i$th successor and $i$th predecessor, we write $v{+}_{\gamma }i$ and $v{-}_{\gamma }i$, respectively.
$w_0,w_0^{\prime }$	We fix two vertices $w_0,w_0^{\prime }\in V_0\left(T\right)$ with $d_T(w_0,w_0^{\prime })\ge 5k$. (Such points exist since we can always assume $V_0\left(T\right)$ to be unbounded with respect to $d_T$ by replacing $T$ with its barycentric subdivision.)
${\theta }_0$	Let ${\theta }_0:=\max \left\{d_{E^k}(\sigma ,{\sigma }^{\prime })|\sigma ,{\sigma }^{\prime }\in E^k\left(T\right)\cap [w_0,w_0^{\prime }]\right\}$.
$m_{ϵ}$	Choose your favourite . For the next two lines denote by $m$ the midpoint of $[w_0,w_0^{\prime }]$. We denote (throughout the rest of this paper) by $m_{ϵ}$ the unique point on $[w_0,m]$ that has distance $ϵ$ to $m$.
$a(g,\xi )$	Now for every $(g,\xi )\in G\times \Delta \left(T\right)$ only one of the two geodesics $[g{w}_0,\xi ]$ and $[g{w}_0^{\prime },\xi ]$ contains the point $g{m}_{ϵ}$. We denote by $a(g,\xi )\in \{g{w}_0,g{w}_0^{\prime }\}$ the vertex such that $[a(g,\xi ),\xi ]$ contains $g{m}_{ϵ}$. Note that, for all $g,h\in G$, we have $ha(g,\xi )=a(hg,h\xi )$.

$L_0$	The set $B_{\alpha }\left(1\right)$ is finite. As is the set $M:=\left\{h{w}_0,h{w}_0^{\prime }\right|h\in B_{\alpha }\left(1\right)\}$. Let $L_0\subset T$ be the subtree of $T$ spanned by $M$. In particular, $L_0$ is a finite subtree of $T$.
${{\rm Y}}_0$	We define ${{\rm Y}}_0$ as the diameter of the set $E^k\left(T\right)\cap L_0$ with respect to the metric $d_{E^k}$. Thus, for all $\sigma ,{\sigma }^{\prime }\in E^k\left(T\right)\cap L_0$ the inequality $d_{E^k}(\sigma ,{\sigma }^{\prime })\le {{\rm Y}}_0$ holds. Note that ${{\rm Y}}_0$ automatically has to be $\ge {\theta }_0$ as $[w_0,w_0^{\prime }]\subset L_0$.
$L_k\left(\chi \right)$	For $\chi >0$ we construct the complex $L_k\left(\chi \right)$ by enlarging $L_0=:L_0\left(\chi \right)$ iteratively $k$-times as follows: For $1\le \nu \le k$ let $L_{\nu }\left(\chi \right)$ be the union of $L_{\nu -1}\left(\chi \right)$ and all $\sigma \in E^k\left(T\right)$ such that there is ${\sigma }^{\prime }\in E^k\left(T\right)\cap L_{\nu -1}\left(\chi \right)$ with $\sigma \cap {\sigma }^{\prime }\ne \varnothing$ and $d_{E^k}(\sigma ,{\sigma }^{\prime })\le \chi$. Since, to obtain $L_{\nu }\left(\chi \right)$ from $L_{\nu -1}\left(\chi \right)$, we only add segments that have non-trivial intersection with $L_{\nu -1}\left(\chi \right)$, each $L_{\nu }\left(\chi \right)$ is again a subtree of $T$. Since $L_0$ is finite and $d_{E^k}$ is proper, the tree $L_{\nu }\left(\chi \right)$ is finite for all $1\le \nu \le k$.
${{\rm Y}}_1\left(\chi \right)$	We define ${{\rm Y}}_1\left(\chi \right)$ as the diameter of the set $E^k\left(T\right)\cap L_k\left(\chi \right)$ with respect to the metric $d_{E^k}$. Thus, for all $\sigma ,{\sigma }^{\prime }\in E^k\left(T\right)\cap L_k\left(\chi \right)$ the inequality $d_{E^k}(\sigma ,{\sigma }^{\prime })\le {{\rm Y}}_1\left(\chi \right)$ holds. Note that ${{\rm Y}}_1\left(\chi \right)\ge {{\rm Y}}_0$.
${{\rm Y}}_m\left(\chi \right)$	Let $\chi >0$. For $m\ge 0$ we define ${{\rm Y}}_m\left(\chi \right)$ iteratively by ${{\rm Y}}_m\left(\chi \right):={{\rm Y}}_1\left({{\rm Y}}_{m-1}\left(\chi \right)\right)$.

${\gamma }_h$	For $h\in B_{\alpha }\left(1\right)$ let ${\gamma }_h:=[a(h,\xi ),\xi ]$.
$v_h$	For $h\ne 1$ let $v_h$ be the unique first vertex that lies on both ${\gamma }_1$ and ${\gamma }_h$. Hence, we have ${\gamma }_1\cap {\gamma }_h=[v_h,\xi ]$. Note that the case $v_h=\xi$ is possible.
$B_v^{\text{inn}}$	For any vertex $v$ of ${\gamma }_1$ we set $B_v^{\text{inn}}:=\left\{h\in B_{\alpha }\left(1\right)\right|v\in V\left({\gamma }_h\right)\}$. Thus, $h\in B_{\alpha }\left(1\right)$ is an element of $B_v^{\text{inn}}$ if and only if ${\gamma }_h$ contains $v$, which is the case if and only if $[v,\xi ]\subseteq [v_h,\xi ]$.
$B_v^{\text{ext}}$	Furthermore, let $B_v^{\text{ext}}:=B_{\alpha }\left(1\right)\setminus B_v^{\text{inn}}$. Thus, $h\in B_{\alpha }\left(1\right)$ is an element of $B_v^{\text{ext}}$ if and only if $v\notin [v_h,\xi ]$.
$v_0\left(\chi \right)$	For any $\chi >{\theta }_0$ denote—if it exists—by $v_0\left(\chi \right)$ the first vertex on ${\gamma }_1$ such that $[a(1,\xi ),v_0\left(\chi \right)]$ is $\chi$-small, but $[a(1,\xi ),v_0\left(\chi \right){+}_{{\gamma }_1}1]$ is not. In particular, the existence of $v_0\left(\chi \right){+}_{{\gamma }_1}1$ is necessary for the existence of $v_0\left(\chi \right)$. Note that $d_T(a(1,\xi ),v_0\left(\chi \right))\ge 2k$: Since $[w_0,w_0^{\prime }]$ is ${\theta }_0$-small, it follows that $m_{ϵ}\in [a(1,\xi ),v_0\left(\chi \right)]$ and $[a(1,\xi ),m_{ϵ}]$ has length $\ge 2k$.

Equations

$$\dots \to K_n(RC)\to K_n(RA)\oplus K_n(RB)\to K_n(RG)\to K_{n-1}(RC)\to \dots .$$

$$\ldots\to K_{n}(RC)\to K_{n}(RA)\oplus K_{n}(RB)\to K_{n}(RG)/\widetilde{\text{Nil}}_{n-1}\to K_{n-1}(RC)\to\ldots\$$

$${\widetilde{\text{Nil}}}_{n-1}(RC;$$

$$\cong \underset{L\in \mathcal{L}}{⨁}\mathrm{coker}(H_n^{G_L}(E_{\mathcal{F}\mathcal{I}\mathcal{N}}{G}_L;{\mathbf{K}}_R)\to H_n^{G_L}(\mathrm{pt};{\mathbf{K}}_R))$$

$${\text{asmb}}_n(G,\mathcal{F},\mathcal{A}):H_n^G(E_{\mathcal{F}}G;{\mathbf{K}}_{\mathcal{A}})\to H_n^G(\mathrm{pt};{\mathbf{K}}_{\mathcal{A}})\cong K_n\left({\int }_G\mathcal{A}\right),$$

$${\text{asmb}}_n(G,\mathcal{F},\mathcal{A}):H_n^G(E_{\mathcal{F}}G;{\mathbf{L}}_{\mathcal{A}}^{-\infty })\to H_n^G(\mathrm{pt};{\mathbf{L}}_{\mathcal{A}}^{-\infty })\cong L_n^{⟨-\infty ⟩}\left({\int }_G\mathcal{A}\right),$$

$$\{M(z,A)|z\in T\cup \partial T,A\subset T\text{finite}\},$$

$$\{M(z,A)|z\in Q,A\subset Q\text{finite}\}$$

$$\{B_{\text{nicefrac}ϵ2}(x)|x\in {\overline{T}}^{obs}\}$$

$$\{M(x,A_x)|x\in {\overline{T}}^{obs}\}$$

$$H(x,t)={\gamma }_{p(x),x}\left(\frac{t}{1-t}\right)$$

$${\widetilde{\text{Nil}}}_{n-1}(RC;$$

$$\cong \underset{L\in \mathcal{L}}{⨁}\mathrm{coker}(H_n^{G_L}(E_{\mathcal{F}\mathcal{I}\mathcal{N}}{G}_L;{\mathbf{K}}_R)\to H_n^{G_L}(\mathrm{pt};{\mathbf{K}}_R))$$

$$\coprod _{L\in \mathcal{L}}G{\times }_{G_L}L$$

$${\text{asmb}}_n(G,{\mathcal{F}}_T,R):H_n^G(E_{{\mathcal{F}}_T}G;{\mathbf{K}}_R)\to H_n^G(\mathrm{pt};{\mathbf{K}}_R).$$

$$H_n^G(T;{\mathbf{K}}_R)=H_n^G(E_{{\mathcal{F}}_T}G;{\mathbf{K}}_R)\to H_n^G(E_{\mathcal{F}}G;{\mathbf{K}}_R),$$

$$\dots \to \underset{L\in \mathcal{L}}{⨁}{H}_n^G(G{\times }_{G_L}L)\to H_n^G(T)\oplus \underset{L\in \mathcal{L}}{⨁}{H}_n^G(G{\times }_{G_L}\text{cone}(L))\to H_n^G(E_{\mathcal{F}}G)\to \dots .$$

$$\dots \to \underset{L\in \mathcal{L}}{⨁}{H}_n^{G_L}(E_{\mathcal{F}\mathcal{I}\mathcal{N}}{G}_L)\stackrel{(f_1,f_2)}{⟶}{H}_n^G(T)\oplus \underset{L\in \mathcal{L}}{⨁}{H}_n^{G_L}(\mathrm{pt})\to H_n^G(E_{\mathcal{F}}G)\to \dots .$$

$$H_{\ast }^{G^{\prime }}(E_{\mathcal{F}\mathcal{I}\mathcal{N}}H)\to H_{\ast }^{G^{\prime }}(E_{\mathcal{V}\mathcal{C}\mathcal{Y}\mathcal{C}}H)$$

$$0\to \underset{L\in \mathcal{L}}{⨁}{\mathcal{H}}_n^{G_L}(E_{\mathcal{F}\mathcal{I}\mathcal{N}}{G}_L)\stackrel{(f_1,f_2)}{⟶}{\mathcal{H}}_n^G(T)\oplus \underset{L\in \mathcal{L}}{⨁}{\mathcal{H}}_n^{G_L}(\mathrm{pt})\to {\mathcal{H}}_n^G(E_{\mathcal{F}}G)\to 0$$

$$0\to V_0\stackrel{(i,j)}{⟶}{V}_2\oplus V_1\to V_3\to 0$$

$$0\to H_n^G(T;{\mathbf{K}}_R)\stackrel{i_{\ast }}{⟶}{H}_n^G(E_{\mathcal{F}}G;{\mathbf{K}}_R)⟶\mathrm{coker}\underset{L\in \mathcal{L}}{⨁}{\text{asmb}}_n(G_L,\mathcal{F}\mathcal{I}\mathcal{N},R)\to 0$$

$$H_n^{G_L}(E_{\mathcal{F}\mathcal{I}\mathcal{N}}{G}_L;{\mathbf{K}}_R)\to H_n^{G_L}(E_{\mathcal{V}\mathcal{C}\mathcal{Y}\mathcal{C}}{G}_L;{\mathbf{K}}_R)$$

$$G{\times }_{\Theta }\partial T:=\{(g,\xi )\in G\times \partial T|[a(g,\xi ),\xi ]\text{is}\Theta \text{-small}\}.$$

$$\displaystyle W(v,\Theta):=\{(g,\xi)\in G\times\Delta(T)\ |\$$

$$v=\xi \text{or}{d}_{E^k}({\sigma }_v(\gamma ),{\sigma }_{v{+}_{\gamma }1}(\gamma ))>\Theta ,$$

$$\text{where}\gamma :=[a(g,\xi ),\xi ]\}.$$

$$[a(g,\xi ),v{+}_{\gamma }1]=[a(g,{\xi }^{\prime }),v{+}_{\gamma }1]\subseteq [a(g,{\xi }^{\prime }),{\xi }^{\prime }]$$

$$U:=M(v,\{t({\sigma }_1^{\prime }),\dots ,t({\sigma }_m^{\prime })\})\cap \Delta (T).$$

$$d({\sigma }_{v_0(\chi )}({\gamma }_h),{\sigma }_{v_0(\chi )+1}({\gamma }_h)).$$

$$B_{\alpha }(1)\times \{\xi \}\subset W(v_0(\chi ),{{\rm Y}}_1(\chi ))$$

$$B_{v_0({{\rm Y}}_2(\chi )){-}_{{\gamma }_1}(r+s)}^{\text{ext}}=B_{v_0(\chi ){-}_{{\gamma }_1}r}^{\text{ext}}=\varnothing \text{and}$$

Full text

This work was supported by the SFB 878 “Groups, Geometry & Actions”.

1. 1 Background on the Farrell–Jones Conjecture
2. 2 The observers’ topology on a tree
3. 3 Acylindrical actions on trees
4. 4 The proof of Theorem A modulo $\mathcal{F}$-amenability of $G\curvearrowright\Delta(T)$
5. 5 The proof of Theorem B
6. 6 The tactic for showing finitely $\mathcal{F}$-amenability
7. 7 Defining $\Theta$-small geodesics
8. 8 Covers for $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$
9. 9 The segment-flow space $FS_{\Theta}$
10. 10 Covers for $G\times_{\Theta}\partial T$

Introduction

The Farrell–Jones Conjecture (for a group $G$ and relative to the family $\mathcal{VCYC}$ of virtually cyclic subgroups) predicts an isomorphism between the $K$- resp. $L$-groups of a group ring $RG$ and the value of a certain $G$-homology theory on a certain classifying space. Such an isomorphism can be seen as a way to (potentially) compute $K_{\ast}(RG)$ resp. $L_{\ast}^{\langle-\infty\rangle}(RG)$ from smaller ‘building blocks’, namely the $K$- resp. $L$-groups of the group rings of the virtually cyclic subgroups of $G$. The conjecture has many implications. For example, if the $K$-theoretic version of the conjecture holds for a torsion-free group $G$, then both the Whitehead group $\text{Wh}(G)$ and the reduced projective class group $\tilde{K}_{0}(\mathbb{Z}G)$ vanish. Also, if both the $K$- and $L$-theoretic version hold for a group $G$, then the Borel Conjecture for $G$ follows. The Farrell–Jones Conjecture has been studied intensively over the past decade and is known for a large class of groups including hyperbolic groups [BLR08a, BL12a], $\text{CAT}(0)$-groups [BL12a, Weg12], virtually solvable groups [Weg15], $\text{GL}_{n}(\mathbb{Z})$ [BLRR14] and mapping class groups [BB16]. So far, no counter-example is known.

Proving the Farrell–Jones Conjecture for a group $G$ often starts with considering a suitable action of $G$ on a nice enough space $X$. Then it can be useful to regard the Farrell–Jones Conjecture not relative to $\mathcal{VCYC}$, but relative to a (larger) family $\mathcal{F}$ that includes all point-stabilisers of $G\curvearrowright X$: If one shows the conjecture for $G$ relative to $\mathcal{F}$ and all point-stabilisers are known to satisfy the conjecture relative to $\mathcal{VCYC}$, the Farrell–Jones Conjecture for $G$ relative to $\mathcal{VCYC}$ follows. Recently Bartels used this approach on relatively hyperbolic groups [Bar17], and the first objective of the present paper is to apply the same tactic to groups acting acylindrically on trees.

The precise formulation of and necessary background on the Farrell–Jones conjecture are given in Section 1. A detailed exposition on the numerous application of the Farrell–Jones Conjecture can be found in [BLR08b], [LR05], [Lüc10]. A survey on the various methods used to prove the conjecture in various cases can be found in [Bar14].

A discrete group $G$ acts $k$-acylindrically on a simplicial tree $T$ if the pointwise stabiliser of every geodesic segment of length $k$ in $T$ is finite (we do not require these stabilisers to be trivial or to have uniform cardinality). An action $G\curvearrowright T$ is called acylindrical if it is $k$-acylindrical for some $k$. Examples of acylindrical actions can be given by considering an amalgamated free product $G=A\ast_{C}B$, where $C$ is an almost malnormal subgroup in $A$ or $B$.

In general, for a group action $G\curvearrowright T$ on a simplicial tree, denote by $\mathcal{F}_{T}$ the family of subgroups of $G$ which are subconjugate to vertex stabilisers, and by $\mathcal{F}_{\partial}$ the family of subgroups of $G$ that fix a pair of boundary points of $T$ pointwise.

Theorem A**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$ and let the family $\mathcal{F}:=\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$ be as above. Let $\mathcal{F}_{2}$ be the family of subgroups of $G$ that contain a group in $\mathcal{F}$ as a subgroup of index $\leq 2$. Then

1. a)

$G$* satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}$;* 2. b)

$G$* satisfies the $L$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}_{2}$.*

In the special cases of [math]- or $1$-acylindrical actions this theorem was already known: If $G$ acts [math]-acylindrically on a tree $T$ (and $G$ is finitely generated), then $G$ is hyperbolic and the Farrell–Jones Conjecture relative to $\mathcal{VCYC}\subset\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$ holds [BLR08a, BL12a]. A more recent result of Bartels [Bar17] for relatively hyperbolic groups covers, in particular, the $1$-acylindrical case. We will see in Section 3 that it is easy to construct actions that are acylindrical but not $1$-acylindrical.

If the vertex stabilisers of the action $G\curvearrowright T$ are known to satisfy the Farrell–Jones Conjecture relative to $\mathcal{VCYC}$, we obtain the following corollary and its $L$-theoretic analogue, which is formulated in Corollary 4.3.

Corollary 4.2**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$. If all vertex stabilisers of $G\curvearrowright T$ satisfy the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$, then $G$ satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$.

