Trees are 1-Transfer

TL;DR

This paper constructs an explicit 1-transfer space to advance the understanding of the Farrell-Jones conjecture for hyperbolic groups, leveraging geometric group properties.

Contribution

It provides a concrete construction of a 1-transfer space, enhancing tools for proving the Farrell-Jones conjecture for hyperbolic groups.

Findings

Explicit 1-transfer space construction for hyperbolic groups

Improved techniques for Farrell-Jones conjecture proofs

Enhanced understanding of geometric properties in group theory

Abstract

The K-theoretic Farrell-Jones isomorphism conjecture for a group ring $R[G]$ has been proved for several groups. The toolbox for proving the Farrell-Jones conjecture for a given group depends on some geometric properties of the group as it is the case of hyperbolic groups. The technique used to prove it for hyperbolic groups $G$ relies in the concept of an $N$-transfer space endowed with a $G$ action. In this work, we give an explicit construction of a $1$-transfer space.

Figures

Equations

$$X⟶H_{\ast }^G(X;{\mathbf{K}}_R)$$

$${\alpha }_{\mathcal{F}}:H_{\ast }^G(E_{\mathcal{F}}G;{\mathbf{K}}_R)\to H_{\ast }^G(G/G;{\mathbf{K}}_R)=K_{\ast }(R[G])$$

$$(x|y)=\frac{1}{2}(d(x)+d(y)-d(x,y))$$

$$(x|y)\ge min\{(x|z),(z|y)\}.$$

$$\underset{i,j\to \infty }{\lim }inf\{(x_i|y_j)\}=1$$

$$(\chi ,{\chi }^{\prime })=sup\underset{i,j\to \infty }{\lim }\{(x_i|y_j)\}$$

$$d(\chi ,{\chi }^{\prime })=l(C_{\chi })+l(C_{{\chi }^{\prime }})$$

$$1-\delta \le \sum _{i=1}^N\frac{1}{2^i}\le 1$$

$$P_{{\sigma }_N}:\overline{T}\to \overline{B}({\sigma }_N,x_0).$$

$$\overline{T}\stackrel{i}{\to }K\stackrel{P_{{\sigma }_N}}{\to }\overline{T}$$

$$d(P_{{\sigma }_N,x},x)=1-{\sigma }_N$$

Full text

1. Introduction

The present work examine one tool using to prove the $K$-theoretic Farrell-Jones Isomorphism Conjecture. The conjecture itself is not the aim of this work. However, to give our result the right dimension we start with a short review of the formulation and results on the Farrell-Jones conjecture.

Let $G$ be a group. A family of subgroups of $G$ is a non-empty collection $\mathcal{F}$ of subgroups that is closed under conjugation and taking subgroups.

Example 1.1** (Family of subgroups).**

Let $G$ be a group. Useful examples of families of subgroups of $G$ are the family of finite subgroups $\mathcal{FIN}$, the family of cyclic subgroups $\mathcal{FCY}$ and the family of virtually cyclic subgroups $\mathcal{VCY}$.

Definition 1.2**.**

A $G$-CW-Complex $E$ is called a classifying space for the family $\mathcal{F}$ of subgroups of $G$, if $E^{H}$ (fixed points) is always contractible for all $H\in\mathcal{F}$ and empty otherwise.

It is a well-known result that the for any family $\mathcal{F}$ always exists a classifying space for the family up to $G$-equivariant homotopy. A model for the classifying space of a family is usually denoted by $E_{\mathcal{F}}G$.

Following the construction of [3], given a ring $R$ and a group $G$, they construct a homology theory for $G$-spaces

[TABLE]

with the property $H_{*}^{G}(G/H;{\mathbf{K}}_{R})=K_{*}(R[H])$.

Definition 1.3** ($\mathcal{F}$-assembly map).**

Let $\mathcal{F}$ be a family of subgroups of $G$. The projection $E_{\mathcal{F}}G\twoheadrightarrow G/G$ to the one-point $G$-space induces the $\mathcal{F}$-assembly map

[TABLE]

Conjecture 1.4** (Farrell-Jones).**

For all groups $G$ and all rings $R$, the assembly map $\alpha_{\mathcal{VCY}}$ is an isomorphism.

The Farrell-Jones conjecture happens to be true for a large class of groups. Examples for which the conjecture is true are the family of Hyperbolic groups [2] and Fundamental groups of graphs of Virtually Cyclic groups [8]. Also, all of them illustrate how diverse the techniques to prove this conjecture are.

