Actions of certain groups on the Gromov boundary of their Cayley graphs

The object of study of this very short post are groups of the form $G = \Z_{n_1}\star ... \Z_{n_m}$ where $n_i \in \N\cup \{\infty\}$ . We will stick to the standard choice of generators $S = \{(0, ..., 1, .., 0) ~|~ \}$ and look at their associated (undirected) Cayley graph $\Gamma = \Gamma_S(G)$ and boundary space $\partial \Gamma$ . Recall that the Cayley graph of $G$ is the graph given the

Recall that the Cayley graph of $G$ is the graph given the following data

Vertices $V(\Gamma) = G$
Edges $~~E(\Gamma)$ are those $(a,b)\in V(\Gamma)\times V(\Gamma)$ for which there is a $s\in S$ such that $as = b$ .

It turns out that the Cayley graph of $G$ is a tree on which $G$ acts by left multiplication. Heuristically, one could think of this tree as a combinatorial object where the vertices are reduced words in $G$ . Increasing the word length corresponds to going outward on the branches of the tree, while there is only one way to reduce the word length (by multiplying from the left by the inverse of the first element) and corresponds to moving closer to the root of the tree.

Exercise) Show that in the special case when $m=2$ one can write $V(\Gamma)$ as the union of all cosets $V(\Gamma) = G/\Z_{n_1} \cup G/\Z_{n_2}$ . Can we do the same for $m\geq 3$ ?

Let $H = \langle s_1,..., s_n | r_1, ..., r_k\rangle$ be a finitely presented group in $n$ -generators. The word metric on $H$ (with respect to the generators $s_1,..., s_n$ ) is a metric on $H$ determined by

$h\mapsto \text{minimum length of all words expressing }h. \text{ with elements in }S$

This turns out to be a metric which depends upon the choice of generating set $S$ only up to quasi-isometric isomorphisms of $H$ , hence properties which are invariant with respect to this relation are in a sense intrinsic to the group itself (this line of reasoning is at the heart of geometric group theory). Similarly one can define a length function on the graph $\Gamma$ by saying each edge has length one and the distance between two vertices equals the shortest path between them.

Exercise) Show that the identity map $G\rightarrow V(\Gamma)$ is is an isometry with respect to the length metric on $\Gamma$ and the word metric on $G$ if we pick the same generating set $S$ in both cases and a quasi-isometric imbedding regardless of the generating set.

Gromov boundaries of free products

With $G$ as above, the Cayley graph is a tree with vertices of valence

$\frac{\prod_{i\in \{1,2.., m\}} (n_i - 1)}{(n_k - 1)} \qquad 1 \leq k\leq m.$

With respect to the length metric, any tree is a hyperbolic space in the sense that all geodesic triangles are “slim” (we don’t go into details here), hence a Gromov boundary can be defined. This boundary consists of all infinite geodesic rays, that is non repeating paths $\gamma: \mathbb{R}^+ \to \Gamma_S(G)$ , starting from a common fixed point $x_0$ .

In our case the boundary $\partial \Gamma$ consists of all geodesic rays starting form the identity vertex $e\in G = V(\Gamma).$ Any choice of vertex would work here, as the resulting space is invariant of choice of base point up to $G$ -equivariant homeomorphism. In the combinatorial view, the space $\partial \Gamma$ consists of all infinite sequences of reduced words of $S$ . The topology of $\partial \Gamma$ is determined by the local neighborhood basis at a point $(s_1s_2s_3.... )$ given by

$V(s_1)\supset V(s_1s_2)\supset V(s_1s_2s_3)\supset ...$

where

$V(s_1...s_k) = \{(v_1...) \in \partial \Gamma ~|~ v_i = s_i \forall 1\leq i \leq k\}.$

The action of $G$ on $\partial \Gamma$ is by left multiplication and is easily seen to be continuous with respect to the above topology.

The boundary can be decomposed into two disjoint sets $\Gamma = E_1\sqcup E_2$ where $E_1$ consists of those sequences which eventually repeat a fixed reduced ad infinitum, while $E_2$ is its complement.

Exercise) Show that $GE_2\subset E_2$ and that $GE_1\subset E_1$ .

Exercise) Show that $E_1$ and $E_2$ are both dense in $\partial \Gamma$ . Conclude that the sets are neither open nor closed in $\partial \Gamma$ hence cannot be Cantor sets themselves.