The Farrell–Jones Conjecture (with coefficients in additive categories) relative to $\mathcal{VCYC}$ has various useful inheritance properties. For example, if two groups $A$ and $B$ satisfy the conjecture, then so do $A\times B$ and $A\ast B$. If $A$ and $B$ are abelian, then an amalgamated free product $G=A\ast_{C}B$ satisfies the Farrell–Jones Conjecture relative to $\mathcal{VCYC}$ by [GMR15], but for a general amalgamated free product this inheritance property is not known. In this context the last corollary can also be interpreted as follows: Any amalgamated free product $G=A\ast_{C}B$ with $C$ being almost malnormal in either $A$ or $B$ acts acylindrically on its Bass-Serre tree. Thus, the above corollary implies that the class of groups satisfying the Farrell–Jones Conjecture relative to $\mathcal{VCYC}$ is closed under taking amalgamated free products over an almost malnormal subgroup.

Another angle on amalgamated free products and the computation of algebraic $K$-theory is the following: For a group $G=A\ast_{C}B$ one can ask whether there is a Mayer-Vietoris type exact sequence

[TABLE]

In general this is not the case, and Waldhausen famously introduced an exact category $\mathfrak{Nil}(RC;R[A-C],R[B-C])$ encoding to what extent exactness of the above sequence fails in [Wal78a, Wal78b]. In particular, Waldhausen Nil-groups $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])$ associated to the amalgamated free product $G=A\ast_{C}B$ are direct summands in the homotopy groups of a certain non-connective spectrum version of $\mathfrak{Nil}(RC;R[A-C],R[B-C])$, and the sequence

[TABLE]

becomes exact. Naturally, one is interested in identifying $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])$ with something that is known to vanish often. For a precise, yet concise definition of the groups $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])$ we refer the reader to [DQR11, Section 3].

For an amalgamated free product $G=A\ast_{C}B$ that acts acylindrically on the associated Bass-Serre tree and is already known to satisfy the Farrell–Jones Conjecture relative to $\mathcal{VCYC}$, Lafont and Ortiz obtained an identification in [LO09]. Building on A, we are able to lift their second assumption. Namely, we obtain the following generalisation of their theorem.

Theorem B**.**

Let $G=A\ast_{C}B$ act acylindrically on the associated Bass-Serre tree $T$. Let $\mathcal{L}$ be a set of representatives for the orbits of the action $G\curvearrowright\partial T\times\partial T\setminus\operatorname{diag}$. Then for any ring $R$ there are isomorphisms

[TABLE]

for $n\in\mathbb{Z}$.

More information on the right hand side of the above isomorphisms–in particular, when they are known to vanish–can be found in Section 5.

The structure of this paper is as follows: Section 1 contains the background on the Farrell–Jones Conjecture necessary for us. In particular, we review the method to prove the Farrell–Jones Conjecture for a group $G$ by establishing the existence of a suitable compact space $X$ and intricate $G$-equivariant covers of $G\times X$. In our case of a group acting on a tree $T$, the space $X$ will be the Bowditch compactification of $T$ and this is the content of Section 2. Section 3 then gives some context on groups acting acylindrically on trees. To make the paper more accessible to readers less familiar with the covering-construction-culture around the Farrell–Jones Conjecture, Section 4 contains the proof of A modulo the lengthy and technical part and Section 5 contains the application to Waldhausen Nil-groups. Afterwards, the rest of the paper describes the plan to construct suitable covers of $G\times X$ in Section 6 and carries it out in Sections 7-10.

Conventions. Any countable group in this paper is assumed to be equipped with an implicitly chosen proper (left-)invariant metric. Given a metric space $(X,d)$, for $r>0$ and $x\in X$, we denote—if not explicitly stated otherwise—by $B_{r}^{d}(x)=B_{r}(x)$ the closed ball of radius $r$ around $x$. By a generalised metric on $X$ we mean a function $d:X\times X\to\mathbb{R}_{\geq 0}\cup\{\infty\}$ which is symmetric, satisfies the triangle inequality and $d(x,y)=0$ if and only if $x=y$. Finally, we stress that it is hard to draw an unbounded geodesic ray or a tree which is not locally finite. So all figures in this paper have to be regarded as ‘incomplete illustrations’ that hopefully still capture the essence of the notion or proof at hand.

Acknowledgements. The results of this paper stem from my PhD thesis and I am indepted to my advisor Arthur Bartels for many helpful discussions, for his encouragement and his support. This work was supported by the SFB 878 “Groups, Geometry & Actions”.

1. Background on the Farrell–Jones Conjecture

The conjecture goes back to Farrell and Jones [FJ93], and since then, its formulation has been further conceptualised and broadly generalised. Its present form (with coefficients in additive categories) was introduced by Bartels and Reich [BR07] (in the $K$-theory case) and Bartels and Lück [BL10] (in the $L$-theory case). The existence of an axiomatic formulation of geometric conditions implying the Farrell–Jones Conjecture for a group $G$, which was also introduced by Bartels, Lück and Reich (cf. Section 1.1), allows us—for almost all of this paper—to display (or feign) ignorance about the precise nature of most objects appearing in the Farrell–Jones Conjecture. Thus, we restrict ourselves to stating the conjecture (as formulated in [BR07] and [BL10]) essentially without further comment on the objects involved.

A family $\mathcal{F}$ of subgroups of $G$ is a collection of subgroups of $G$ closed under conjugation and taking subgroups. Examples being the family $\mathcal{ALL}$ of all subgroups of $G$, the family $\mathcal{FIN}$ of all finite subgroups of $G$, and most importantly, the family $\mathcal{VCYC}$ of all virtually cyclic subgroups of $G$.

Definition 1.1**.**

($K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}$) A group $G$ satisfies the $K$-theoretic Farrell–Jones Conjecture relative to the family of subgroups $\mathcal{F}$ if for all additive $G$-categories $\mathcal{A}$ the $K$-theoretic assembly map

[TABLE]

induced by the projection $E_{\mathcal{F}}G\to\operatorname{pt}$, is an isomorphism for all $n\in\mathbb{Z}$.

Definition 1.2**.**

($L$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}$) A group $G$ satisfies the $L$-theoretic Farrell–Jones Conjecture relative to the family of subgroups $\mathcal{F}$ if for all additive $G$-categories $\mathcal{A}$ with involution the $L$-theoretic assembly map

[TABLE]

induced by the projection $E_{\mathcal{F}}G\to\operatorname{pt}$, is an isomorphism for all $n\in\mathbb{Z}$.

As of now we use the term Farrell–Jones Conjecture (relative to $\mathcal{F}$) when talking about both of the above definitions simultaneously. So in particular, and contrary to the existing literature, we do not take $\mathcal{VCYC}$ to be the default family.

1.1. A geometric condition implying the Farrell–Jones Conjecture

Along with the proof of the $K$-theoretic Farrell–Jones Conjecture for hyperbolic groups, Bartels, Lück and Reich showed that the existence of a particular group action $G\curvearrowright X$ admitting intricate $G$-invariant covers implies the Farrell–Jones Conjecture (cf. [BLR08a, Theorem 1.1]). Since then these geometric conditions have been reformulated, generalised and adapted in several ways (cf. [BL12a, Section 1], [Weg12, Theorem 1.1 and Sections 2 and 3], [Bar17, Definition 0.1 and Theorem 4.3]). We review here the formulation that suits us most while mainly following [Bar17].

Definition 1.3**.**

Let $\mathcal{F}$ be a family of subgroups of a group $G$. A subset $U\subset X$ of a $G$-space $X$ is called an $\mathcal{F}$-subset if there is $F\in\mathcal{F}$ such that $gU=U$ for $g\in F$ and $gU\cap U=\emptyset$ for $g\not\in F$. A collection $\mathcal{U}$ of subsets of $X$ is said to be $G$-invariant if $gU\in\mathcal{U}$ for all $g\in G$ and $U\in\mathcal{U}$. An (open) $\mathcal{F}$-cover of $X$ is a cover of $X$ consisting of (open) $\mathcal{F}$-subsets.

Definition 1.4**.**

Let $\mathcal{U}$ be a collection of subsets of some space $X$. The order of $\mathcal{U}$ is $\leq N$ if each $x\in X$ is contained in at most $N+1$ members of $\mathcal{U}$. If the collection $\mathcal{U}$ is a cover of $X$, the order of $\mathcal{U}$ will also be called the dimension of $\mathcal{U}$ and then is denoted by $\dim\mathcal{U}$.

For the connection of the next definition to amenable actions we refer the reader to [Bar17, Remark 0.4].

Definition 1.5**.**

(cf. [Bar17, Definition 0.1]) Let $G$ be a countable group (with a chosen proper (left-)invariant metric) and let $\mathcal{F}$ be a family of subgroups. An action of $G$ on a space $X$ is $N-\mathcal{F}$-amenable if for any $\alpha>0$ there exists an open $G$-invariant $\mathcal{F}$-cover $\mathcal{U}_{\alpha}$ of $G\times X$ (equipped with the diagonal action) with the following properties:

1. a)

the dimension of $\mathcal{U}_{\alpha}$ is at most $N$; 2. b)

for all $(g,x)\in G\times X$ there is $U\in\mathcal{U}_{\alpha}$ with $B_{\alpha}(g)\times\{x\}\subset U$.

An action is finitely $\mathcal{F}$-amenable if it is $N-\mathcal{F}$-amenable for some $N$.

Since we will not construct those types of covers in one go, we introduce the following useful terminology reminiscent of [BLR08a, Assumption 1.4].

Definition 1.6**.**

Let $\alpha>0$ be given. A collection $\mathcal{U}$ of subsets of $G\times X$ is wide for $Y$, where $Y\subset G\times X$ is a subset, if for all $(g,x)\in Y$ there is $U\in\mathcal{U}$ with $B_{\alpha}(g)\times\{x\}\subset U$. The collection $\mathcal{U}$ is wide if it is wide for $Y=G\times X$.

Definition 1.7**.**

([BL12a, Definition 1.5]) Let $X$ be a metric space and $N\in\mathbb{N}$. Then $X$ is controlled $N$-dominated if, for every $\epsilon>0$, there are a finite CW-complex $K_{\epsilon}$ of dimension at most $N$, maps $i:X\to K_{\epsilon},p:K_{\epsilon}\to X$ and a homotopy $H:X\times[0,1]\to X$ between $p\circ i$ and $\text{id}_{X}$ such that for every $x\in X$ the diameter of $\{H(x,t)\ |\ t\in[0,1]\}$ is at most $\epsilon$. $X$ is controlled finitely-dominated if $X$ is controlled $N$-dominated for some $N$.

Theorem 1.8**.**

(cf. [Bar17, Theorem 4.3]) Let $G$ be a group and $\mathcal{F}$ be a family of subgroups. Let $\mathcal{F}_{2}$ be the family of subgroups of $G$ that contain a group in $\mathcal{F}$ as a subgroup of index $\leq 2$. If $G$ admits a finitely $\mathcal{F}$-amenable action on a compact contractible controlled finitely-dominated metric space $X$, then

1. a)

$G$* satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}$;* 2. b)

$G$* satisfies the $L$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}_{2}$.*

{proof}

b) is the conclusion of [BL12a, Theorem 1.1 (ii)]. The assumptions of [BL12a, Theorem 1.1] were phrased as ‘$G$ is transfer reducible over $\mathcal{F}$’ (cf. [BL12a, Definition 1.8]). The notion of transfer reducibility is more general than $N-\mathcal{F}$-amenability of an action on a suitable space. (Transfer reducibility neither demands a strict action of $G$ on $X$, nor that $X$ is independent of the given $\alpha>0$.) Moreover, transfer reducibility demands $G$-invariant covers for the action of $G$ on $G\times X$ that is given by $h(g,x):=(hg,x)$. Nevertheless, using that $(g,x)\mapsto(g,g^{-1}x)$ is a $G$-invariant homeomorphism from $G\times X$ with the diagonal action to $G\times X$ with the action described above, it is straightforward to check from the respective definitions that the existence of a finitely $\mathcal{F}$-amenable action of $G$ on a compact contractible controlled finitely-dominated metric space $X$ implies that $G$ is transfer reducible over $\mathcal{F}$ (cf. the comment behind [BL12a, Definition 1.4]).

a) is the conclusion of [Weg12, Theorem 1.1]. The assumptions of [Weg12, Theorem 1.1] were phrased as ‘$G$ is strongly transfer reducible over $\mathcal{F}$’ (cf. [Weg12, Definition 3.1]). Again, this notion is more general than $N-\mathcal{F}$-amenability and one can check, by translation of terminology, that the existence of a finitely $\mathcal{F}$-amenable action of $G$ on a compact contractible controlled finitely-dominated metric space $X$ implies that $G$ is strongly transfer reducible over $\mathcal{F}$.

1.2. The Farrell–Jones Conjecture and directed colimits

The Farrell–Jones Conjecture described above has various useful inheritance properties, see for instance [BEL08, Section 5] and [BFL14, Section 2.3]. These sources have mostly been interested in the conjecture relative to the family $\mathcal{VCYC}$, and subsequently the author could not find the following lemma in the literature.

Lemma 1.9**.**

Let $\{G_{i}\ |\ i\in I\}$ be a directed system of groups (with not necessarily injective structure maps). Let $G=\operatorname{colim}_{i\in I}G_{i}$, with structure maps $\phi_{i}:G_{i}\to G$, and let $\mathcal{F}$ be a family of subgroups of $G$. If for all $i\in I$ the group $G_{i}$ satisfies the $K$-theoretic (resp. $L$-theoretic) Farrell–Jones Conjecture relative to $\phi_{i}^{\ast}\mathcal{F}:=\{H\leq G_{i}\ |\ \phi_{i}(H)\in\mathcal{F}\}$, then $G$ satisfies the $K$-theoretic (resp. $L$-theoretic) Farrell–Jones Conjecture relative to $\mathcal{F}$.

{proof}

In the $L$-theoretic case, if $\mathcal{F}=\mathcal{VCYC}$ and $\phi_{i}^{\ast}\mathcal{F}$ is replaced by $\mathcal{VCYC}$ as well ($\mathcal{VCYC}(G_{i})$ that is), then this lemma is [BL10, Corollary 0.8]. This corollary is indicated in [BL10] to come from [BEL08, Theorem 5.6], which is formulated generally enough to also conclude the lemma in the $K$- and $L$-theoretic case provided the families $\mathcal{F}$ and $\phi_{i}^{\ast}\mathcal{F}$ of subgroups come from a class of groups closed under isomorphisms and taking quotients–a restriction that is not acceptable for our purposes. Luckily, using [BEL08, Theorem 5.2] instead, in combination with a straightforward generalisation of [BL10, Theorem 0.7] to arbitrary families, allows to deduce [BL10, Corollary 0.8] from [BEL08] in a way that works verbatim for arbitrary families and the $K$-theoretic case as well.

We wish to employ Theorem 1.8 in order to prove A. The next section introduces the right compact contractible controlled finitely-dominated metric space for this.

2. The observers’ topology on a tree

For proper geodesic metric hyperbolic spaces (so in particular for a locally finite tree $T$) adding the Gromov boundary results in a compact metrisable space (see [BH99, III.H.3, in particular 3.7, 3.18 (4)]). Since we are interested mostly in trees that are not locally finite, changing the topology on $T$ itself becomes necessary.

We denote by $\partial T$ the geodesic/Gromov boundary of $T$, by $V(T)$ the set of vertices of $T$, by $V_{0}(T)$ the set of vertices of finite degree/valency of $T$ and set $V_{\infty}(T):=V(T)\setminus V_{0}(T)$. Furthermore, $d_{T}$ will always denote the path-metric on $T$.

Notation 2.1**.**

For two points $x,y$ in $T\cup\partial T$ we denote both the geodesic from $x$ to $y$ and its image by $[x,y]$. In particular, our terminology does not distinguish between geodesics, geodesic rays and bi-infinite geodesic rays. Moreover, we use $(x,y]$ and $[x,y)$ for $[x,y]\setminus\{x\}$ and $[x,y]\setminus\{y\}$, respectively (provided $x\in T$ and $y\in T$, respectively).

Bowditch’s construction of the observers’ topology amounts to compactifying $\Delta(T):=V_{\infty}(T)\cup\partial T$. We will need to use a compactification—via the observers’ topology—of all of $T\cup\partial T$ instead. Written in the more general language of $\mathbb{R}$-trees this can be found in [CHL07, Section 1].

Definition 2.2**.**

The topology on $T\cup\partial T$ given by the basis

[TABLE]

where $M(z,A):=\{y\in T\cup\partial T\ |\ A\cap[z,y]=\emptyset\}$, is called the observers’ topology. The resulting space is the Bowditch compactification of $T$ and will be denoted by $\overline{T}^{obs}$. The Bowditch boundary of $T$ is the subspace $\Delta(T)=V_{\infty}(T)\cup\partial T\subset\overline{T}^{obs}$. We further set $\Delta_{+}(T):=V(T)\cup\partial T\subset\overline{T}^{obs}$.

The above naming is justified since it is compatible with Bowditch’s construction, i.e. the subspace topology on $\Delta(T)\subset\overline{T}^{obs}$ is the topology constructed by Bowditch in [Bow12, Section 8, p. 51ff]. Note also that $M(z,A)$ consists of all points in $T\cup\partial T$ that can be reached from $z$ via a geodesic that does not run through a point of $A$. Therefore, if $x\in M(z,A)$, we have $M(x,A)=M(z,A)$.

Lemma 2.3**.**

[CHL07, Proposition 1.13]** The space $\overline{T}^{obs}$ is Hausdorff. Moreover, if $T$ is separable, then the space $\overline{T}^{obs}$ is separable and compact.

Lemma 2.4**.**

If $T$ is countable (i.e. has only countably many edges), then $\overline{T}^{obs}$ is second-countable and metrisable. In particular, (any subspace of) $\overline{T}^{obs}$ is separable.

{proof}

If $T$ is countable, then $T$ is separable. Choose a countable dense subset $Q$ of $T$ which contains all vertices of $T$. Then

[TABLE]

is countable and still a basis for the observers’ topology, as $M(x,A)=M(z,A)$ holds for all $z\in M(x,A)$. So $\overline{T}^{obs}$ is second-countable. Since, by the above lemma, $\overline{T}^{obs}$ is also compact Hausdorff, the space $\overline{T}^{obs}$ is metrisable. Finally, for metric spaces being second-countable and being separable are equivalent properties, so any subspace of $\overline{T}^{obs}$ is separable as well.

We proceed to show that $\overline{T}^{obs}$ is contractible and controlled 1-dominated (see Definition 1.7). We split the proof into two lemmata.