While it is still an open conjecture and the proves does not show a recognizable pattern some attempts to find general arguments has been done. On Proofs of the Farrell–Jones Conjecture [1] the statement of Theorem A has this objective. The statement is

Theorem 1.5** (Theorem A, [1]).**

Suppose $G$ is finitely generated by $S$. Let $\mathcal{F}$ be family of subgroups of $G$. Assume that there is $N\in{\mathbb{N}}$ such that for any $\epsilon>0$ there are

1. (a)

an $N$-transfer space $X$ equipped with a $G$-action, 2. (b)

a simplicial $(G,\mathcal{F})$-complex $E$ of dimension at most $N$, 3. (c)

a map $f:X\to E$ that is $G$-equivariant up to $\epsilon$: $d^{\prime}(f(sx),sf(x))\leq\epsilon$ for all $s\in S$, $x\in X$.

Then $\alpha_{\mathcal{F}}:H_{*}^{G}(E_{\mathcal{F}}G;K_{R})\rightarrow K_{*}(R[G])$ is an isomorphism

A simplicial $(G,\mathcal{F})$-complex a simplicial complex $E$ with a simplicial $G$-action whose isotropy groups $G_{x}=\{g\in G|gx=x\}$ belongs to $\mathcal{F}$ for all $x\in E$.

Remark 1.6**.**

Theorem A applies to Hyperbolic groups and $\mathcal{F}=\mathcal{VCY}$. the family of virtually cyclic sugroups. This theorem is a minor formulation of [Bartels-Lueck-Reich] work.

The concept of $N$-transfer is our main concern. In [Bar], an example of a $1$-transfer space is given. Namely, the compactification $\overline{T}$ of a locally finite simplicial tree $T$ by geodesic rays. The result is elementary but by no means trivial. There are no explicit prove of this result and the author provides a proof of this fact.

2. Preliminaries

In this section, we give the necessary definitions to establish our result.

Definition 2.1** ($N$-transfer).**

An $N$-transfer space $X$ is a compact contractible metric space such that the following holds. For any $\delta>0$ there exists a simplicial complex $K$ of dimension at most $N\in{\mathbb{N}}$, continuous maps $i:X\to K$, $p:K\to X$ and homotopy $H:p\cdot i\to Id_{X}$ such that for any $x\in X$, $\text{diam}\{H(t,x)|t\in[0,1]\}\leq\delta$.

The definition of an $N$-transfer recall us that of a dominated complex. The difference is the control requirement on the diameter being less or equal to $\delta$.

Definition 2.2** (Abstract simplicial complex).**

An abstract simplicial complex $K$ consists of a non-empty set $V$ of vertices and a collection $\mathcal{S}$ of non-empty subsets of $V$ such that

• •

for every $v\in V$, $\{v\}\in\mathcal{S}$,

• •

if $S\in\mathcal{S}$, then $\emptyset\neq T\subseteq\mathcal{S}$ is in $\mathcal{S}$.

Remark 2.3**.**

We call the elements of $\mathcal{S}$ the simplices of $K$. An element $S\in\mathcal{S}$ is an $n$-simplex if $|S|=n+1$ and we set $dim(S)=n$. The single elements of $n$-simplex $S$ are called vertices and each proper subset of $S$ is a face.

Definition 2.4** (Simplicial Tree).**

A simplicial tree $T$ is a connected simply connected $1$-simplex.

Remark 2.5**.**

Working with an abstract simplicial complex whose simplex are $1$-simplex might be undue. Instead, we use its geometric realization. Every statement we made on the geometric realization of $T$ is valid for the abstract simplicial tree.

Definition 2.6** (Simplicial Metric).**

A metric $d$ on a simplicial complex $K$ is called simplicial if

• (a)

the restriction, $d_{i}=d|_{S_{i}}$, to each simplex $S_{i}$ is euclidean,

• (b)

$d$ is maximal for the condition $d_{i}=d|_{S_{i}}$ for each $S_{i}$.

We give a simplicial metric to a tree $T$ as follows.

Let $x_{0}\in T$ be an arbitrary fixed vertex of $T$. A vertex $y\in T$ is adjacent to $x_{0}$ if it is joined with $x_{0}$ by just one edge. Give length $1/2$ to all edges joining adjacent $y$ to $x_{0}$. For the next step, consider all the adjacent vertices to $x_{0}$ and give length $1/4$ to all edges joining adjacent vertices to $y$ but that joining $x_{0}$ and $y$ which has already length $1/2$.

In general, an edge between $x_{1}$ and $x_{2}$ adjacent vertices has length $1/2^{n}$ if the (minimum) number of vertices from $x_{1}$ to $x_{0}$ is $n-1$ and the number of vertices from $x_{2}$ to $x_{0}$ is $n$.

Our goal is to define a metric on the geometric realization of $T$. Call a pair of vertices $x,y\in T$ adjacent if they are joined by just one edge. We now give each edge on the geometric realization of $T$ a length.

