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Yes. This is essentially an immediate consequence of the Folner sets definition of amenability. You can find references at Theorem 10.23 of the article by Ghys and de la Harpe, "Infinite groups as geo …

In the context of that proof, $p$ is a base point for $X$, $q$ is a base point for $Y$, and the function $f : X \to Y$ is defined on each $x \in X$ by first choosing $g \in G$ so that $d(x,gp) \le R$ …

The theorem of Papasoglu and Whyte, which says the following. Given a finitely presented group $G$ and any graph of groups presentation as in the theorem you quote, let $V(G)$ denote the set of quasi- …

Let $G$ be a finitely generated subgroup of a product of two finite rank free groups $F_m \times F_n$. If there is a Lipschitz retraction $F_m \times F_n \to G$ with respect to word metrics, then $G$ …

The gap is easily fixable in the context of the paper. Let me explain the fix after first explaining the critique of Kapovich and Lustig. The first paragraph of the proof starts by choosing a leaf $ …

Actually, for free abelian groups something does in fact happen along the lines of what you ask. It is just that the semidirect product idea is a little bit of a red herring. In the $Z^n$ case, what …

Here's an idea for a proof in hyperbolic groups which generalizes to relatively hyperbolic groups. Actually, this will prove something slightly weaker but which easily strengthens to what you want: na …

There is also a simple geometric proof using Bass-Serre theory, which does not require much in the way of calculation. Let $G$ act on the Bass-Serre tree $T$ of the amalgamated free product $A*_CB$. T …

The cusp in $G$ can be described as the conjugacy class of the infinite cyclic group generated by $b a^{-1}$. The intersection of this cyclic group with $H$ is described in two cases. When $n$ is odd …

Let me add another class of examples: virtual hyperbolic surface groups, by which I mean groups that have some $\pi_1(S_g)$ as a finite index subgroup, where $S_g$ is the closed, oriented surface of g …

Here is a counterexample, coming from the theory of hyperbolic 2-orbifold groups. Consider the fundamental group $\pi_1(S_2)$ of a closed, oriented surface of genus $2$ (its minimum number of generato …

While there's no single answer to this question, there are a lot of partial results. To start with, there is the still unresolved Kropholler-Roller conjecture, although various special cases are know …

The answer is yes. For the proof, the ray $R$ contains three distinct points in order, $y_0,y_1,y_2$, such that $y_1 = g_1 y_0$ and $y_2 = g_2 y_1$ for some elements $g_1,g_2 \in G$. If either of $g_1 …

There is a proof attributed to Peter Scott in Lemma 6.0.6 of "The Tits alternative for Out(F_n) I: Dynamics of exponentially growing automorphisms", MR1765705. The proof uses the Kurosh subgroup theor …

I don't have a general answer, but I have a specific and interesting special case. Let $\Phi : F_n \to F_n$ be an automorphism of the finite rank free group $F_n$ which is atoroidal meaning that no …