Lemma 2.5**.**

Let $d$ be a metric on $\overline{T}^{obs}$ compatible with the observers’ topology. For every $\epsilon>0$ there is a finite subtree $K_{\epsilon}\subset T$ and a continuous projection $p:\overline{T}^{obs}\to K_{\epsilon}$ such that $d(x,p(x))<\epsilon$ for all $x\in\overline{T}^{obs}$.

{proof}

For this proof we denote the open ball of radius $r$ around $x\in\overline{T}^{obs}$ with respect to $d$ by $B_{r}(x)$. Certainly,

[TABLE]

is an open cover of $\overline{T}^{obs}$. Since the $M(x,A)$ with $x\in T\cup\partial T$ and $A$ a finite subset of $T$ form an open neighbourhood basis at $x$ for the observers’ topology, there is for each $B_{\nicefrac{{\epsilon}}{{2}}}(x)$ an open set $M(x,A_{x})\subset B_{\nicefrac{{\epsilon}}{{2}}}(x)$. So the collection

[TABLE]

is an open cover of $\overline{T}^{obs}$ as well. Since $\overline{T}^{obs}$ is compact, we can find a finite subcover $M(x_{1},A_{1}),\ldots,M(x_{l},A_{l})$. Without loss of generality we can assume that the $x_{i}$ lie in $V(T)$. Let $K_{\epsilon}$ be the finite subtree of $T$ spanned by $x_{1},\ldots,x_{l}$. We define $p:\overline{T}^{obs}\to K_{\epsilon}$ as follows: Fix some point $b\in K_{\epsilon}$ and define $p:\overline{T}^{obs}\to K_{\epsilon}$ by sending $x$ to the last vertex on $[b,x]$ that is contained in $K_{\epsilon}$.

To show $d(x,p(x))<\epsilon$, we argue as follows: For $x\in K_{\epsilon}$ we have $d(x,p(x))=0$, since $p_{|K_{\epsilon}}=\operatorname{id}_{K_{\epsilon}}$. For any point $x\not\in K_{\epsilon}$ there is an $x_{i_{x}}$ such that $x\in M(x_{i_{x}},A_{i})$. Since $K_{\epsilon}$ is the tree spanned by $x_{1},\ldots,x_{l}$, the point $p(x)$ lies on $[x,x_{i_{x}}]$, and hence $p(x)\in M(x_{i_{x}},A_{i})$. Therefore, $d(p(x),x)<\epsilon$. The proof that $p$ is indeed continuous is not very iluminating and only uses the fact that inside each $B_{\delta}(x)$ one can find suitable $M(x,A)$ that are guaranteed to be maped into the given $B_{\tilde{\epsilon}}(p(x))$.

Lemma 2.6**.**

The space $\overline{T}^{obs}$ is controlled 1-dominated (by the $K_{\epsilon}$ of Lemma 2.5). In particular, $\overline{T}^{obs}$ is contractible.

{proof}

Let $d$ be a metric on $\overline{T}^{obs}$ compatible with the observers’ topology. Again, for this proof denote the open ball of radius $r$ around $x\in\overline{T}^{obs}$ with respect to $d$ by $B_{r}(x)$. Let $\epsilon>0$ and let $p:\overline{T}^{obs}\to K_{\epsilon}$ be as in the previous lemma, i.e. we have a finite subtree $K_{\epsilon}$ of $T$, fixed a point $b\in K_{\epsilon}$ and defined $p(x)$ as the last vertex on $[b,x]$ that also lies in $K_{\epsilon}$. We now have to show that there is a homotopy $H:\overline{T}^{obs}\times[0,1]\to\overline{T}^{obs}$ between $\operatorname{id}_{\overline{T}^{obs}}$ and $i\circ p:\overline{T}^{obs}\to K_{\epsilon}\to\overline{T}^{obs}$, where $i:K_{\epsilon}\to\overline{T}^{obs}$ is the inclusion, such that the set $\{H(x,t)\ |\ t\in I\}$ has diameter $\leq 2\epsilon$ for all $x\in\overline{T}^{obs}$.

For $x\in\overline{T}^{obs}$ let $\gamma_{p(x),x}:[0,\infty)\to T$ be the unique generalised geodesic in $T$ from $p(x)$ to $x$, i.e. if $x\in T$, then $\gamma(t)$ is stationary for $t\geq d_{T}(p(x),x)$. We define

[TABLE]

where for $t=1$ and $x\in T$ this is to be read as $H(x,1)=x$, and for $t=1$ and $x\in\partial T$ this is to be read as $H(x,1)=[\gamma_{p(x),x}]$. So $H(-,0)=p$ and $H(-,1)=id_{\overline{T}^{obs}}$. Furthermore, note that the set $\{H(x,t)\ |\ t\in I\}$ is exactly the image of the generalised geodesic $\gamma_{p(x),x}$. Any point on $\gamma_{p(x),x}$ is mapped by $p$ to $p(x)$. Thus, this set has diameter $\leq 2\epsilon$ by the previous lemma. Again the proof that $H$ is indeed contiunous is an exercise in choosing suitable $M(-,A)$ inside the appropiate $B_{\delta}(x)$.

We will also need to know that the (small inductive) dimension111Recall that for separable metrisable spaces, the small inductive dimension coincides with the covering dimension (and the strong inductive dimension) [Nag65, Theorem IV.1, p. 90]. of $\Delta_{+}(T)$ is 0.

Definition 2.7**.**

Let $X$ be a regular space. The small inductive dimension $\operatorname{ind}X$ is $\leq n$ if for all $x\in X$ and all neighbourhoods $V\subset X$ of $x$ there is an open set $U\subset X$ such that $x\in U\subset V$ and $\operatorname{ind}\partial U\leq n-1$. The small inductive dimension of the empty set is defined as $\operatorname{ind}\emptyset:=-1$.

Lemma 2.8**.**

The space $\Delta_{+}(T)$ has (small inductive) dimension 0.

{proof}

By definition, $\mathcal{B}:=\{M(z,A)\ |\ z\in T\cup\partial T,A\subset T\ \text{finite}\}$ is a basis for the topology of $\overline{T}^{obs}$. So if $V$ is a neighbourhood of some $y$ in $\Delta_{+}(T)$, then there is a set $M(z,A)\in\mathcal{B}$ with $y\in M(z,A)\cap\Delta_{+}(T)=:U\subset V$. In particular $U$ is open in $\Delta_{+}(T)$ and we wish to show that the boundary of $U$ (in $\Delta_{+}(T)$) is empty.

Since $y\in M(z,A)$, we can write $U=M(y,A)\cap\Delta_{+}(T)$. Now let $x_{n}\to x$ be a convergent sequence in $\Delta_{+}(T)$ with $x_{n}\in U$ for all $n\in\mathbb{N}$. If the $x_{n}$ are contained in a finite subtree $S$ of $T$, then–since $x_{n}\in\Delta_{+}(T)$–the sequence must have a constant subsequence and $x\in U$ follows. So assume from now on that the sequence is not contained in a finite subtree of $T$. If $x\not\in U$, then there must be an $a\in A$ lying in the interior of $[z,x]$. Thus $M(x,a)\cap\Delta_{+}(T)$ is an open set containing $x$, but none of the $x_{n}$. A contradiction to $x_{n}\rightarrow x$.

3. Acylindrical actions on trees

The original definition of a $k$-acylindrical action of a group on a simplicial tree is due to Sela [Sel97, p. 528]. And in recent years Osin established Bowditch’s more general notion of an acylindrical action of a group on a path-metric space [Bow08, p. 284] as part of his definition of acylindrically hyperbolic groups [Osi16]. We will however use Delzant’s definition of a $(k,\mathcal{FIN})$-acylindrical action (see [Del99, Definition, p. 1215]), which in the case of simplicial actions on trees generalises both Sela’s and Bowditch’s definition.

Definition 3.1**.**

Let $k\geq 0$. An action $G\curvearrowright T$ on a simplicial tree is $k$-acylindrical if the pointwise stabiliser of any geodesic segment of length $k$ is finite.

We say the action is uniformly $k$-acylindrical, if there is a uniform bound on the cardinality of the pointwise stabiliser of geodesic segments of length $k$. The action is strictly $k$-acylindrical, if the pointwise stabiliser of any geodesic segment of length $k$ is trivial. And we say an action is (strictly/uniformly) acylindrical, if it is (strictly/uniformly) $k$-acylindrical for some $k\in\mathbb{N}$.

Remark 3.2**.**

In Sela’s original definition of $k$-acylindricity the action was required to be strictly $k+1$-acylindrical in the above sense. And Bowditch’s definition applied to a simplicial tree is equivalent to a uniformly $k$-acylindrical action in the above sense.

In the case $k=0$ the action $G\curvearrowright T$ has finite point stabilisers. Hence, if $G$ is finitely generated, then the existence of a [math]-acylindrical action on a simplicial tree implies that $G$ is word hyperbolic (since $G$ is finitely generated we can assume without loss of generality that $G$ acts cocompactly on $T$).

In the case $k=1$ the action $G\curvearrowright T$ has finite edge stabilisers. Since any tree is, in particular, a fine hyperbolic graph in the sense of Bowditch (cf. [Bow08, Definition after Proposition 2.1, p. 11]), if $G$ is finitely generated the existence of a $1$-acylindrical action on a simplicial tree with finitely generated vertex stabilisers implies that $G$ is strongly relatively hyperbolic, i.e. relatively hyperbolic in the sense of Bowditch (cf. [Bow08, Definition 2]). Examples of $1$-acylindrical actions on simplicial trees can be readily given: For groups $A$ and $B$ with an (up to isomorphism) common finite subgroup $C$, the amalgamated free product $A\ast_{C}B$ acts $1$-acylindrically on the Bass-Serre tree associated to $A\ast_{C}B$.

Plenty of examples for $k$-acylindrical actions that are not $1$-acylindrical can be given using the classical notion of an (almost) malnormal subgroup.

Definition 3.3**.**

A subgroup $H\leq G$ is malnormal resp. almost malnormal if for all $g\in G\setminus H$ the intersection $H\cap gHg^{-1}$ is trivial resp. finite. We call a subgroup $H\leq G$ uniformly almost malnormal if there is a bound $K>0$ such that for all $g\in G\setminus H$ the intersection $H\cap gHg^{-1}$ has at most $K$ elements.

From the explicit description of the Bass-Serre tree associated to an amalgamated free product (see [Ser80, p. 33]) the following lemma follows readily.

Lemma 3.4**.**

(cf. [Sel97, p. 528]) Let $C\leq A$ be (uniformly) almost malnormal and let $C\leq B$. Then the action of $G:=A\ast_{C}B$ on the Bass-Serre tree associated to $G$ is (uniformly) 3-acylindrical. In particular, if $C$ is infinite, then $G$ does not act 1-acylindrical on its Bass-Serre tree.

Similar statements can be formulated for HNN-extensions and graphs of groups using the notion of an almost malnormal collection of subgroups.

There are plenty of examples for (almost) malnormal subgroups (for an overview on results giving such groups see for instance [dlHW14]). In particular, peripheral subgroups in relatively hyperbolic groups are uniformly almost malnormal (see for instance [DS08, Lemma 4.20]). On the other hand, it is worth noting that not all acylindrical actions arise in this fashion from (almost) malnormal subgroups. As can be seen (with some translation of terminology) from the type of example given in [KS71, p. 946 and p. 951], there are amalgamated free products $A\ast_{C}B$ that act 3-acylindrically on their associated Bass-Serre tree, with $C$ not being almost malnormal in $A$ or $B$.

Notation 3.5**.**

Let $G\curvearrowright T$ be an action of a group on a tree. We denote by $\mathcal{F}_{T}$ the family of subgroups $\mathcal{F}_{T}:=\{H\leq G\ |\ \exists\ x\in T:\ H\leq G_{x}\}$ and by $\mathcal{F}_{\partial}$ the family of subgroups $\mathcal{F}_{\partial}:=\{H\leq G\ |\ \exists\ (x,y)\in\partial T\times\partial T\setminus\operatorname{diag}:\ H\leq G_{x}\cap G_{y}\}$. Here $G_{x}$ denotes the stabiliser of a point $x\in T\cup\partial T$.

It is known that $\mathcal{F}_{\partial}\subset\mathcal{VCYC}$ holds for acylindrical amalgamations (see [LO09, Claim 2]). The author expects the next lemma also to be well-known, but could not find a reference for this particular statement either.

Lemma 3.6**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$. Then each element in $\mathcal{F}_{\partial}$ is either finite or virtually cyclic of type I (i.e. surjects onto $\mathbb{Z}$).

{proof}

For $g\in G$, if the translation length $||g||$ of $g$ is $>0$, denote by $C_{g}$ its translation axis. If $||g||=0$, denote by $C_{g}$ the subtree of $T$ which is pointwise fixed by $g$. Let $\xi\in\partial T$. Then for all $g\in G_{\xi}$ the set $C_{g}$ has unbounded intersection with each geodesic ray $\rho$ representing $\xi$. ($C_{g}\cap\rho$ being empty or bounded immediately gives a contradiction to $g\in G_{\xi}$, since $T$ is a tree; cf. Figure 1 for an exemplary case).

Let $(\xi_{1},\xi_{2})\in\partial T\times\partial T\setminus\operatorname{diag}$. Then any $g\in G_{\xi_{1}}\cap G_{\xi_{2}}$ must satisfy $[\xi_{1},\xi_{2}]\subset C_{g}$ (and equality holds if $||g||>0$). Hence, $G_{\xi_{1}}\cap G_{\xi_{2}}$ can only contain finitely many elements with translation length [math]. In particular, if the group $G_{\xi_{1}}\cap G_{\xi_{2}}$ consists only of elements with translation length [math], this group is finite. So from now on we assume that there is at least one element $\overline{g}\in G_{\xi_{1}}\cap G_{\xi_{2}}$ with $||\overline{g}||>0$. Since $||g||\in\mathbb{N}$ there is $g_{0}\in G_{\xi_{1}}\cap G_{\xi_{2}}$ with $||g_{0}||>0$ minimal.

The translation length $||h||$ of any $h\in G_{\xi_{1}}\cap G_{\xi_{2}}$ with $||h||>0$ must be divisible by $||g_{0}||$: Assume $||g_{0}||$ does not divide $||h||$. Then there are integers $\alpha,\beta$ such that $||g_{0}||>\text{gcd}(||g_{0}||,||h||)=\alpha||g_{0}||+\beta||h||$. Hence, the element $g_{0}^{\alpha}h^{\beta}$ acts as a shift on $[\xi_{1},\xi_{2}]$ and therefore lies in $G_{\xi_{1}}\cap G_{\xi_{2}}$. Since $g_{0}^{\alpha}h^{\beta}$ has translation length less than $||g_{0}||$, this is a contradiction.

Let $\{g_{1},\ldots,g_{K}\}$ be the set of all elements in $G_{\xi_{1}}\cap G_{\xi_{2}}$ with translation length [math]. If $h\in G_{\xi_{1}}\cap G_{\xi_{2}}$ and $||h||=0$, then $h=g_{i}$ for some $1\leq i\leq K$ and, in particular, $h=g_{i}g_{0}^{0}\in g_{i}\langle g_{0}\rangle$. If $h\in G_{\xi_{1}}\cap G_{\xi_{2}}$ with $||h||=\alpha||g_{0}||>0$, then one of the two elements $hg_{0}^{\pm\alpha}$ fixes $[\xi_{1},\xi_{2}]$ pointwise and thus must lie in $\{g_{1},\ldots,g_{K}\}$. In other words, we have $h=g_{i}g_{0}^{\alpha}$ for some $\alpha\in\mathbb{Z}$ and $1\leq i\leq K$. Hence, $\mathbb{Z}=\langle g_{0}\rangle$ is of finite index in $G_{\xi_{1}}\cap G_{\xi_{2}}$. Moreover, sending $g_{i}g_{0}^{\alpha}$ to $g_{0}^{\alpha}$ gives the desired surjection onto $\mathbb{Z}$.

4. The proof of Theorem A modulo

$\mathcal{F}$-amenability of $G\curvearrowright\Delta(T)$

Theorem A**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$ and let the family $\mathcal{F}:=\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$ be as in 3.5. Let $\mathcal{F}_{2}$ be the family of subgroups of $G$ that contain a group in $\mathcal{F}$ as a subgroup of index $\leq 2$. Then

1. a)

$G$* satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}$;* 2. b)

$G$* satisfies the $L$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}_{2}$.*

We would like to use the geometric conditions from Section 1.1 to conclude the above theorem by basically showing that $G$ acts finitely $\mathcal{F}$-amenable on $\overline{T}^{obs}$. The major work, which also turns out to be lenghty and technical, towards this is to establish the following proposition.

Proposition 4.1**.**

Let $G$ be a countable group and $T$ be a countable tree. Let $G\curvearrowright T$ be a (not necessarily strictly or uniformly) acylindrical action without a global fixed point and let $\mathcal{F}:=\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$ (as in 3.5). Then the action $G\curvearrowright\Delta(T)$ of $G$ on the Bowditch boundary of $T$ is finitely $\mathcal{F}$-amenable.

To make this paper more accessible to readers unfamiliar with the covering-construction-culture around the Farrell–Jones Conjecture, we postpone the proof of the above proposition to the Sections 6-10.

{proof}

[Proof of A (modulo the proof of Proposition 4.1)] By Lemma 1.9, it suffices to prove the theorem for finitely generated $G$. Furthermore, if $G\curvearrowright T$ had a global fixed point, then $\mathcal{F}=\mathcal{ALL}$ and the conclusion of the theorem is imminent. Thus, we can assume without loss of generality that there is a minimal $G$-invariant subtree $T_{\min}$ of $T$ such that $G$ acts cocompactly on $T_{\min}$. In particular, $T_{\min}$ has only countably many edges and the restricted action is still acylindrical. If the Farrell–Jones Conjecture holds for a group $G$ relative to a family $\mathcal{G}$, then it also holds for $G$ relative to any larger family $\mathcal{G}^{\prime}\supset\mathcal{G}$ (this is a special case of the transitivity principle [BFL14, Theorem 2.10]). Hence, it suffices to show the theorem for countable $G$ and $T$ and fixed point free acylindrical actions $G\curvearrowright T$.

Assuming Proposition 4.1, we proceed to show that the action of $G$ on $\overline{T}^{obs}$ is finitely $\mathcal{F}$-amenable: By Proposition 4.1, there is $N^{\prime}$ such that the action of $G$ on the Bodwitch boundary $\Delta(T)$ is $N^{\prime}-\mathcal{F}$-amenable. Let $N:=N^{\prime}+2$. To produce, for all $\alpha>0$, an open $G$-invariant $\mathcal{F}$-cover of $G\times\overline{T}^{obs}$ that is wide and of dimension at most $N$, we first tend to covering $G\times\overline{T}^{obs}\setminus G\times\Delta(T)$.