1. (i)

Let $x_{0}\in T$ be an arbitrary but fixed vertex. 2. (ii)

The length of an edge joining $x_{0}$ with an adjacent vertex $y$ is $1/2$. 3. (iii)

Recursively, an edge joining $y$, adjacent to $x_{0}$, with any other adjacent vertex different from $x_{0}$ has length $1/2^{2}$. 4. (iv)

Repeat steps (ii) and (iii).

In general, a vertex $x\in T$ can be reached from $x_{0}$ by a minimum sequence of vertices $x_{1},x_{2},...,x_{n}$ with $x_{i}$ adjacent to $x_{i+1}$ and $x_{n}$ adjacent to $x$. By definition of lengths it is easy to see that the edge joining $x$ with $x_{n}$ has length $1/2^{n+1}$.

Defining the length of each edge as we did induces a metric $d$ on $T$. Indeed, this metric makes $T$ into a geodesic metric space and, by definition of a tree, into a uniquely geodesic metric space. We denote this metric space by $(T,d)$.

Proposition 2.7**.**

The metric tree $(T,d)$ is an ${\mathbb{R}}$-tree. Moreover, (T,d) is a $CAT(k)$-spaces for every $k\in{\mathbb{R}}$.

The proof of Proposition 2.7 and precise definition of ${\mathbb{R}}$-tree are in Gromov’s work [5]. For us it is enough to consider $(T,d)$ as $CAT(0)$-space because the compactification we consider depends only on the properties of $CAT(0)$-spaces. We refer the reader to ([2] Chap II. 8) for a detailed exposition of this proof.

Definition 2.8** (Gromov Product).**

Let $(T,d)$ be a metric space with a distinguished point $x_{0}$ and $d(x)$ denote $d(x,x_{0})$. The Gromov product is

[TABLE]

We can think of the Gromov product as a way to measure the distance from $x_{0}$ to the vertex joining $x$ to $y$ in $T$. It is straightforward to see that $(T,d)$ is a [math]-hyperbolic space, that is, for every $x,y,z\in T$ and fixed $x_{0}$ we have

[TABLE]

Definition 2.9** (Geodesic ray and convergence).**

A geodesic ray $C:[0,\infty]\rightarrow T$ (based at $x_{0}$) is a sequence $\{x_{i}\}$ of vertices such that each of them belongs to the image of $C$. Let $\{x_{i}\}$ be a sequence of points in $(T,d)$. We say that $\{x_{i}\}\rightarrow\infty$ converges to the infinity if $(x_{i}|x_{j})\rightarrow 1$ for $i,j\rightarrow\infty$

Consider the set ${\mathcal{C}}$ of all sequences that converges to the infinity in a simplicial metric tree. A pair $\{x_{i}\}$, $\{y_{i}\}$ in ${\mathcal{C}}$ is equivalent if

[TABLE]

This equivalence defines an equivalence relation in ${\mathcal{C}}$ since $(T,d)$ is [math]-hyperbolic.

Definition 2.11** (The boundary $\partial T$).**

The hyperbolic boundary $\partial T$ of a tree $(T,d)$ is the set of equivalent classes of ${\mathcal{C}}$ modulo the relation 2.10.

We can think of two equivalent rays a pair of rays having in common a long sequence of vertex and just differing by a small geodesic segment.

By construction of $(T,d)$ with $x_{0}$ fix, any element $\chi\in\partial T$ has a unique sequence issuing from $x_{0}$. Write $x_{i}\to\chi$ for this sequence.

Now we extend Gromov’s product to the boundary $\partial T$ using the ideas in [4] and [6] defining

[TABLE]

where $x_{i}\to\chi$ and $y_{j}\to\chi^{\prime}$ are the unique sequences on their respective equivalence class.

Immediate properties of 2.12 are

• (1)

For all $\chi,\chi^{\prime}\in\partial T$, $(\chi|\chi^{\prime})=1$ if and only if $\chi=\chi^{\prime}$.

• (2)

For all $\chi,\chi^{\prime}$, $(\chi,\chi^{\prime})=(\chi^{\prime},\chi)$.

• (3)

For all $\chi,\chi^{\prime},\chi^{{}^{\prime\prime}}$, $(\chi,\chi^{\prime})\geq min\{(\chi,\chi^{{}^{\prime\prime}})(\chi^{{}^{\prime\prime}},\chi^{\prime})\}$.

The properties enlisted above are valid for all points in $T$, except for the first property valid only for points in the boundary of $T$. Let $\overline{T}=T\cup\partial T$.

In Figure 1 we illustrate a geometric realization of $\overline{T}=T\bigcup\partial T$. We point out the tree $T$ in the circle with center $x_{0}$. The boundary $\partial T$ is then the circumference. We only highlight points in $\partial T$ in the upper half of the circumference.

Our goal is now to define a distance in $\partial T$ extending $d$ to all $\overline{T}$.

