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Well, every action of $F$ corresponds to a subgroup $H\leq F$ in the standard way. Specifically, the "standard" action on the interval corresponds to the stabilizers of various points in the interval …

Here is a construction for a group similar to the braided Thompson group $BV$ that ought to have this property. Define the $n$th ribbon group to be the semidirect product $$ R_n = \mathbb{Z}^n \rtime …

Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x_0$ be a basepoint for $X$. There seem to be two different definitions of "horofunction" for $X$, …

I don't know whether this was known before, but Collin Bleak, Francesco Matucci, and I have settled this question in the course of our work on our recent paper [1]. The answer is that any horofunctio …

Just to make the method as concrete as possible, I'll compute the growth rate for the fundamental group $G$ of a surface of genus two. The Cayley graph of $G$ is the 1-skeleton of a tiling of the hyp …

This is not exactly an answer to the question, but is instead essentially a comment that was way too long for the comment space. The OP mentioned that he doesn't have a good sense for the shapes of F …

The answer is already no for $\mathbb{Z}$, assuming the question is whether this holds for every meaure. Let $n\in\mathbb{N}$, and let $$ S \,=\, \{(a_1,\ldots,a_n)\in\mathbb{Z}^n \mid \gcd(a_1,\ldot …

First, it is easy to see that $\mathrm{Comm}(\mathbb{Z}^n)$ must be isomorphic to a subgroup of $\mathrm{GL}(n,\mathbb{Q})$. In particular, every finite-index subgroup of $\mathbb{Z}^n$ is an $n$-dim …

This is false. The archetypical family of hyperbolic Coxeter groups are the hyperbolic triangle groups $$ T(l,m,n) = \langle a,b,c \mid a^2=b^2=c^2=(ab)^l=(bc)^m=(ca)^n=1\rangle $$ where $l,m,n\geq 2 …

Given a geodesic metric space $X$ and a point $p\in X$, the injectivity radius $\mathrm{injrad}(p)$ is the maximum value of $r$ such that every point in the open ball $B(p,r)$ is connected to $p$ by a …

Is the following statement true? If so, can anyone provide a reference? Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected subcomplex of $X$. Then the following are equivalent: …