For a moment we consider $T$ with the path-metric topology and define two kinds of open sets of $T$. Let $\mathcal{I}$ be all open edges of $T$ and $\mathcal{B}$ be all open balls of radius $\epsilon$ (i.e. some fixed small positive number) around vertices of $T$. Then, both $\mathcal{I}$ and $\mathcal{B}$ are $G$-invariant collections of open $\mathcal{F}$-subsets of $T$ (with the path-metric topology). Since the observers’ topology coincides with the path-metric topology on finite subtrees, it follows that $\mathcal{V^{\prime}}:=\{U\setminus V_{\infty}(T)\ |\ U\in\mathcal{I}\cup\mathcal{B}\ \}$ consists of open sets of $\overline{T}^{obs}$. Moreover, $\mathcal{V^{\prime}}$ is a $G$-invariant collection of open $\mathcal{F}$-subsets of $\overline{T}^{obs}$, since $V_{\infty}(T)$ is $G$-invariant. As $\mathcal{I}\cup\mathcal{B}$ is of order 1, so is $\mathcal{V^{\prime}}$. Define $\mathcal{V}:=\{G\times U^{\prime}\ |\ U^{\prime}\in\mathcal{V}^{\prime}\}$. This is a $G$-invariant collection of open $\mathcal{F}$-subsets of $G\times\overline{T}^{obs}$ which still has order 1. For $\xi\in\overline{T}^{obs}\setminus\Delta(T)$ there is an $U^{\prime}\in\mathcal{V}^{\prime}$ with $\xi\in U^{\prime}$. Thus—independent of $\alpha>0$—for all $(g,\xi)\in G\times(\overline{T}^{obs}\setminus\Delta(T))$ there is an element of $\mathcal{V}$ that contains $B_{\alpha}(g)\times\{\xi\}$. In other words, $\mathcal{V}$ is wide for $G\times(\overline{T}^{obs}\setminus\Delta(T))$.

Now, let $\alpha>0$ be fixed. By Proposition 4.1, there is an open $G$-invariant $\mathcal{F}$-cover $\mathcal{U}_{\alpha}$ of $G\times\Delta(T)$ of dimension at most $N^{\prime}$ which is wide. Of course the sets in $\mathcal{U}_{\alpha}$ are not open sets of $G\times\overline{T}^{obs}$, but it is possible to thicken the collection $\mathcal{U}_{\alpha}$ to an open collection $\mathcal{U}_{\alpha}^{+}$ of $G\times\overline{T}^{obs}$ without losing any of the desired properties ($G$-invariace, order, wideness) of $\mathcal{U}_{\alpha}$ (see [Bar17, Appendix B], [BL12b, Lemma 4.14]). Let $\mathcal{U}_{\alpha}^{+}$ be the result of this thickening process. So $\mathcal{U}_{\alpha}^{+}$ is a $G$-invariant collection of open $\mathcal{F}$-subsets of $G\times\overline{T}^{obs}$ that is wide for $G\times\Delta(T)$ and has order at most $N^{\prime}$. Defining $\mathcal{V}_{\alpha}:=\mathcal{V}\cup\mathcal{U}^{+}_{\alpha}$ gives the desired open $G$-invariant $\mathcal{F}$-cover of $G\times\overline{T}^{obs}$ which is wide for all of $G\times\overline{T}^{obs}$ and is of dimension at most $N=N^{\prime}+2$. Since we have now established finitely $\mathcal{F}$-amenability for the action $G\curvearrowright\overline{T}^{obs}$, to apply Theorem 1.8 we merely have to recall that $\overline{T}^{obs}$ is a compact contractible controlled 1-dominated metric space (see Lemma 2.3, Lemma 2.4 and Lemma 2.6).

Inheritance properties now allow us to deduce the Farrell–Jones Conjecture relative to $\mathcal{VCYC}$ if it is known (relative to $\mathcal{VCYC}$) for all groups in $\mathcal{F}$.

Corollary 4.2**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$. If all vertex stabilisers of $G\curvearrowright T$ satisfy the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$, then $G$ satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$.

{proof}

By A, $G$ satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$. Hence, by Lemma 3.6, $G$ satisfies the $K$-theoretic Farrell–Jones Conjecture relative to the larger family $\mathcal{F}_{T}\cup\mathcal{VCYC}$. Since the Farrell–Jones Conjecture relative to $\mathcal{VCYC}$ is closed under taking subgroups (see [BFL14, Theorem 2.8]), if $H\in\mathcal{F}_{T}$, then $H$ satisfies the $K$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$ by assumption. If $H\in\mathcal{VCYC}$, $H$ satisfies the $K$-theoretic Farrell–Jones Conjecture trivially. Thus, by the transitivity principle for the Farrell–Jones Conjecture (see [BFL14, Theorem 2.10]) the claim follows.

The $L$-theoretic case works analogously, but we have to take overgroups of index 2 of vertex stabilisers of $G\curvearrowright T$ into account.

Corollary 4.3**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$. Assume that all (subgroups of $G$ that are) overgroups of index at most 2 of vertex stabilisers of $G\curvearrowright T$ satisfy the $L$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$. Then $G$ satisfies the $L$-theoretic Farrell–Jones Conjecture relative to $\mathcal{VCYC}$.

These two corollaries have a counterpart for the Farrell–Jones Conjecture with finite wreath products: A group $G$ is said to satisfy the Farrell–Jones Conjecture with finite wreath products relative to a family $\mathcal{F}$ if for all finite groups $F$ the wreath product $G\wr F$ satisfies the Farrell–Jones Conjecture relative to $\mathcal{F}$. Recalling that we proved A basically by showing that the group $G$ involved is strongly transfer reducible over $\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$ (cf. the proof of Theorem 1.8), we can obtain the following variant of the last two corollaries by using [BLRR14, Theorem 5.1] and the transitivity principle.

Corollary 4.4**.**

Let $G$ be a group acting acylindrically on a simplicial tree $T$. Assume that all vertex stabilisers of $G\curvearrowright T$ satisfy the Farrell–Jones Conjecture with finite wreath products relative to $\mathcal{VCYC}$. Then $G$ satisfies the Farrell–Jones Conjecture with finite wreath products relative to $\mathcal{VCYC}$.

5. The proof of Theorem B

Theorem B**.**

Let $G=A\ast_{C}B$ act acylindrically on the associated Bass-Serre tree $T$. Let $\mathcal{L}$ be a set of representatives for the orbits of the action $G\curvearrowright\partial T\times\partial T\setminus\operatorname{diag}$. Then for any ring $R$ there are isomorphisms

[TABLE]

for $n\in\mathbb{Z}$.

We now need some knowledge on the objects appearing in the $K$-theoretic Farrell–Jones Conjecture. In general, for any family $\mathcal{G}$ of subgroups of $G$, $E_{\mathcal{G}}(G)$ denotes the classifying space for the family $\mathcal{G}$. Any $G$-CW-complex $X$ with $X^{H}\simeq\operatorname{pt}$ for $H\in\mathcal{G}$ and $X^{H}=\emptyset$ for $H\not\in\mathcal{G}$ is a model for $E_{\mathcal{G}}G$.

We next give a suitable model for $E_{\mathcal{F}}G$, where $\mathcal{F}=\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$ is again defined as in 3.5. In particular, we do not yet require $G$ to be an amalgamated product or the action of $G$ on $T$ to be acylindrically. The author expects the following construction to be well known, but could not find a reference in the literature.

Let $\mathcal{L}$ be a set of representatives for the orbits of the action $G\curvearrowright\partial T\times\partial T\setminus\operatorname{diag}$. We think of $\mathcal{L}$ as a collection of bi-infinite geodesic rays $L$ in $T$ and we denote by $G_{L}$ the stabiliser of the element in $\mathcal{L}$ given by $L$. In particular, each $G_{L}$ is an element of $\mathcal{F}_{\partial}$. Denote by $\text{cone}(L)$ the simplicial complex (and hence CW-complex) obtained by taking the simplicial join $p_{L}\ast L$. Both $L$ and $\text{cone}(L)$ are $G_{L}$-CW-complexes, and thus $G\times_{G_{L}}L$ and $G\times_{G_{L}}\text{cone}(L)$ are $G$-CW-complexes. Each $G\times_{G_{L}}L$, when viewed as a CW-complex, is the disjoint union of lines. The complex $G\times_{G_{L}}L$ contains one line for each element in the set $\{gL\ |\ g\in G\}$. Similarly, $G\times_{G_{L}}\text{cone}(L)$ viewed as a CW-complex is the disjoint union of coned off lines and contains one coned off line for each element in the set $\{\text{cone}(gL)\ |\ g\in G\}$.

Since $T$ is a $G$-CW-complex as well, we can form the following pushout of $G$-CW-complexes, where $i$ is the canonical inclusion induced by the inclusion of each line into its cone and $j_{L}$ is the $G$-map given by sending $(g,t)\in G\times_{G_{L}}L$ to $gt\in gL\subset T$.

[TABLE]

Lemma 5.1**.**

The $G$-CW-complex $Y$ given by the above pushout is a model for $E_{\mathcal{F}}G$.

{proof}

The $G$-CW-complex $Y$ is obtained from $T$ by equivariantly gluing coned off lines onto $T$ along their ‘baselines’. Thus, the intersection of (the images of) different cones in $Y$ lies in $T$. In particular, if $L\neq L^{\prime}$, then the cone point $p_{L}$ over $L$ is distinct from the cone point $p_{L^{\prime}}$ over $L^{\prime}$. It follows that each $p_{L}$ is fixed exactly by the elements in $G_{L}$. Moreover, since $Y$ is a $G$-CW-complex, elements of $G$ that do not lie in some $G_{L}$ can only fix points in $T$. Let $H\in\mathcal{F}=\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$. If $H\not\in\mathcal{F}_{\partial}$, then $H$ fixes only points in $T$, and $T^{H}$ is non-empty and contractible.

If $H\in\mathcal{F}_{\partial}$ and $H$ contains an element of positive translation length (with respect to $G\curvearrowright T$), then $H$ can not fix a point in $T$. Furthermore, such a group $H$ fixes exactly one cone point $p_{L}$ for a unique line $L$ and—since it does not fix any point in $T$—it does not fix any other point in $\text{cone}(L)$. Hence, in this case $Y^{H}=p_{L}$.

If $H\in\mathcal{F}_{\partial}$ and all $h\in H$ have translation length zero, then any line $L$ for which $Hp_{L}=p_{L}$ holds must be fixed pointwise by $H$. Hence, for any such line $\text{cone}(L)$ must be fixed pointwise by $H$ as well. There might be more than one such line, but $Y^{H}$ is still contractible: Since different cones can only intersect in $T$, we can first contract $Y^{H}$ to $T^{H}$ by simultaneously retracting all cones $\text{cone}(L)$ in $Y^{H}$ to their respective lines $L$, which all lie in $T^{H}$. Then we contract $T^{H}$ to a point.

Conversely, if for some $H\leq G$ the set $Y^{H}$ is non-empty, then (since $Y$ is a $G$-CW-complex) $H$ must fix at least one 0-cell of $Y$. Thus, $H$ either fixes at least one vertex of $T$ or is a subgroup of at least one of the $G_{L}$. Then, by definition, $H$ lies in $\mathcal{F}_{T}\cup\mathcal{F}_{\partial}=\mathcal{F}$.

In the sequel we will also use the fact that for every ring $R$ there is an equivariant homology theory $H_{\ast}^{?}(-;\mathbf{K}_{R})$ such that for all groups $G$ the $G$-homology theory $H_{\ast}^{G}(-;\mathbf{K}_{R})$ is the $G$-homology theory appearing in the formulation of the $K$-theoretic Farrell–Jones Conjecture with coefficients in the ring $R$ (see [BEL08, Theorem 6.1]). In particular, we assume the reader is comfortable with the properties of an equivariant homology theory (see [Lüc02, Section 1] for a definition).

{proof}

[Proof of B] Davis, Quinn and Reich, in [DQR11, Lemma 3.1], identified the group $\widetilde{\text{Nil}}_{n-1}(RC;R[A-C],R[B-C])$ with the cokernel of the map

[TABLE]

By A a), this cokernel is isomorphic to the cokernel of the map

[TABLE]

which can be determined by exploiting the properties of $H_{\ast}^{?}(-;\mathbf{K}_{R})$ and the explicit model for $E_{\mathcal{F}}G$ constructed above. Applying the $G$-homology theory $H_{\ast}^{G}(-;\mathbf{K}_{R})$ (abbreviated to $H_{\ast}^{G}(-)$ in the following) to the pushout (1) yields a Mayer-Vietoris sequence

[TABLE]

The space $\text{cone}(L)$ is $G_{L}$-equivariantly contractible and $L$ is a model for $E_{\mathcal{FIN}}G_{L}$. Thus, using the induction structure of $H_{\ast}^{?}$ for the $G$-CW-complexes $G\times_{G_{L}}L=\text{ind}_{G_{L}}^{G}L$ and $G\times_{G_{L}}\text{cone}(L)=\text{ind}_{G_{L}}^{G}(\text{cone}(L))$, the above sequence becomes

[TABLE]

Furthermore, the map $f_{2}$ is induced by the projections $\text{pr}_{L}:E_{\mathcal{FIN}}G_{L}\to\operatorname{pt}$. In other words, $f_{2}=\bigoplus_{L\in\mathcal{L}}\text{asmb}_{n}(G_{L},\mathcal{FIN},R)$. Since the relative assembly map

[TABLE]

is split injective for all groups $G^{\prime}$ and rings $R$ [Bar03, Theorem 1.3] and all $G_{L}$ are virtually cyclic by Lemma 3.6, the maps $\text{asmb}_{n}(G_{L},\mathcal{FIN},R)$ are split injective. Hence, $f_{2}$ is split injective as well and the long exact sequence (2) gives rise to a short exact sequence

[TABLE]

for every $n\in\mathbb{Z}$. By basic yoga with short exact sequences one can obtain from any short exact sequence of abelian groups

[TABLE]

with $j$ injective, a short exact sequence of the form $0\to V_{2}\to V_{3}\to\operatorname{coker}(i)\to 0$. Thus, we obtain for all $n\in\mathbb{Z}$ the short exact sequence

[TABLE]

from which the claim follows.

Exploiting what is known about the relative assembly map, one can now collect the following vanishing results for Waldhausen Nil-groups.

Corollary 5.2**.**

Let $G=A\ast_{C}B$ act acylindrically on its Bass-Serre tree $T$ and let $R$ be a regular ring. Then $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])$ vanishes rationally. Furthermore,

1. a)

if the action $G\curvearrowright T$ is strictly acylindrical (e.g. when $C$ is malnormal in $A$ or $B$), then $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])=0$; 2. b)

if $\mathbb{Q}\subset R$, then $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])=0$.

{proof}

By [LS16, Theorem 0.3], the relative assembly map

[TABLE]

is rationally bijective. Since all $G_{L}$ are virtually cyclic, $\widetilde{Nil}_{n}(RC;R[A-C],R[B-C])$ vanishes rationally by B.

a) If $G\curvearrowright T$ is strictly acylindrical, then any $G_{L}$ is either trivial or $\mathbb{Z}$. Since $R$ is assumed to be regular, the fundamental theorem of algebraic $K$-theory [Ros94, 5.3.30 Theorem] implies that all $H_{n}^{G_{L}}(E_{\mathcal{FIN}}G_{L};\mathbf{K}_{R})\to H_{n}^{G_{L}}(\operatorname{pt};\mathbf{K}_{R})$ are isomorphism.

b) By [KL05, Lemma 21.24], the relative assembly map for $G_{L}$ is integrally an isomorphism. Since all $G_{L}$ are virtually cyclic, the claim follows.

In light of Waldhausen’s notion of regular coherent groups (see [Wal78b, Section 19]) a remark on Corollary 5.2 a) is in order.

Remark 5.3**.**

For all regular coherent groups $C$ and all regular rings $R$, the group $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])$ already vanishes integrally by [Wal78b, Theorem 11.2]. However, it is easy to find examples of amalgamated free products $G=A\ast_{C}B$ acting strictly acylindrically on their Bass-Serre tree, such that $C$ is not regular coherent, because it contains torsion: By [Kap99, Theorem C] any torsion-free hyperbolic group $A^{\prime}$ (that is not virtually cyclic) contains a subgroup $H\cong F_{2}$ that is malnormal in $A^{\prime}$. Then $C=\mathbb{Z}/2\mathbb{Z}\times F_{2}$ is malnormal in $A:=D_{\infty}\times A^{\prime}$, but is not regular coherent.

Remark 5.4**.**

In the cases where $\widetilde{\text{Nil}}_{n}(RC;R[A-C],R[B-C])$ does not vanish by any of the above statements, the right hand side of the isomorphisms in B has been further identified: By Lemma 3.6 the groups $G_{L}$ that contribute to the direct sum are virtually cyclic of type I. For any such finite-by-$\mathbb{Z}$ group $V=H\rtimes_{\alpha}\mathbb{Z}$ it is known (see [DQR11, Lemma 3.1]) that the cokernel of the relative assembly map $H_{n}^{V}(E_{\mathcal{FIN}}V;\mathbf{K}_{R})\to H_{n}^{V}(E_{\mathcal{VCYC}}V;\mathbf{K}_{R})$ is isomorphic to the direct sum of two Farrell Nil-groups associated to $RH$ and $\alpha$. (These type of Nil-groups were introduced by Farrell in his PhD thesis [Far71] and a modern definition encompassing all degrees can be found in [Gru08]). In the case present, where $H$ is finite, these Farrell Nil-groups are known to be either trivial or infinitely generated: in lower degrees, this was shown independently by Grunewald [Gru07] and Ramos [Ram07]. Very recently, Lafont, Prassidis and Wang extended this result to all degrees [LPW16] and obtained a structure result for these groups (provided they have finite exponent).

6. The tactic for showing finitely

$\mathcal{F}$-amenability

The rest of this paper is concerned with the proof of Proposition 4.1. Thus, from now on, $G$ is a countable group acting $k$-acylindrically and without a global fixed point on a countable tree $T$. Without loss of generality we can assume $k\geq 1$, which spares us some notational exceptions.

Showing $\mathcal{F}$-amenability of the action $G\curvearrowright\Delta(T)$ amounts to providing intricate ‘wide’ covers for $G\times\Delta(T)$. A group $G$ acting 1-acylindrically on a tree is relatively hyperbolic, so in this case the desired covers are already given by Bartels’ methods in [Bar17] and our tactic is to adapt those methods for the case of a $k$-acylindrical action.