Exercise) Show that $G$ acts freely on $E_2$ (meaning if $g v = v$ then $g = e$ ). Show that every point in $E_1$ is a fixed point of some $g\in G$ .

Spoiler) If $(s_1....)\in E_1$ then there is a $g\in G$ and a reduced word $(v_1,...,v_k)$ such that $v_i\in S$ and $(s_1,...) = g (v_1,...,v_k,v_1,...v_k,v_1...)$ . It should be clear that the element $g(v_1...,v_k)g^{-1}$ fixes $(s_1,...)$ and that

$Stab(s_1, ...) = \{o^n~|~ o = g(v_1...,v_k)g^{-1} \} \simeq \Z.$

For my own sanity, I like to think of $E_1$ as representing the rational numbers while $E_2$ the irrationals. The justification is manifold (in the non-mathematical sense). For one $E_1$ is countable, while $E_2$ is not. Secondly, the rational numbers are precisely those that can be expressed with a finite (possibly infinitely repeating) pattern in its decimal expansion (or in any base for that mater). For instance, if $G = \Z_2\star \Z_2$ , then $E_1$ correspondence to the rational numbers between in $[0,1]$ and $E_2$ is in a 1-1 with the irrationals through binary expansions. For all irrationals this correspondence is 1-1, but for certain rationals this correspondence will be 2-1, for others it is 1-1. A point $x$ has two base $b$ expansions if and only if $x b^n$ is an integer for some $n\in \N$ which happens if and only if the expansion is finite in base $b$ . For the decimal expansion this amounts to $1/b$ with $b$ divisible by $5$ or $2$ . In the case of $PSL_n(\Z)= \Z_2\star \Z_3$ one can find a concrete map $PSL_n(\Z) \to \R \cup \{\infty\} = S^1 = \partial \H^2_\R$ to the geodesic boundary of the real hyperbolic plane which is surjective, equivariant (with respect to the action of $PSL_2$ by Mobius transformations), and sends $E_1$ 2-1 to the rational numbers while sending $E_2$ 1-1 to the irrational numbers. I’m struggling to find the article where I read this fact, so for the moment I will leave it without a reference.

Dynamic asymptotic dimension

One says that an action of a discrete group $G$ on a compact space $X$ has dynamic asymptotic dimension less then or equal to $d$ if for any finite set $H\subset G$ there exists an open cover $\{U_1,.., U_d\}$ of $X$ for which the $E$ orbits entirely contained within some $U_i$ are all finite. More explicitly, for any $x\in U_i$ , sets

$O_x := \{ g\in G ~|~ \thereis g_1, ..., g_n \in E, \text{ s.t. } g_1...g_n = g\text{ and } g_1..g_k x \in U_i \forall 1\leq k\leq n \}$

are all finite. Dynamic asymptotic dimension generalizes the notion of asymptotic dimension introduced by Gromov in the sense that a countable group $G$ has finite asymptotic dimension if and only if the action of $G$ on its Stone-Cech compactification has finite asymptotic dynamic dimension (p. 158 of this wonderful book). There seems to be two main use cases for finite dynamic asymptotic dimension- one is to determine the K-theory the associated crossed product C*-algebra and the other is to bound the nuclear dimension of said algebra (see for instance this article). I’m sure many more exist, but these are the ones I have seen.

Exercise) Prove that the dynamic asymptotic dimension of the system $(C(\partial \Gamma_S(G))\rtimes \G$ is not finite.

Spoiler) Here one can use either that the fact that we have stabilizer subgroups which are not locally finite (i.e. $\Z$ ), which is sufficient to prove the claim. Alternatively one can construct a counterexample explicitly as follows – for two elements $s,t\in S$ the sets

$V(st)\supset V(stst)\supset V(ststst)\supset...$

forms a local neighborhood basis at the point $v=(ststststst... ) \in \partial \Gamma_S(G)$ . Now use $E=\{st\}$ and show that each such local basis element contains an infinite $E$ -orbit. Since for any open cover one of these basis elements would have to be entirely contained in one of the covering sets, this would be enough to show the dynamic asymptotic dimension is not finite.

Gromov boundaries of free products

Dynamic asymptotic dimension

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