Let $x_{i}\rightarrow\chi$ and $y_{j}\rightarrow\chi^{\prime}$ points in $\partial T$ and $z_{\chi}^{\chi^{\prime}}$ denote the common vertex of $\{x_{i}\}$ and $\{y_{j}\}$ at distance $(\chi|\chi^{\prime})$ from $x_{0}$. Denote by $C_{\chi}$ and $C_{\chi^{\prime}}$ the truncated rays obtained from $\{x_{i}\}$ and $\{y_{j}\}$ starting at the common vertex $z_{\chi}^{\chi^{\prime}}$ going to $\partial T$ ignoring the previous finite sets. Let $l(C_{\chi})$ and $l(C_{\chi^{\prime}})$ denote the lengths of the respective rays and define

[TABLE]

this is $d(\chi,\chi^{\prime})=2(1-(\chi|\chi^{\prime})$,the projection over a convex set given in [2].

In Figure 2 we illustrate the Gromov product $(\chi|\chi^{\prime})$ of two finite rays. The point $z_{\chi}^{\chi^{\prime}}$ marks what vertices are common. In this example, if assume that vertex adjacent to $z_{\chi}^{\chi^{\prime}}$ has length $1/2^{n}$, then $l(C_{\chi})$ has length $1/2^{n}+1/2^{n+1}$ and $l(C_{\chi^{\prime}})$ has length $1/2^{n}+1/2^{n+1}+1/2^{n+2}$. Thus $d(\chi,\chi^{\prime})=l(C_{\chi})+l(C_{\chi^{\prime}})=1/2^{n-1}+1/2^{n}+1/2^{n+2}$.

Proposition 2.13**.**

The definition $(\partial T,d)$ given above satisfies

1. (i)

$d(\chi,\chi^{\prime})=0$* if and only if $(\chi|\chi^{\prime})=1$ if and only if $\chi=\chi^{\prime}$* 2. (ii)

$d(\chi,\chi^{\prime})\geq 0$* and $d(\chi,\chi^{\prime})=d(\chi^{\prime},\chi)$* 3. (iii)

$d(\chi,\chi^{\prime})\leq d(\chi,\chi^{{}^{\prime\prime}})+d(\chi^{{}^{\prime\prime}},\chi^{\prime})$** 4. (iv)

for all $\chi\in\partial T$, $d(\chi,x_{0})=1$ 5. (v)

$d(\chi,\chi^{\prime})=2$* if $\chi$, $\chi^{\prime}$ belong to different connected component of $T\setminus\{x_{0}\}$*

Remark 2.14**.**

Formally, we have defined $d$ only for points in the boundary. However, $d(\chi,x_{0})$ is defined in the same way we defined $d$ on $\partial T$ but consider finite sequences for points in $T$.

3. Trees are 1-Transfer

In this section we prove that the space $\overline{T}$ is a $1$-transfer.

Proposition 3.1**.**

The space $\overline{T}$ given the cone topology [2] is compact and compatible with $(\overline{T},d)$ given as before.

Proof.

The neighbourhood basis given in [2] coincides with a basis of balls in the metric $d$. ∎

Proposition 3.2**.**

The space $(\overline{T},d)$ is contractible.

Proof.

In [6] notation, each ball $\overline{B}(r,x_{0})$ centered at $x_{0}$, with $r\in[0,2]$ is a sub-continuum and hence a tree. ∎

Finally, we construct the homotopy between $\overline{T}$ and a $1$-simplex $K$ and verify the control conditions. Let $\delta>0$, hence $1-\delta<1$ and because $\lim_{n\to\infty}(\sum_{i}^{n}1/2^{i})=1$ there exists $N$ such that

[TABLE]

denote $\sigma_{N}=\sum_{i=1}^{N+1}\frac{1}{2^{i}}$. Using the structure maps, necessaries for the construction of the cone topology as a direct limit, we have a map

[TABLE]

If we consider $\overline{T}=K$ as $1$-simplex then we obtain

[TABLE]

which is a contraction of $K$ over the subtree $\tau=\overline{B}(\sigma_{N},x_{0})$. According to [7] there is a homotopy $H$ between $\tau$ and $\overline{T}$ such that $H(x,0)=Id_{\overline{T}}$ and $H(x,1)=P_{\sigma_{N}}$.

Moreover, for each $x\in\tau$ we have $\{H(x,t)|t\in[0,1]\}=\{x\}$ and hence it has diameter [math]. For any $x\in\overline{T}\setminus\tau$ we have $H(x,0)=x$ and $H(x,1)=P_{\sigma_{N}}(x)$ as extreme points of the homotopy. Since $P_{\sigma_{N}}(x)\in\tau$ we have

[TABLE]

and $1-\sigma_{n}\leq\delta$ we have $diam\{H(t,x)|t\in[0,1]\}\leq\delta$. Hence $\overline{T}$ is a $1$-transfer space.