In [Bar17], first a notion of $\Theta$-small angles ‘between edges’ is introduced. Its crucial feature is that it functions like a proper $G$-invariant metric on the set of edges $E(T)$ incident to a given vertex. Then, a geodesic is called $\Theta$-small if the angle between any two incident edges lying on this geodesic is $\Theta$-small. In particular, any $\Theta$-small geodesic ending in a vertex can only be extended in finitely many directions without loosing the property of being $\Theta$-small. Bartels uses the notion of $\Theta$-small geodesics to divide $G\times\Delta(T)$ in two subsets. For trees, the first one, $G\times_{\Theta}\partial T$, consists of all pairs $(g,\xi)\in G\times\partial T$ such that, starting from a fixed vertex in $T$ of finite valence, the point $g^{-1}\xi$ can be reached with a $\Theta$-small geodesic ray. So this set is of the form $G\cdot(\{1\}\times\partial T^{\prime})$ for a locally finite subtree $T^{\prime}\subset T$. Furthermore, by definition of a relatively hyperbolic group, $G$ acts cocompactly on $E(T)$ and each edge stabiliser is finite. In the proof of Bartels, these three facts play an essential role in showing that $G\times_{\Theta}\partial T$ admits suitable covers. The second set, the rest, is covered by specially tailored sets that are constructed ad hoc, using the same notion of $\Theta$-small angles.

In our case of groups acting $k$-acylindrically on trees, edge stabilisers are in general not finite. The entities that have finite stabilisers instead are the geodesic segments of length $k$ in $T$. In trying to define a notion of $\Theta$-small geodesics by measuring ‘angles’ between ‘incident segments of length $k$’ on a geodesic, several difficulties arise. First, from a naïve geometric point of view, it is not clear when two segments of length $k$ should be called incident. The second problem appears when determining which points $g^{-1}\xi\in\partial T$ can be reached from a given fixed vertex $v_{0}$ via a small geodesic. Since on a geodesic of length $<k$ there is no segment of length $k$, and thus no ‘angle’, any such geodesic is automatically $\Theta$-small. Hence, one can start from $v_{0}$ in infinitely many directions along a $\Theta$-small geodesic, and the set $G\times_{\Theta}\partial T$ would be of the form $G\cdot(\{1\}\times\partial T^{\prime})$ where $T^{\prime}$ is a tree that is not locally finite. Moreover, the action of $G$ on the set $E^{k}(T)$ of geodesic segments of length $k$ is not cocompact anymore. We resolve these difficulties as follows.

In Section 7, a proper $G$-invariant metric on $E^{k}(T)$ is defined. This metric is constructed to have the additional property that the action of $G$ on $E^{k}(T)$ is ‘cocompact along $\Theta$-small geodesics’, and this property is sufficient for our purposes. Using this metric we can measure the distance between any two segments of length $k$. Our notion of $\Theta$-small geodesics is also defined in Section 7, where ‘incident segments of length $k$’ on a geodesic are segments on this geodesic whose midpoints have distance 1 in $T$. For the set $G\times_{\Theta}\partial T$, instead of starting to measure from a single vertex, we start to measure from one of two fixed geodesic segments of length $k$. These segments are ensured to be connected by a $\Theta$-small geodesic. This way $G\times_{\Theta}\partial T$ can again be written as $G\cdot(\{1\}\times\partial T^{\prime})$ for a locally finite subtree $T^{\prime}$ of $T$, and wide covers for $G\times_{\Theta}\partial T$ are constructed in Sections 9–10 utilising the long thin covers for coarse flow spaces provided by [Bar17, Theorem 1.1].

Covers for the remaining set $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$ are defined ad hoc in Section 8. In general, a single such cover will not be wide, but we show that one can combine $k+2$ of these covers to obtain a wide cover for $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$. Since ‘incident segments of length $k$’ overlap, this is more involved than in the case $k=1$, and we elaborate on the difficulties that arise from overlapping segments at the beginning of Section 8.

Finally, formally sewing together the results of Sections 7-10 gives Proposition 4.1:

{proof}

[Proof of Proposition 4.1] The proposition follows immediately from Proposition 8.1 and Proposition 10.5 as for any given $\alpha>0$ we can take the union of the cover $\mathcal{W}_{\alpha}$ from Proposition 8.1 and the cover $\mathcal{U}_{\Theta_{-1}(\alpha),\alpha}$ from Proposition 10.5.

7. Defining $\Theta$-small

geodesics

In order to define $\Theta$-small geodesics we need a $G$-invariant proper metric on the set $E^{k}(T)$ of all geodesic segments of length $k$ in $T$ whose endpoints are vertices of $T$. Since the action of $G$ on $E^{k}(T)$ is, in general, not cocompact, we have to construct the metric in a way that the action on $E^{k}(T)$ becomes ‘cocompact along small geodesics’.

Lemma 7.1**.**

Let $G$ be a countable group and $X$ be a countable discrete space. Furthermore, let $G\curvearrowright X$ have finite point stabilisers. Let $\pi:X\to G\backslash X$ be the canonical projection and let $F=\{x_{0},x_{1},\ldots\}$ be a set of representatives for the orbits of the action $G\curvearrowright X$. Then there is a $G$-invariant proper metric $d_{X}$ on $X$ compatible with the discrete topology such that the following holds: For all $\Theta>0$ there is $n_{0}\in\mathbb{N}$ such that if $x,y\in X$ with $d_{X}(x,y)\leq\Theta$, then $\pi(x)=[x_{j}],\pi(y)=[x_{j^{\prime}}]$ for some $j,j^{\prime}\leq n_{0}$. In (other) words: If $x$ and $y$ belong to different orbits and their distance is $\leq\Theta$, then the orbits of $x$ and $y$ both are among the first $n_{0}=n_{0}(\Theta)$ orbits.

{proof}

Abels, Manoussos and Noskov showed that for a locally compact group $H$ acting properly on a locally compact, $\sigma$-compact metrisable space $Z$, there is an $H$-invariant proper metric on $Z$ compatible with the topology on $Z$ [AMN11, Theorem 1.1]. Thus, in the situation of the lemma, the existence of a $G$-invariant proper metric on $X$ follows directly from [AMN11, Theorem 1.1]. Their construction is split in several steps and one of those steps [AMN11, Section 7] assures that the constructed metric is orbitwise proper222For discrete spaces this amounts to saying that for all $r>0$ and $x\in X$ the image of $B_{r}^{d}(x)$, the (closed) ball of radius $r$ around $x$, under $\pi:X\to G\backslash X$ is finite.. This is obtained by taking any proper continuous function $f:G\backslash X\to[0,\infty)$ and adding $d^{\prime}(x,y):=|f(\pi(x))-f(\pi(y))|$ to the result of their previous construction step. Since in our case $G\backslash X$ is indeed countable (instead of only $\sigma$-countable), we can choose $f$ such that orbits are forced further and further apart to obtain the additonally desired property the lemma claims. The rest of the construction of Abels, Manoussos and Noskov retains this property. A detailed elaboration on this modification can be found in the author’s PhD thesis [Kno16, Lemma 2.3].

Recall that we assume $G$ to act $k$-acylindrically on $T$ for some $k\geq 1$, that the term geodesic refers to both finite geodesics as well as (bi-infinite) geodesic rays in $T$ and that $d_{T}$ denotes the path-metric on $T$. We set up the following long list of carefully crafted choices and notations, some of which are illustrated in Figure 2.

Next we use the metric $d_{E^{k}}$ to define when a geodesic is $\Theta$-small.

Definition 7.2**.**

*Let $\Theta>0$ and let $\gamma$ be a geodesic in $T$. Any term of the form $d_{E^{k}}(\sigma,\sigma^{\prime})$ for $\sigma,\sigma^{\prime}\in E^{k}(T)\cap\text{im}(\gamma)$ with $\text{diam}_{d_{T}}(\sigma\cap\sigma^{\prime})=k-1$ is a measurement (on $\gamma$). For a vertex $v$ on $\gamma$ the term $d_{E^{k}}(\sigma_{v}(\gamma),\sigma_{v+1}(\gamma))$ is called the measurement at $v$ (on $\gamma$). This is illustrated in Figure 3.

A measurement is $\Theta$-small if it is $\leq\Theta$ and it is $\Theta$-large if it is $>\Theta$. Finally, the geodesic $\gamma$ is $\Theta$-small if all measurements on $\gamma$ are $\Theta$-small. In particular, any geodesic $\gamma$ of length $\leq k$ is automatically $\Theta$-small, since in this case there are no measurements on $\gamma$.*

Notation 7.3**.**

For $\Theta>0$ we define

[TABLE]

We will find covers for $G\times_{\Theta}\partial T$ and its complement in $G\times\Delta(T)$ separately.

8. Covers for $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$

The goal of this section is to establish the following proposition.

Proposition 8.1**.**

Retain the assumptions of Proposition 4.1. Then there is $N\in\mathbb{N}$ such that for all $\alpha>0$ there is a $\Theta_{-1}=\Theta_{-1}(\alpha)\in\mathbb{R}$ and a $G$-invariant collection $\mathcal{W}_{\alpha}$ of open $\mathcal{F}_{T}$-subsets of $G\times\Delta(T)$ such that the order of $\mathcal{W}_{\alpha}$ is at most $N$ and $\mathcal{W}_{\alpha}$ is wide for $G\times\Delta(T)\setminus G\times_{\Theta_{-1}(\alpha)}\partial T$.

We start by defining ad hoc the sets that will later on be used to build the collection $\mathcal{W}_{\alpha}$.

Definition 8.2**.**

Let $\Theta\geq\theta_{0}$ and $v\in V(T)$. Define the set $W(v,\Theta)\subset G\times\Delta(T)$ to be

[TABLE]

This definition is illustrated below in Figure 4. Furthermore, note that, if $v\in[gm_{\epsilon},\xi]$, then the geodesic $[a(g,\xi),v]$ is of length $\geq 2k$ since we have chosen $w_{0},w_{0}^{\prime}$ with $d_{T}(w_{0},w_{0}^{\prime})\geq 5k$. Hence, if $v\in[gm_{\epsilon},\xi]$, then the segment $\sigma_{v}(\gamma)$ is always defined and, if additionally $v\neq\xi$, so is the segment $\sigma_{v+_{\gamma}1}(\gamma)$.

If we were to drop the requirement of wideness in Proposition 8.1, then one collection of the form $\mathcal{W}_{\Theta}:=\{W(v,\Theta)\ |\ v\in V(T)\}$ would be sufficient as the following two lemmata show.

Lemma 8.3**.**

Let $\Theta\geq\theta_{0}$.

1. a)

For all $v\in V(T)$, the set $W(v,\Theta)\subset G\times\Delta(T)$ is open. 2. b)

For all $g\in G$ and $v\in V(T)$, we have $gW(v,\Theta)=W(gv,\Theta)$. 3. c)

If $v\neq w\in V(T)$, then $W(v,\Theta)\cap W(w,\Theta)=\emptyset$.

{proof}

a) Let $(g,\xi)\in W(v,\Theta)$ and let $\gamma:=[a(g,\xi),\xi]$. First, let $v\neq\xi$, and thus $[a(g,\xi),v+_{\gamma}1]\subset[a(g,\xi),\xi]$. Note that in this case any point $(g,\xi^{\prime})$ with

[TABLE]

lies in $W(v,\Theta)$ as well (since the measurements relevant for $(g,\xi)\in W(v,\Theta)$ are the same measurements relevant for $(g,\xi^{\prime})\in W(v,\Theta)$). Set $U:=M(\xi,v)\cap\Delta(T)$. Then, for all $\xi^{\prime}\in U$, the geodesic $[a(g,\xi),\xi^{\prime}]$ contains $v+_{\gamma}1$, and hence $gm_{\epsilon}$ as well. Therefore, $[a(g,\xi),v+_{\gamma}1]=[a(g,\xi^{\prime}),v+_{\gamma}1]$ and $(g,\xi)\in W(v,\Theta)$ follows. If $v=\xi$, then we construct $U$ as follows: Let $\sigma:=\sigma_{v}([a(g,\xi),v])$. Since $d_{E^{k}}$ is proper, there are only finitely many $\sigma^{\prime}$ with $d_{E^{k}}(\sigma,\sigma^{\prime})\leq\Theta$ and $\text{diam}_{d_{T}}(\sigma\cap\sigma^{\prime})=k-1$. Denote them by $\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{m}$. Set

[TABLE]

Since $[a(g,\xi),v]$ is $\Theta$-small by assumption, the segment $\sigma_{v-_{\gamma}1}(\gamma)$ is some $\sigma^{\prime}_{i}$, and hence $v-_{\gamma}1$ is $t(\sigma_{i}^{\prime})$. Therefore, for any $\xi^{\prime}\in U$, the geodesic $[a(g,\xi),\xi^{\prime}]$ contains $[a(g,\xi),v]$ and, in particular, $gm_{\epsilon}$. It follows that $[a(g,\xi^{\prime}),v]=[a(g,\xi),v]$, which is $\Theta$-small by assumption. And our choice of the $\sigma_{i}^{\prime}$ guarantees that the measurement at $v$ on $[a(g,\xi^{\prime}),\xi^{\prime}]$ is $\Theta$-large.

b) This follows from the $G$-invariance of $d_{E^{k}}$.

c) Assume we have $v\neq w$ and $(g,\xi)\in W(v,\Theta)\cap W(w,\Theta)$. Then, by definition of $W(v,\Theta)$ and $W(w,\Theta)$, both $v$ and $w$ lie on $[gm_{\epsilon},\xi]$. Hence, without loss of generality we may assume $v$ lies on $[gm_{\epsilon},w]$. Now, on the one hand, $(g,\xi)\in W(w,\Theta)$ implies that the measurement at $v$ on $[a(g,\xi),w]$ is $\Theta$-small. On the other hand, $(g,\xi)\in W(v,\Theta)$ implies that the measurement at $v$ on $[a(g,\xi),\xi]$ is $\Theta$-large.

Lemma 8.4**.**

Let $\Theta\geq\theta_{0}$. Then $\mathcal{W}_{\Theta}:=\{W(v,\Theta)\ |\ v\in V(T)\}$ is a $G$-invariant collection of open $\mathcal{F}_{T}$-subsets of $G\times\Delta(T)$ and covers $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$. Furthermore, the order of $\mathcal{W}_{\Theta}$ is 0.

{proof}

By Lemma 8.3 it is clear that $\mathcal{W}_{\Theta}$ is an open, $G$-invariant collection of order 0. Moreover, by combining Lemma 8.3 b) and c), it follows that each $W(v,\Theta)$ is an $\mathcal{F}_{T}$-subset of $G\times\Delta(T)$. The only property left to show is that the collection $\mathcal{W}_{\Theta}$ covers the set $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$: Let $(g,\xi)\in G\times\Delta(T)\setminus G\times_{\Theta}\partial T$ and $\gamma:=[a(g,\xi),\xi]$. If $\gamma$ is $\Theta$-small, then $\xi\in V(T)$, and hence $(g,\xi)\in W(\xi,\Theta)$. If $\gamma$ is not $\Theta$-small, then there is a unique vertex $v$ on $\gamma$ such that $[a(g,\xi),v]$ is $\Theta$-small, but $[a(g,\xi),v+_{\gamma}1]$ is not. Note that $v$ might lie on $[gw_{0},gw_{0}^{\prime}]$, but due to our choice of $\theta_{0}$ the vertex $v$ can not lie between $a(g,\xi)$ and $gm_{\epsilon}$. Thus, $v\in[gm_{\epsilon},\xi]$ and $(g,\xi)\in W(v,\Theta)$ holds.

To obtain—for a given $\alpha>0$—a threshold $\Theta_{-1}(\alpha)$ and a collection $\mathcal{W}_{\alpha}$ that is wide for $G\times\Delta(T)\setminus G\times_{\Theta_{-1}(\alpha)}\partial T$, it is necessary to combine $k+2$ collections of the form $\mathcal{W}_{\Theta_{i}}$. The verification that we can indeed find suitable $\Theta_{i}$ such that $\mathcal{W}_{\alpha}:=\bigcup_{i=0}^{k+1}\mathcal{W}_{\Theta_{i}}$ is wide for $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$ is rather technical, so we outline its underlying ideas first: Since all $\mathcal{W}_{\Theta}$ are $G$-invariant and all $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$ are as well, it suffices to consider wideness for points of the form $(1,\xi)$.

For any $\Theta\geq\theta_{0}$ we already know that for each point $(1,\xi)\in G\times\Delta(T)\setminus G\times_{\Theta}\partial T$ we can find a vertex $v\in V(T)$ by Lemma 8.4 such that $(1,\xi)\in W(v,\Theta)$ holds. We do not expect $[a(h,\xi),v]$ to be $\Theta$-small for $h\in B_{\alpha}(1)$ as well, but we wish to calculate how small the geodesics $[a(h,\xi),v]$ are, given that $[a(1,\xi),v]$ is $\Theta$-small.

In the case $k=1$ this results in the following type of picture:

v$$a(1,\xi)$$\xi\in\Delta(T)$$a(h,\xi)$$\sigma_{0}$$\sigma_{2}$$\sigma_{1}

Let $L_{0}$ be the tree spanned by $\{a(h,\xi)\ |\ h\in B_{\alpha}(1)\}$. Since $B_{\alpha}(1)$ is finite, $L_{0}$ is as well. Thus, there is some $\Upsilon$ independent of $h$ such that $d_{E^{1}}(\sigma_{0},\sigma_{1})\leq\Upsilon$. Since $d_{E^{1}}(\sigma_{0},\sigma_{2})\leq\Theta$ by virtue of $(1,\xi)\in W(v,\Theta)$, we can estimate $d_{E^{1}}(\sigma_{1},\sigma_{2})$ to be $\leq\Theta+\Upsilon$. Hence, for all $h\in B_{\alpha}(1)$ the geodesic $[a(h,\xi),v]$ is $\Theta+\Upsilon$-small. If the measurement at $v$ on $[a(1,\xi),\xi]$ is not only $\Theta$-large, but in fact $\Theta+\Upsilon$-large, we can conclude $B_{\alpha}(1)\times\{\xi\}\subset W(v,\Theta+\Upsilon)$. However, in general, the measurement at $v$ on $[a(1,\xi),\xi]$ is not $\Theta+\Upsilon$-large and this requires that we ‘push $v$ further to the right’. More precisely, we search for the first vertex $w$ on $[a(1,\xi),\xi]$ such that $[a(1,\xi),w]$ is $\Theta+\Upsilon$-small, but the measurement at $w$ is not. If such $w$ exists, then $B_{\alpha}(1)\times\{\xi\}\subset W(w,\Theta+\Upsilon)$ follows. If such $w$ does not exist, then $[a(1,\xi),\xi]$ is $\Theta+\Upsilon$-small. If $\xi\in V_{\infty}(T)$, our tailored definition of the sets $W(v,\Theta+\Upsilon)$ implies that $B_{\alpha}(1)\times\{\xi\}\subseteq W(\xi,\Theta+\Upsilon)$ holds. And if $\xi\in\partial T$, we do not require our collection of sets to be wide at $(1,\xi)$ anyway. Since we only used sets of the form $W(-,\Theta+\Upsilon)$, we thus obtain a wide collection for $G\times\Delta(T)\setminus G\times_{\Theta}\partial T$ and this basically concludes the proof of Proposition 8.1 for the case $k=1$.

In the case $k\geq 2$, however, examining how small the geodesics $[a(h,\xi),v]$ are is more involved since we will have to make statements about pictures of the following type:

v$$a(1,\xi)$$\xi\in\Delta(T)$$a(h,\xi)$$\sigma_{0}$$\sigma_{3}$$\sigma_{1}$$\sigma_{2}

We still know that $L_{0}$ (as defined before) is finite. Thus, we still have a bound $\Upsilon$ on all measurements on $[a(1,\xi),a(h,\xi)]$. However, we do not have a triangle inequality anymore to give an estimate for $d_{E^{k}}(\sigma_{1},\sigma_{2})$ based on $\Upsilon$ and the fact that $d_{E^{k}}(\sigma_{0},\sigma_{3})\leq\Theta$. So in order to obtain some estimate we use that $d_{E^{k}}$ is proper: since $[a(1,\xi),v]$ is $\Theta$-small, has a segment of length $\geq k$ in common with $L_{0}$ and the latter is finite, there are only finitely many possibilities for $\sigma_{3}$. Adding all possible $\sigma_{3}$ to $L_{0}$ still results in a finite tree $L_{1}$ and we find some bound $\Upsilon^{\prime}$ on the measurements lying on $L_{1}$. This obviates the need for a triangle inequality, since now both $\sigma_{1}$ and $\sigma_{2}$ lie on $L_{1}$. Hence, $d_{E^{k}}(\sigma_{1},\sigma_{2})$ is $\Upsilon^{\prime}$-small. The idea of ‘pushing $v$ further to the right if necessary’ stays the same, but various possible constellations of how the resulting vertex $w$ lies relative to $v$ necessitates more than one iteration step and careful case distinctions, both of which will be carried out in the following.

We start by making precise how to define an enlargement of $L_{0}$ suitable for all iteration steps.

Choosing Constants 8.5**.**

Let $\alpha>0$. The following notions all depend on $\alpha$. However, since we will never compare different values of $\alpha$, this dependence is not reflected in the notation.

The proof of Proposition 8.1 requires some more notation.

Notation 8.6**.**

For a given $\alpha>0$ and $(1,\xi)\in G\times\Delta(T)$ we fix the following notation that depends on $\alpha$ and $\xi$ (cf. Figure 5 for an illustration of some of the notation). The naming will not always reflect both of these dependencies, though.

We proceed by showing a series of technical lemmata, whose sole purpose is to increase readability of the proof of Proposition 8.1. For all statements we will use extensively the constants defined in 8.5 and assume that $\alpha$ and $(1,\xi)$ are fixed. While looking at the pictures in the proofs it is also important to keep in mind the following observation: Our definition of $W(v,\Theta)$ allows for $\xi$ to lie on $[w_{0},w_{0}^{\prime}]$. However, in what follows we often assume the existence of a $v_{0}(\chi)$ for some $\chi\geq\Upsilon_{0}$, which forces $\xi$ to lie outside of $[w_{0},w_{0}^{\prime}]$.

Lemma 8.7**.**

Let $\chi\geq\Upsilon_{0}$. If $v_{0}(\chi)$ exists, then $B^{\text{inn}}_{v_{0}(\chi)}=B_{\alpha}(1)$.

{proof}

Assume there is $h\in B_{\alpha}(1)$ such that $h\not\in B^{\text{inn}}_{v_{0}(\chi)}$. Thus, $v_{h}\in(v_{0}(\chi),\xi]$. Then by uniqueness of geodesics in $T$ and the construction of $\Upsilon_{0}$ it follows that $[a(1,\xi),v_{0}(\chi)+_{\gamma_{1}}1]$ is $\Upsilon_{0}$-small (cf. Figure 5 with $h=h_{4}$ and $v_{0}(\chi)=v$). A contradiction.

In general, the ‘concatenation’ of two $\chi$-small geodesics $[x,y]$ (ending in a vertex of $T$) and $[y,z]$ (starting in a vertex of $T$) is not $\chi$-small. However, if two $\chi$-small geodesics overlap in a sufficiently large terminal segment, we can conclude in this way. This is made precise in the following lemma, whose proof is entirely encapsulated in Figure 6.

Lemma 8.8**.**

Let $\chi\geq 0$ and let $[x,z]$ be a geodesic. If there are $y,y^{\prime}$ on $[x,z]$ such that the intersection $[x,y]\cap[y^{\prime},z]=[y^{\prime},y]$ has length $\geq k$ and both $[x,y]$ and $[y^{\prime},z]$ are $\chi$-small, then $[x,z]$ is $\chi$-small.

In the sequel we will use this way of reasoning without explicitly referencing the above lemma.

Lemma 8.9**.**

Let $\chi\geq\Upsilon_{0}$ and $x$ be a vertex on $\gamma_{1}$ such that $[a(1,\xi),x]$ is $\chi$-small. Then the geodesic $[a(h,\xi),x]$ is $\Upsilon_{1}(\chi)$-small for all $h\in B_{\alpha}(1)$.

{proof}

If $h\in B^{\text{ext}}_{x}$, then $[a(h,\xi),x]\subset[a(h,\xi),a(1,\xi)]\subset L_{0}$. Hence, $[a(h,\xi),x]$ is $\Upsilon_{0}$-small. If $h\in B^{\text{inn}}_{x}$, then we distinguish cases upon where $v_{h}$ lies on $[a(1,\xi),x]$:

1. a)

This case is clear:

a(1,\xi)$$x$$\xi\in\Delta(T)$$a(h,\xi)=v_{h}

Thus, we assume from now on, that $a(h,\xi)\not\in[a(1,\xi),x]$. 2. b)

This one is clear as well:

a(1,\xi)$$x=v_{h}$$\xi\in\Delta(T)$$a(h,\xi)$$\text{Contained in}\ L_{0}\text{ and thus }\Upsilon_{0}\text{-small.} 3. c)

If $v_{h}$ lies in the interior of $[a(1,\xi),x]$ and $[a(1,\xi),v_{h}]$ has length $\geq k$,

a(1,\xi)$$v_{h}$$\xi\in\Delta(T)$$a(h,\xi)$$x

$\geq k$

then the first $k$ edges on $[v_{h},x]$ (if there are this many) are added to $L_{0}$ in the process of building $L_{k}(\chi)$, since $[a(1,\xi),v_{h}]$ is $\chi$-small (cf. 8.5). Thus, $[a(h,\xi),v_{h}+_{\gamma_{1}}k]$ (resp. $[a(h,\xi),x]$ if $[v_{h},x]$ is of length $<k$) is $\Upsilon_{1}(\chi)$-small. Since $[v_{h},x]$ was $\chi$-small by assumption, it follows that $[a(h,\xi),x]$ is $\Upsilon_{1}(\chi)$-small. 4. d)

If $v_{h}$ lies in the interior of $[a(1,\xi),x]$ and $[a(1,\xi),v_{h}]$ has length $<k$, there are two subcases. The subcase where $x\in[a(1,\xi),m_{\epsilon}]$ is trivial:

a(1,\xi)$$v_{h}$$\xi\in\Delta(T)$$a(h,\xi)$$m_{\epsilon}$$x

$<k$

$\text{Contained in}\ L_{0}\text{ and thus }\Upsilon_{0}\text{-small.}$

The subcase where $x\not\in[a(1,\xi),m_{\epsilon}]$ looks as follows:

a(1,\xi)$$v_{h}$$\xi\in\Delta(T)$$a(h,\xi)$$x$$m_{\epsilon}

$<k$

$\geq k$

$\text{Contained in}\ L_{0}\text{ and thus }\Upsilon_{0}\text{-small.}$

Since $[v_{h},m_{\epsilon}]$ has length $\geq k$, $[a(h,\xi),m_{\epsilon}]$ is $\Upsilon_{0}$-small and $[v_{h},x]$ is $\chi$-small, the geodesic $[a(h,\xi),x]$ is $\chi$-small. 5. e)

This case is trivial again:

a(1,\xi)=v_{h}$$x$$\xi\in\Delta(T)$$a(h,\xi)$$m_{\epsilon}$$\text{Contained in}\ L_{0}\text{ and thus }\Upsilon_{0}\text{-small.} 6. f)

And finally:

a(1,\xi)=v_{h}$$x$$\xi\in\Delta(T)$$a(h,\xi)$$m_{\epsilon}

$\geq 2k$

$\text{Contained in}\ L_{0}\text{ and thus }\Upsilon_{0}\text{-small.}$

Again, the geodesic $[a(h,\xi),x]$ is $\chi$-small since $[a(h,\xi),m_{\epsilon}]$ is $\Upsilon_{0}$-small, the geodesic $[a(1,\xi),x]$ is $\chi$-small and $[a(1,\xi),m_{\epsilon}]$ has length $\geq k$.

Lemma 8.10**.**

Let $\chi\geq\Upsilon_{0}$. If $v_{0}(\chi)$ exists and $B^{\text{inn}}_{v_{0}(\chi)-_{\gamma_{1}}k}=B_{\alpha}(1)$, then one of the following statements holds:

$B_{\alpha}(1)\times\{\xi\}\subset W(v_{0}(\chi),\Upsilon_{1}(\chi))$; 2. 2)

$v_{0}(\Upsilon_{1}(\chi))$* does not exist, i.e. $[a(1,\xi),\xi]$ is $\Upsilon_{1}(\chi)$-small;* 3. 3)

$v_{0}(\Upsilon_{1}(\chi))$* exists and $B_{\alpha}(1)\times\{\xi\}\subset W(v_{0}(\Upsilon_{1}(\chi)),\Upsilon_{1}(\chi))$.*

{proof}

In this proof we abbreviate $d_{E^{k}}$ to $d$. For all $h\in B_{\alpha}(1)$, the geodesic $[a(h,\xi),v_{0}(\chi)]$ is $\Upsilon_{1}(\chi)$-small by Lemma 8.9 (applied to $x=v_{0}(\chi)$). Since $B_{\alpha}(1)=B^{\text{inn}}_{v_{0}(\chi)-_{\gamma_{1}}k}$, the geodesics $[v_{h},v_{0}(\chi)]\subset\gamma_{h}$ all have length $\geq k$. Hence, for all $h\in B_{\alpha}(1)$, the measurement at $v_{0}(\chi)$ on $\gamma_{1}$ is the measurement at $v_{0}(\chi)$ on $\gamma_{h}$. Thus, if the measurement at $v_{0}(\chi)$ on $\gamma_{1}$ is $\Upsilon_{1}(\chi)$-large, so is the measurement at $v_{0}(\chi)$ on $\gamma_{h}$ and $B_{\alpha}(1)\times\{\xi\}\subset W(v_{0}(\chi),\Upsilon_{1}(\chi))$ follows, which is statement 1) above.

Assume from now on that the measurement at $v_{0}(\chi)$ on $\gamma_{1}$ is $\Upsilon_{1}(\chi)$-small. If $v_{0}(\Upsilon_{1}(\chi))$ does not exist, then $[a(1,\xi),\xi]$ is $\Upsilon_{1}(\chi)$-small, which is statement 2) above. If $v_{0}(\Upsilon_{1}(\chi))$ exists, it lies in $(v_{0}(\chi),\xi]$. Thus, the intersection of the $\Upsilon_{1}(\chi)$-small geodesic $[a(h,\xi),v_{0}(\chi)]$ with the $\Upsilon_{1}(\chi)$-small geodesic $[v_{0}(\chi)-_{\gamma_{1}}k,v_{0}(\Upsilon_{1}(\chi))]$ has length $\geq k$ for all $h\in B_{\alpha}(1)$. Thus, all $[a(h,\xi),v_{0}(\Upsilon_{1}(\chi))]$ are still $\Upsilon_{1}(\chi)$-small and the measurement at $v_{0}(\Upsilon_{1}(\chi))$ on $\gamma_{1}$ is the measurement at $v_{0}(\Upsilon_{1}(\chi))$ on $\gamma_{h}$. Hence, $B_{\alpha}(1)\times\{\xi\}\subset W(v_{0}(\Upsilon_{1}(\chi)),\Upsilon_{1}(\chi))$, which is statement 3) above.

Lemma 8.11**.**

Let $\chi\geq\Upsilon_{0}$. Assume that $v_{0}(\chi)$ exists and let $0\leq r\leq k-1$ such that $B^{\text{ext}}_{v_{0}(\chi)-_{\gamma_{1}}r}=\emptyset$, but $B^{\text{ext}}_{v_{0}(\chi)-_{\gamma_{1}}(r+1)}\neq\emptyset$. Then one of the following statements holds:

$B_{\alpha}(1)\times\{\xi\}\subset W(v_{0}(\chi),\Upsilon_{1}(\chi))$; 2. 2)

$v_{0}(\Upsilon_{2}(\chi))$* does not exist, i.e. $[a(1,\xi),\xi]$ is $\Upsilon_{2}(\chi)$-small;* 3. 3)

$v_{0}(\Upsilon_{2}(\chi))$* exists and there is an $s\geq 1$ such that $B^{\text{ext}}_{v_{0}(\Upsilon_{2}(\chi))-_{\gamma_{1}}(r+s)}=\emptyset$, but*

$B^{\text{ext}}_{v_{0}(\Upsilon_{2}(\chi))-_{\gamma_{1}}(r+s+1)}\neq\emptyset$**.

{proof}

In this proof we abbreviate $d_{E^{k}}$ to $d$ again. As in the previous lemma, for all $h\in B_{\alpha}(1)$ the geodesics $[a(h,\xi),v_{0}(\chi)]$ is $\Upsilon_{1}(\chi)$-small by Lemma 8.9 (applied to $x=v_{0}(\chi)$). The measurement at $v_{0}(\chi)$ on $[a(h,\xi),\xi]$ is

[TABLE]

Note that, since $B^{\text{inn}}_{v_{0}(\chi)}=B_{\alpha}(1)$ by Lemma 8.7, $v_{0}(\chi)+_{\gamma_{1}}1=v_{0}(\chi)+_{\gamma_{h}}1$. So the $+1$ in the second argument above is unambiguous. However, in general $\sigma_{v_{0}(\chi)+1}(\gamma_{h})\neq\sigma_{v_{0}(\chi)+1}(\gamma_{1})$ and this is the point were we have to put in a little work.

If for all $h\in B_{\alpha}(1)$ the measurement at $v_{0}(\chi)$ on $\gamma_{h}$ is $\Upsilon_{1}(\chi)$-large, then

[TABLE]

by definition of $W(v_{0}(\chi),\Upsilon_{1}(\chi))$, and this is statement 1) above. So assume from now on that there is an $h_{1}\in B_{\alpha}(1)$ such that the measurement at $v_{0}(\chi)$ on $\gamma_{h_{1}}$ is $\Upsilon_{1}(\chi)$-small. Moreover, we can assume that $v_{0}(\Upsilon_{2}(\chi))$ does exist, since the case that $v_{0}(\Upsilon_{2}(\chi))$ does not exist, is exactly statement 2) above.

Since $B^{\text{ext}}_{v_{0}(\chi)-_{\gamma_{1}}r}=\emptyset$, but $B^{\text{ext}}_{v_{0}(\chi)-_{\gamma_{1}}(r+1)}\neq\emptyset$, there is $h_{0}\in B_{\alpha}(1)$ with $v_{h_{0}}=v_{0}(\chi)-_{\gamma_{1}}r$. So for this $h_{0}$ the geodesic $[a(1,\xi),a(h_{0},\xi)]$ runs through $v_{0}(\chi)-_{\gamma_{1}}r$. Since, for all $h\in B_{\alpha}(1)$, the vertex $v_{h}$ is contained in $[a(1,\xi),v_{h_{0}}]$, it follows that $[a(h,\xi),v_{0}(\chi)-_{\gamma_{1}}r]\subset L_{0}$ holds for all $h$. Assume $h_{1}m_{\epsilon}\not\in[a(h_{1},\xi),v_{0}(\chi)]$. Then we must have $[a(1,\xi),v_{0}(\chi)+1]\subset L_{0}$ - a contradiction to the definition of $v_{0}(\chi)$, since $\chi\geq\Upsilon_{0}$. Thus, $h_{1}m_{\epsilon}\in[a(h_{1},\xi),v_{0}(\chi)]$, and hence $d_{T}(a(h_{1},\xi),v_{0}(\chi)-_{\gamma_{1}}r)\geq k$. Combining the facts that $[a(h_{1},\xi),v_{0}(\chi)]$ is $\Upsilon_{1}(\chi)$-small (by Lemma 8.9) and that $r+1\leq k$ with our assumption that the measurement at $v_{0}(\chi)$ on $\gamma_{h_{1}}$ is $\Upsilon_{1}(\chi)$-small, it follows that the segment $[v_{0}(\chi)-_{\gamma_{1}}r,v_{0}(\chi)+_{\gamma_{1}}1]$ is contained in $L_{k}(\Upsilon_{1}(\chi))$ (cf. Figure 7 and 8.5). Note that, if $r=k-1$, then we indeed need all $k$ steps in the iteration process used to build $L_{k}(\Upsilon_{1}(\chi))$ from $L_{0}$ in order to add this whole segment.

Now recall that $\Upsilon_{2}(\chi)=\Upsilon_{1}(\Upsilon_{1}(\chi))$. Hence, $[a(h,\xi),v_{0}(\chi)+1]\subset L_{k}(\Upsilon_{1}(\chi))$ is $\Upsilon_{2}(\chi)$-small for all $h\in B_{\alpha}(1)$. In particular, the vertex $v_{0}(\Upsilon_{2}(\chi))$ has to lie on $(v_{0}(\chi),\xi]$ and subsequently $s:=d_{T}(v_{0}(\chi),v_{0}(\Upsilon_{2}(\chi)))$ is $\geq 1$. Moreover, it is

[TABLE]

We are now ready to prove Proposition 8.1.

{proof}

[Proof of Proposition 8.1] Set $N:=k+1$ and let $\alpha>0$ be given. Further define the following constants depending on $\alpha$:

• •

Let $\Upsilon_{0}$ be as in 8.5.

• •

For $0\leq i\leq k$ define $\Theta_{i}:=\Upsilon_{2i+1}(\Upsilon_{0})$ (also cf. 8.5).

• •

Choose $\Theta_{-1}$ such that $\Theta_{-1}>\Upsilon_{2(k+1)}(\Upsilon_{0})$. Note that the iterative construction of $\Upsilon_{2(k+1)}(\Upsilon_{0})$ guarantees $\Theta_{-1}\geq\Upsilon_{i}(\Upsilon_{0})$ for $1\leq i\leq 2k+1$ and, therefore, also $\Theta_{-1}>\Upsilon_{0}$.

• •

Define $\Theta_{k+1}:=\Upsilon_{1}(\Theta_{-1})$.

By Lemma 8.4, $\mathcal{W}_{\alpha}:=\bigcup_{i=0}^{k+1}\mathcal{W}_{\Theta_{i}}$ is a $G$-invariant collection of open $\mathcal{F}_{T}$-subsets of $G\times\Delta(T)$ and has order at most $k+1$. It remains to verify the wideness claim. That is, for each $(1,\xi)\in G\times\Delta(T)$ we either have to find $0\leq i\leq k+1$ such that $B_{\alpha}(1)\times\{\xi\}\subseteq W(v,\Theta_{i})$ for some $v\in V(T)$ or conclude that $[a(1,\xi),\xi]$ is $\Theta_{-1}$-small and $\xi$ lies in $\partial T$.

Let $(1,\xi)\in G\times\Delta(T)$ and recall the notation $\gamma_{h},B^{\text{inn}}_{v}$ and $B^{\text{ext}}_{v}$ from 8.6.

Case 1: Assume that $\gamma_{1}$ is $\Theta_{-1}$-small. If $\xi\in\partial T$, there is nothing to show. If $\xi\in V_{\infty}(T)$, then by Lemma 8.9 (for $x=\xi$ and $\chi=\Theta_{-1}$) and the definition of $W(\xi,\Upsilon_{1}(\Theta_{-1}))$ it follows that

[TABLE]

Case 2: Assume that $\gamma_{1}$ is not $\Theta_{-1}$-small. In particular, $\gamma_{1}$ is not $\Upsilon_{0}$-small and $v_{0}(\Upsilon_{0})$ exists. Thus, by Lemma 8.7, $B^{\text{ext}}_{v_{0}(\Upsilon_{0})}=\emptyset$ and there is a unique $r_{0}\geq 0$ such that

[TABLE]

If $r_{0}\geq k$, we can apply Lemma 8.10 with $\chi=\Upsilon_{0}$. Since $\gamma_{1}$ is not $\Upsilon_{1}(\Upsilon_{0})$-small, it either follows that

[TABLE]

or that

[TABLE]

If $r_{0}\leq k-1$, we can apply Lemma 8.11 with $r=r_{0}$ and $\chi=\Upsilon_{0}$. Since $\gamma_{1}$ is not $\Upsilon_{2}(\Upsilon_{0})$-small, it either follows that

[TABLE]

or that $v_{0}(\Upsilon_{2}(\Upsilon_{0}))$ exists and that there is an $s_{0}\geq 1$ such that

[TABLE]

This process can be iterated for $0\leq i\leq k$. Assume that we have already constructed an $r_{i}\geq 0$ with

[TABLE]

where for $i=0$ we interpret $\Upsilon_{0}(\Upsilon_{0})$ as $\Upsilon_{0}$. If $r_{i}\geq k$, we can apply Lemma 8.10 with $\chi=\Upsilon_{2i}(\Upsilon_{0})$. Since $\gamma_{1}$ is not $\Upsilon_{2i+1}(\Upsilon_{0})$-small (as long as $i\leq k$), it either follows that

[TABLE]

or that

[TABLE]

If $r_{i}\leq k-1$, we can apply Lemma 8.11 with $r=r_{i}$ and $\chi=\Upsilon_{2i}(\Upsilon_{0})$. Since $\gamma_{1}$ is not $\Upsilon_{2(i+1)}(\Upsilon_{0})$-small (as long as $i\leq k$), it either follows that

[TABLE]

or that $v_{0}(\Upsilon_{2(i+1)}(\Upsilon_{0}))$ exists and that there is $s_{i}\geq 1$ such that

[TABLE]

In this case we define $r_{i+1}:=r_{i}+s_{i}$ and repeat the process. Since in each step $s_{i}\geq 1$, there is $0\leq j\leq k$ such that $r_{j}\geq k$. Hence, all constants $\Upsilon_{2i+1}(\Upsilon_{0})$ considered in the process are among $\Theta_{0},\ldots,\Theta_{k}$.

This concludes the construction of suitable $G$-invariant collections of open $\mathcal{F}$-subsets that are wide for $G\times\Delta(T)\setminus G\times_{\Theta_{-1}}\partial T$ (with $\Theta_{-1}$ depending on $\alpha>0$).

9. The segment-flow

space $FS_{\Theta}$

The aim of this and the following section is to construct—for all $\Theta$—a $G$-invariant collection of open $\mathcal{F}$-subsets that is wide for $G\times_{\Theta}\partial T$. Our construction will be analogously to the one in [Bar17] for relatively hyperbolic groups. For this, we first define another proper $G$-invariant metric on $E^{k}(T)$ which essentially coincides with the path-metric $d_{T}$ when restricted to segments lying on a small geodesic.

From now on assume that $k$ is even and recall that in this case the midpoint $m(\sigma)$ of any $\sigma\in E^{k}(T)$ is a vertex of $T$.

Definition 9.1**.**

Let $\Theta>0$ and let $\sigma,\sigma^{\prime}\in E^{k}(T)$. The geodesic $\gamma=[m(\sigma),m(\sigma^{\prime})]$ is called extended $\Theta$-small, if the following holds:

• •

$\gamma$* is $\Theta$-small (as in Definition 7.2).*

• •

If $d_{T}(m(\sigma),m(\sigma^{\prime}))\geq k$, then

[TABLE]

and

[TABLE]

• •

If $d_{T}(m(\sigma),m(\sigma^{\prime}))\leq k-1$, then $d_{E^{k}}(\sigma,\sigma^{\prime})\leq\Theta$.

Remember that the $+_{\gamma}$-operation on vertices of $\gamma$ is meant as introduced in Section 7.

Definition 9.2**.**

An oriented segment (of length $k$) in $T$ is an element $\sigma\in E^{k}(T)$ together with a vertex $b\in\{o(\sigma),t(\sigma)\}$. For $\sigma,\sigma^{\prime}\in E^{k}(T)$ denote by $\mathcal{M}_{\Theta}(\sigma,\sigma^{\prime})$ the set of all sequences $(\sigma_{0},b_{0}),(\sigma_{1},b_{1}),\ldots,(\sigma_{n},b_{n})$ of oriented segments of length $k$ in $T$ such that $\sigma=\sigma_{0}$ and $\sigma^{\prime}=\sigma_{n}$ (as unoriented segments) and the geodesics $[m(\sigma_{i}),m(\sigma_{i+1})]$ are extended $\Theta$-small. For brevity, elements of $\mathcal{M}_{\Theta}(\sigma,\sigma^{\prime})$ are called admissible sequences (for the pair $(\sigma,\sigma^{\prime})$) and the vertices $b_{i}$ are dropped from the notation. Finally, we define

[TABLE]

where $d:=d_{T}$ denotes the path-metric on $T$.

Lemma 9.3**.**

$d_{\Theta}$* is a proper $G$-invariant generalised metric on $E^{k}(T)$. Furthermore, if $\sigma,\sigma^{\prime}\in E^{k}(T)$ both lie on the same $\Theta$-small geodesic, then $d_{\Theta}(\sigma,\sigma^{\prime})=3\cdot d_{T}(m(\sigma),m(\sigma^{\prime}))$.*

{proof}

The $G$-invariance of $d_{\Theta}$ follows from the $G$-invariance of both $d_{T}$ and $d_{E^{k}}$. Symmetry follows from $\mathcal{M}_{\Theta}(\sigma,\sigma^{\prime})=\mathcal{M}_{\Theta}(\sigma^{\prime},\sigma)$. The triangle inequality holds, since we can concatenate an admissible sequence realising $d_{\Theta}(\sigma,\sigma^{\prime})$ with an admissible sequence realising $d_{\Theta}(\sigma^{\prime},\sigma^{\prime\prime})$ to obtain an admissible sequence for the pair $(\sigma,\sigma^{\prime\prime})$. Taking mid-, start- and endpoints of segments into account for the definition of $d_{\Theta}(\sigma,\sigma^{\prime})$ guarantees that $d_{\Theta}$ is indeed a generalised metric (instead of a generalised pseudo-metric).

For two elements $\sigma$ and $\sigma^{\prime}$ in $E^{k}(T)$ lying on the same $\Theta$-small geodesic, the sequence $(\sigma,o(\sigma)),(\sigma^{\prime},o(\sigma^{\prime}))$ is an admissible sequence and yields $d_{\Theta}(\sigma,\sigma^{\prime})\leq 3\cdot d_{T}(m(\sigma),m(\sigma^{\prime}))$. Combined use of the triangle inequality for $d_{T}$ and the fact that $\sigma,\sigma^{\prime}$ lie on the same $\Theta$-small geodesic gives that the admissible sequence $(\sigma,o(\sigma)),(\sigma^{\prime},o(\sigma^{\prime}))$ already realises $d_{\Theta}(\sigma,\sigma^{\prime})$. It remains to show that $d_{\Theta}$ is proper:

Let $R>0$ and $\sigma\in E^{k}(T)$ be given. Note that, since the distance between two vertices (with respect to $d_{T}$) is an integer, if $\sigma\neq\sigma^{\prime}$, then $d_{\Theta}(\sigma,\sigma^{\prime})\geq 1$. So, if $\sigma^{\prime}\in B_{R}^{d_{\Theta}}(\sigma)$, then there is an admissible sequence $(\sigma_{0},b_{0}),(\sigma_{1},b_{1}),\ldots,(\sigma_{r},b_{r})$, $r\leq R$ for the pair $(\sigma,\sigma^{\prime})$ such that $d_{T}(m(\sigma_{i}),m(\sigma_{i+1}))\leq R$ for all $0\leq i\leq r-1$.

Since $d_{E^{k}}$ is proper, given $\sigma_{i}$ there are only finitely many segments $\sigma\in E^{k}(T)$ with $d_{E^{k}}(\sigma_{i},\sigma)\leq\Theta$. Thus, since $[m(\sigma_{i}),m(\sigma_{i+1})]$ has to be extended $\Theta$-small and of length $\leq R$ (with respect to $d_{T}$), there are only finitely many possibilities for $\sigma_{1}$. And given any of those possible $\sigma_{1}$, there are only finitely many possibilities for $\sigma_{2}$. Iterating this argument $R$ times we conclude that there are only finitely many possibilities for $\sigma_{r}=\sigma^{\prime}$.

Next we will define a segment-flow space $FS_{\Theta}$ as a certain subset of the space $E^{k}(T)\times\Delta_{+}(T)\times\Delta_{+}(T)$. The definition given here is a slight adaptation of the coarse $\Theta$-flow space in [Bar17, Definition 3.4] to our situation on trees.

Definition 9.4**.**

Let $\Theta>0$. Let $Z:=\Delta_{+}(T)\times\Delta_{+}(T)$ be equipped with the subspace topology. Furthermore, let

[TABLE]

where the notion of a $\Theta$-small geodesic is as in Definition 7.2.

Before we list properties of $FS_{\Theta}$, we cite a definition of Bartels, which we will need briefly while applying [Bar17, Theorem 1.1] to obtain long thin covers for $FS_{\Theta}$.

Definition 9.5**.**

(see [Bar17, p. 750, before Theorem 1.1]) Let $(V,d)$ be a generalised discrete metric space. Let $\alpha\geq 0$. A subset $S\subset V$ is $\alpha$-separated if $d(x,y)\geq\alpha$ for any two distinct points $x,y\in S$. A subspace $V_{0}\subset V$ has the $(D,R)$-doubling property if for all $\alpha\geq R$ the following holds: if $S\subset V_{0}$ is $\alpha$-separated and contained in a $2\alpha$-ball, then the cardinality of $S$ is at most $D$.

The following lemma collects the properties of $E^{k}(T),Z$ and $FS_{\Theta}$ needed to apply [Bar17, Theorem 1.1] later on.

Lemma 9.6**.**

Let $\Theta>0$ and let $d_{\Theta}$ be as in Definition 9.2. Let $Z$ and $FS_{\Theta}$ be as above. Then

1. a)

$(E^{k}(T),d_{\Theta})$* is a discrete countable proper generalised metric space with a proper isometric $G$-action.* 2. b)

$Z$* is a separable metric space with an action of $G$ by homeomorphisms. $FS_{\Theta}$ is separable and metrisable as well.* 3. c)

$FS_{\Theta}$* is a closed $G$-invariant subspace of $E^{k}(T)\times Z$ (equipped with the diagonal action of $G$).* 4. d)

The (small inductive) dimension of $FS_{\Theta}$ is [math]. 5. e)

There are $D,R>0$, independent of $\Theta$, such that for all $(\xi_{-},\xi_{+})\in Z$ the subspace $E^{k}(T)_{(\xi_{-},\xi_{+})}:=\{\sigma\in E^{k}(T)\ |\ (\sigma,\xi_{-},\xi_{+})\in FS_{\Theta}\}$ of $E^{k}(T)$ has the $(D,R)$-doubling property. 6. f)

For all $(\sigma,\xi_{-},\xi_{+})\in FS_{\Theta}$, the isotropy group $G_{(\xi_{-},\xi_{+})}=G_{\xi_{-}}\cap G_{\xi_{+}}$ belongs to the family $\mathcal{F}_{\partial}\cup\mathcal{F}_{(k)}$, where $\mathcal{F}_{(k)}:=\{H\leq G\ |\ \exists\ \sigma\in E^{k}(T):\ H\leq G_{o(\sigma)}\cap G_{t(\sigma)}\}\subset\mathcal{FIN}(G)$. In particular, these isotropy groups all belong to $\mathcal{F}=\mathcal{F}_{T}\cup\mathcal{F}_{\partial}$.

{proof}

a) is immediate since $d_{\Theta}$ only takes values in $\mathbb{N}$ and the action of $G$ on $E^{k}(T)$ has finite point stabilisers.

b) Since we assumed $T$ to be countable, $\Delta_{+}(T)$ is separable and metrisable by Lemma 2.4. So $Z$ is as well. Since $E^{k}(T)$ is discrete and countable, $E^{k}(T)\times Z$ is separable and metrisable and hence is $FS_{\Theta}$.

d) By the product theorem [Nag65, Theorem II.5, p. 20]333This and the following results referenced are formulated for the strong inductive dimension. However, recall that for separable metrisable spaces, the small inductive dimension coincides with the covering dimension (and the strong inductive dimension) [Nag65, Theorem IV.1, p. 90]. and Lemma 2.8 the space $Z=\Delta_{+}(T)\times\Delta_{+}(T)$ has dimension [math]. Since a discrete space $V$ has dimension 0, $E^{k}(T)\times Z$ has dimension 0 by [Nag65, Theorem II.5, p. 20]. Finally, by [Nag65, Theorem II.3, p. 19], the dimension does not increase when taking subspaces and hence $FS_{\Theta}$ has dimension [math].

f) Since $G_{(\xi_{-},\xi_{+})}=G_{\xi_{-}}\cap G_{\xi_{+}}$, if $(\xi_{-},\xi_{+})\in\partial T\times\partial T\setminus\operatorname{diag}$, then $G_{(\xi_{-},\xi_{+})}\in\mathcal{F}_{\partial}$ by definition. If at least one of the points $\xi_{-},\xi_{+}$ lies in $T$, then, by definition of $FS_{\Theta}$, the group $G_{\xi_{-}}\cap G_{\xi_{+}}$ is contained in the pointwise stabiliser $G_{\sigma}$ of some $\sigma\in E^{k}(T)$, and hence is finite. It remains to show c) and e).

c) Since $E^{k}(T)\times Z$ is metrisable, we can work with sequences. Moreover, without loss of generality any convergent sequence has the form

[TABLE]

since $E^{k}(T)$ is discrete. Now, if all $(\sigma,\xi_{-,n},\xi_{+,n})$ lie in $FS_{\Theta}$, then $\sigma\subset[\xi_{-,n},\xi_{+,n}]$ holds for all $n\in\mathbb{N}$. Hence, the topology of $\Delta_{+}(T)$ forces $\sigma$ to lie on $[\xi_{-},\xi_{+}]$ (though $\sigma=[\xi_{-},\xi_{+}]$ is possible). The geodesic $[\xi_{-},\xi_{+}]$ is $\Theta$-small, since the geodesics $[\xi_{-,n},\xi_{+,n}]$ are and any finite subsegment of $[\xi_{-},\xi_{+}]$ is eventually contained in some $[\xi_{-,N},\xi_{+,N}]$. $G$-invariance of $FS_{\Theta}$ is immediate, since the notion of a $\Theta$-small geodesic is $G$-invariant.

e) We will show that for all $(\xi_{-},\xi_{+})$ the subspace $E^{k}(T)_{(\xi_{-},\xi_{+})}$ has the $(5,3)$-doubling property: So let $\alpha\geq R=3$. First, note that $E^{k}(T)_{(\xi_{-},\xi_{+})}$ is empty if the geodesic $[\xi_{-},\xi_{+}]$ is not $\Theta$-small or is of length $\leq k-1$. Otherwise $E^{k}(T)_{(\xi_{-},\xi_{+})}$ is equal to $E^{k}(T)\cap[\xi_{-},\xi_{+}]$. In this case, we can take midpoints of segments to obtain a bijection between $E^{k}(T)_{(\xi_{-},\xi_{+})}$ and (a subset of) $\mathbb{Z}$. Since $d_{\Theta}(\sigma,\sigma^{\prime})=3\cdot d_{T}(m(\sigma),m(\sigma^{\prime}))$ for all $\sigma,\sigma^{\prime}$ on $[\xi_{-},\xi_{+}]$, any set of the form $B_{2\alpha}(\tilde{\sigma})\cap E^{k}(T)_{(\xi_{-},\xi_{+})}$ consists of at most $\frac{4}{3}\alpha+1$ elements of $E^{k}(T)$. Furthermore, if $\sigma,\sigma^{\prime}\in E^{k}(T)_{(\xi_{-},\xi_{+})}$ belong to an $\alpha$-separated subset, then $d_{T}(m(\sigma),m(\sigma^{\prime}))\geq\frac{\alpha}{3}$ follows from $d_{\Theta}(\sigma,\sigma^{\prime})=3\cdot d_{T}(m(\sigma),m(\sigma^{\prime}))$. Thus, there are at most $\frac{3}{\alpha}\cdot(\frac{4}{3}\alpha+1)\leq 5=:D$ elements in any $\alpha$-separated subset of $E^{k}(T)_{(\xi_{-},\xi_{+})}$ which is contained in a $2\alpha$-ball.

Next, we define a family of maps from $G\times_{\Theta}\partial T$ to $FS_{\Theta}$ that formalise the idea of letting an element $\sigma$ of $E^{k}(T)$ ‘flow along a $\Theta$-small geodesic $[a(g,\xi),\xi]$’. Morally, this family of maps could be thought of as a partial flow on $FS_{\Theta}$.

Definition 9.7**.**

Let $\tau\in\mathbb{N}$ and $\Theta^{\prime}\geq\Theta>\frac{k}{2}$. We define

[TABLE]

where $\sigma_{\tau}(g,\xi)$ is the unique geodesic segment of length $k$ on $[a(g,\xi),\xi]$ such that

[TABLE]

Lemma 9.8**.**

$\phi_{\Theta,\Theta^{\prime},\tau}$* is continuous and $G$-equivariant.*

{proof}

$G$-equivariance is clear, since the action of $G$ on $T$ is simplicial. Let $(g,\xi_{n})\in G\times_{\Theta}\partial T$ be a sequence converging to some $(g,\xi)\in G\times_{\Theta}\partial T$. Since $a(g,\eta)\in\{gw_{0},gw_{0}^{\prime}\}$ for arbitrary $\eta$, we can assume without loss of generality that $a(g,\xi_{n})=a(g,\xi)$ for all $n\in\mathbb{N}$. The convergence of $\xi_{n}\to\xi$ then already implies that $\sigma_{\tau}(g,\xi_{n})=\sigma_{\tau}(g,\xi)$ for all but finitely many $n$.

10. Covers for $G\times_{\Theta}\partial T$

Given some threshold $\Theta>0$, this section provides an open $G$-invariant $\mathcal{F}_{\partial}\cup\mathcal{F}_{(k)}$-cover of $G\times_{\Theta}\partial T$ that is wide for $G\times_{\Theta}\partial T$. We start by using [Bar17, Theorem 1.1] to obtain a family of covers for the flow space $FS_{\Theta^{\prime}}$ for any $\Theta^{\prime}$.

Corollary 10.1**.**

Let $\Theta^{\prime}>0$. Let $FS_{\Theta^{\prime}}$ be as before and let $D$ and $R$ be as in Lemma 9.6. Then there is $N\in\mathbb{N}$ independent of $\Theta^{\prime}$, such that for any $\alpha>0$ there is an open $G$-invariant $\mathcal{F}_{\partial}\cup\mathcal{F}_{(k)}$-cover $\mathcal{W}_{\Theta^{\prime},\alpha}$ of $FS_{\Theta^{\prime}}$ satisfying

1. a)

$\dim\mathcal{W}_{\Theta^{\prime},\alpha}\leq N$, 2. b)

for every $(\sigma,\xi_{-},\xi_{+})\in FS_{\Theta^{\prime}}$ there is $W\in\mathcal{W}_{\Theta^{\prime},\alpha}$ such that

[TABLE]

{proof}

This is the conclusion of [Bar17, Theorem 1.1]. That the assumptions of this theorem are satisfied by $FS_{\Theta^{\prime}}$ follows from Lemma 9.6.

Pulling back these covers along $\phi_{\Theta,\Theta^{\prime},\tau}$ gives the following corollary.

Corollary 10.2**.**

There is $N$ such that for all $\alpha>0,\Theta^{\prime}\geq\Theta>\frac{k}{2}$ and $\tau\in\mathbb{N}$ the set

[TABLE]

is an open $G$-invariant $\mathcal{F}_{\partial}\cup\mathcal{F}_{(k)}$-cover of $G\times_{\Theta}\partial T$ of dimension $\leq N$.

Before we show that among these covers we can find one which is wide for $G\times_{\Theta}\partial T$, we will state two technical assertions.

Lemma 10.3**.**

Let $\alpha>0$. Let $v\in V(T)$ and $g_{\tau}v\to\xi\in\partial T$ for $\tau\to\infty$. Then, for all $h\in B_{\alpha}(1)$, the sequence $g_{\tau}hv$ also converges to $\xi$.

{proof}

Fix a basepoint $b\in T$. So $[b,g_{\tau}v]$ converges uniformly on compacta to $[b,\xi]$. Let $C:=\max\{d_{T}(v,hv)\ |\ h\in B_{\alpha}(1)\}$. Since $d_{T}(g_{\tau}v,g_{\tau}hv)=d_{T}(v,hv)\leq C$ for all $\tau$, the sequence $[b,g_{h}\tau v]$ converges uniformly on compacta to $[b,\xi]$ if and only if the sequence $[b,g_{\tau}v]$ does.

Lemma 10.4**.**

Let $\Theta>0$. Let $\xi_{-,n}\to\xi_{-}$ and $\xi_{+,n}\to\xi_{+}$ be convergent sequences in $\Delta_{+}(T)$. Furthermore, assume that there is a segment $\sigma\in E^{k}(T)$ such that $\sigma$ lies on all geodesics $[\xi_{-,n},\xi_{+,n}]$ and that for all $r>0$ the set $B^{d_{T}}_{r}(\sigma):=\{z\in T\ |\ d_{T}(z,\sigma)\leq r\}$ contains only finitely many of the $\xi_{-,n},\xi_{+,n}$. If the geodesics $[\xi_{-,n},\xi_{+,n}]$ are $\Theta$-small, then $\xi_{-},\xi_{+}\in\partial T$ and $[\xi_{-},\xi_{+}]$ is $\Theta$-small.

{proof}

Assume that $\xi_{+}\not\in\partial T$. If $\xi_{+,n}=\xi_{+}$ for all $n\geq n_{0}$, there is—in contradiction to the assumptions of the lemma—some $r>0$ such that $B_{r}^{d_{T}}(\sigma)$ contains infinitely many of $\xi_{+,n}$. So, for $\xi_{+,n}\to\xi_{+}\in V(T)$ to hold, the union of the $[\xi_{+},\xi_{+,n}]$ must contain infinitely many of the edges incident to $\xi_{+}$. Since $d_{E^{k}}$ is proper, this contradicts the other assumption of the lemma, that all $[o(\sigma),\xi_{+,n}]$ are $\Theta$-small. The argument for $\xi_{-}\in\partial T$ is the same and the geodesic $[\xi_{-},\xi_{+}]$ is $\Theta$-small since all geodesics $[\xi_{-,n},\xi_{+,n}]$ are.

Proposition 10.5**.**

Retain the assumptions of Proposition 4.1. Then there is $N\in\mathbb{N}$ such that for all $\alpha>0$ and all $\Theta>0$ there is a $G$-invariant collection $\mathcal{U}_{\Theta,\alpha}$ of open $\mathcal{F}_{\partial}\cup\mathcal{F}_{(k)}$-subsets of $G\times\Delta(T)$ such that the order of $\mathcal{U}_{\Theta,\alpha}$ is at most $N$ and $\mathcal{U}_{\Theta,\alpha}$ is wide for $G\times_{\Theta}\partial T$.

{proof}

(following the proof of [Bar17, Proposition 3.2]) Let $N$ be as in Corollary 10.1 and Corollary 10.2. Let $\alpha,\Theta>0$ be given. Recall that $L_{0}$ is the finite subtree of $T$ spanned by the vertices $\{hw_{0},hw_{0}^{\prime}\ |\ h\in B_{\alpha}(1)\}$. Let $C=C(\alpha)$ be the diameter of $L_{0}$ with respect to $d_{T}$. (In particular, for any $\xi\in\Delta_{+}(T)$, $h\in B_{\alpha}(1)$ and $g\in G$ the distance of $a(g,\xi)$ to $a(gh,\xi)$ is bounded by $C$.) Set $\alpha^{\prime}:=3C$ and $\Theta^{\prime}:=\Upsilon_{1}(\Theta)$ (cf. 8.5). Finally, let $\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}}$ be as in Corollary 10.1.

We will show that there is $\tau\in\mathbb{N}$ such that $\phi_{\Theta,\Theta^{\prime},\tau}^{-1}(\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}})$ is ($\alpha$-)wide for $G\times_{\Theta}\partial T$. Once we have proven this, we can thicken $\phi_{\Theta,\Theta^{\prime},\tau}^{-1}(\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}})$ to a $G$-invariant collection $\mathcal{U}_{\Theta,\alpha}$ of open $\mathcal{F}$-subsets of $G\times\Delta(T)$ that is still ($\alpha$-)wide for $G\times_{\Theta}\partial T$ (see [Bar17, Appendix B], [BL12b, Lemma 4.14]). Since all $\phi_{\Theta,\Theta^{\prime},\tau}^{-1}(\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}})$ have dimension at most $N$ by Corollary 10.2, the collection $\mathcal{U}_{\Theta,\alpha}$ will have order $\leq N$ as well as the thickening process does not increase the order [Bar17, Lemma B.2].

To show that for the given $\alpha,\Theta>0$ there is a $\phi_{\Theta,\Theta^{\prime},\tau}^{-1}(\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}})$ that is ($\alpha$-)wide, start by assuming the contrary.

Assumption: For all $\tau\in\mathbb{N}$ there is $(g_{\tau},\xi_{\tau})\in G\times_{\Theta}\partial T$ such that for all $W\in\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}}$

[TABLE]

Applying $\phi_{\Theta,\Theta^{\prime},\tau}$ to $(g_{\tau},\xi_{\tau})$ we obtain a segment $\sigma_{\tau}(g_{\tau},\xi_{\tau})$ of length $k$ on $[a(g_{\tau},\xi_{\tau}),\xi_{\tau}]$. Let $\sigma_{1}$ be the first segment of length $k$ on $[w_{0},w_{0}^{\prime}]$ and $\sigma_{2}$ be the last segment of length $k$ on $[w_{0},w_{0}^{\prime}]$ (those segments exist since $[w_{0},w_{0}^{\prime}]$ has length $\geq 5k$). Since $[a(g_{\tau},\xi_{\tau}),\xi_{\tau}]$ contains $[a(g_{\tau},\xi_{\tau}),gm_{\epsilon}]$, the first segment of length $k$ on $[a(g_{\tau},\xi_{\tau}),\xi_{\tau}]$ is either $g_{\tau}\sigma_{1}$ or $g_{\tau}\sigma_{2}$. Furthermore, since all $[a(g_{\tau},\xi_{\tau}),\xi_{\tau}]$ are $\Theta$-small and their respective first segments of length $k$ lie in at most two different $G$-orbits of $G\curvearrowright E^{k}(T)$, by our special construction of $d_{E^{k}}$ (cf. Lemma 7.1), it follows that the segments $\sigma_{\tau}(g_{\tau},\xi_{\tau})$ can only belong to a finite number of $G$-orbits of $G\curvearrowright E^{k}(T)$. Since $\phi_{\Theta,\Theta^{\prime},\tau}^{-1}(\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}})$ is a $G$-invariant collection we can assume (after passing to a subsequence) that there is a segment $\sigma\in E^{k}(T)$ such that $\sigma=\sigma_{\tau}(g_{\tau},\xi_{\tau})$ for all $\tau$.

Since $\Delta_{+}(T)$ is a closed subspace of $\overline{T}^{obs}$, it is compact as well and by passing to a further subsequence (twice) we can enforce the existence of $\xi_{+}=\lim_{\tau}\xi_{\tau}$ and $\xi_{-}=\lim_{\tau}a(g_{\tau},\xi_{\tau})$ in $\Delta_{+}(T)$.

Since $\tau$ increases, no $B^{d_{T}}_{r}(\sigma)$ contains infinitely many of the $a(g_{\tau},\xi_{\tau})$. Moreover, by assumption, all $\xi_{\tau}$ lie in $\partial T$. Hence, by Lemma 10.4, we obtain that $\xi_{-}$ and $\xi_{+}$ both lie in $\partial T$ and that $[\xi_{-},\xi_{+}]$ is $\Theta$-small. Furthermore, $\sigma$ lies on $[\xi_{-},\xi_{+}]$. In other words, $(\sigma,\xi_{-},\xi_{+})\in FS_{\Theta}\subseteq FS_{\Theta^{\prime}}$.

Now, by Corollary 10.1, there is $W_{0}\in\mathcal{W}_{\Theta^{\prime},\alpha^{\prime}}$ such that

[TABLE]

As $d_{\Theta^{\prime}}$ is proper, the ball $B_{\alpha^{\prime}}^{d_{\Theta^{\prime}}}(\sigma)$ is finite and so is $M:=B_{\alpha^{\prime}}^{d_{\Theta^{\prime}}}(\sigma)\cap[\xi_{-},\xi_{+}]$. Since $\xi_{-},\xi_{+}\in\partial T$ and $W_{0}$ is open, there are disjoint open neighbourhoods $U_{-}$ and $U_{+}$ in $\Delta_{+}(T)$ of $\xi_{-}$ and $\xi_{+}$, respectively, such that

[TABLE]

Without loss of generality we can assume that $U_{+}$ is of the form $M(\xi_{+},a)$ (cf. Definition 2.2) for some $a\in T$ with $d_{T}(t(\sigma),a)>C$ (the purpose of this technical assumption will only be clear at the very end of this proof).

By Lemma 10.3, we still have $\lim_{\tau}a(g_{\tau}h,\xi_{\tau})=\xi_{-}$ for all $h\in B_{\alpha}(1)$. So we can find $\tau_{0,h}$ such that $a(g_{\tau}h,\xi_{\tau})\in U_{-}$ and $\xi_{\tau}\in U_{+}$ for all $\tau\geq\tau_{0,h}$. Since $B_{\alpha}(1)$ is finite and $B_{\alpha}(g_{\tau})=g_{\tau}B_{\alpha}(1)$, there is a $\tau_{0}$ such that $a(g,\xi_{\tau})\in U_{-}$ and $\xi_{\tau}\in U_{+}$ for all $\tau\geq\tau_{0}$ and $g\in B_{\alpha}(g_{\tau})$. Since $M$ is finite as well, we can (if necessary) enlarge $\tau_{0}$ further to obtain that all elements of $M$ already lie on $[a(g_{\tau_{0}},\xi_{\tau_{0}}),\xi_{\tau_{0}}]$. We now claim that there is some $\tau\geq\tau_{0}$ such that

[TABLE]

which would be a contradiction to the assumption we started with. To prove this claim it is sufficient to find $\tau$ such that

[TABLE]

for all $h\in B_{\alpha}(1)$. \big{(}\sigma_{\tau}(g_{\tau}h,\xi_{\tau}),a(g_{\tau}h,\xi_{\tau}),\xi_{\tau}\big{)}\in FS_{\Theta^{\prime}} holds by Lemma 8.9 and Lemma 8.8. By the way $\tau_{0}$ was defined, it is guaranteed that $\xi_{\tau}\in U_{+}$ and $a(g_{\tau}h,\xi_{\tau})\in U_{-}$ for all $\tau\geq\tau_{0}$. So it only remains to find $\tau\geq\tau_{0}$ large enough such that $\sigma_{\tau}(g_{\tau}h,\xi_{\tau})\in M$ holds for all $h\in B_{\alpha}(1)$.

Let $v_{h}$ be the first vertex on $[a(g_{\tau}h,\xi_{\tau}),\xi_{\tau}]$ that also lies on $[a(g_{\tau},\xi_{\tau}),\xi_{\tau}]$, i.e.

[TABLE]

Since $B_{\alpha}(1)$ is finite, there is a vertex $v=v_{h_{0}}$ which—among the $v_{h}$—has maximal distance to $a(g_{\tau},\xi_{\tau})$. In particular, $[v,\xi_{\tau}]$ is contained in $[a(g_{\tau}h,\xi_{\tau}),\xi_{\tau}]$ for all $h\in B_{\alpha}(1)$. Note that, since all $a(g_{\tau}h,\xi_{\tau})$ lie in $U_{-}$, all $v_{h}$ lie in $U_{-}$ as well. In particular, $v$ lies in $U_{-}$. Moreover, the vertex $v$ lies on the geodesic $[a(g_{\tau}h_{0},\xi_{\tau}),a(g_{\tau}h,\xi_{\tau})]$ for all $h\in B_{\alpha}(1)$. Since the length of $[a(g_{\tau}h_{0},\xi_{\tau}),a(g_{\tau}h,\xi_{\tau})]$ is bounded by $C$, the vertex $v$ has distance at most $C$ to any of the $a(g_{\tau}h,\xi_{\tau})$. Hence, (by choosing $\tau\geq\tau_{0}$ large enough) we can force $\sigma_{\tau}(g_{\tau}h,\xi_{\tau})$ to lie on $[v,\xi_{\tau}]$ for all $h\in B_{\alpha}(1)$. Since $d_{T}(a(g_{\tau},\xi_{\tau}),m(\sigma_{\tau}(g_{\tau},\xi_{\tau})))=\tau$, the bound $C$ and the triangle inequality for $d_{T}$ imply

[TABLE]

Both $\sigma_{\tau}(g_{\tau}h,\xi_{\tau})$ and $\sigma_{\tau}(g_{\tau},\xi_{\tau})$ lie on $[v,\xi_{\tau}]$ and it follows that

[TABLE]

Since the geodesic $[a(g_{\tau},\xi_{\tau}),\xi_{\tau}]$ is $\Theta$- and hence $\Theta^{\prime}$-small and contains both the segments $\sigma_{\tau}(g_{\tau}h,\xi_{\tau})$ and $\sigma_{\tau}(g_{\tau},\xi_{\tau})$, by definition of $d_{\Theta^{\prime}}$ it follows that

[TABLE]

Hence, $\sigma_{\tau}(g_{\tau}h,\xi_{\tau})\in B_{\alpha^{\prime}}^{d_{\Theta^{\prime}}}(\sigma_{\tau}(g_{\tau},\xi_{\tau}))$ for all $h\in B_{\alpha}(1)$. Recall that $\sigma=\sigma_{\tau}(g_{\tau},\xi_{\tau})$ and $U_{+}=M(\xi_{+},a)$ where $a$ satisfies $d_{T}(t(\sigma),a)\geq C$. Therefore, $\sigma_{\tau}(g_{\tau}h,\xi_{\tau})$ indeed lies on $[\xi_{-},\xi_{+}]$ and is contained in $B_{\alpha^{\prime}}^{d_{\Theta^{\prime}}}(\sigma)$, so it is contained in $M$.

This concludes the last technical part that was used in the proof of A.