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Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is i …

I think this is a great question, as there is still a need for an authoritative reference about (word-)hyperbolic groups. Since the textbook doesn't exist, I'd like to take the question in a slightly …

Bridson and Vogtmann proved a much stronger result. From the abstract: 'If $m$ is less than $n$ then [the image of] a homomorphism $\mathrm{Aut}(F_n)\to\mathrm{Aut}(F_m)$ can have cardinality at most …

The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then …

An example of a free-by-cyclic group that is not CAT(0) was given by Gersten. It is constructed from the automorphism of $F_3\cong\langle a,b,c\rangle$ that sends $a\mapsto a,~b\mapsto ba,~ c\mapsto …

For some reason, the question seems to be asking for an algebraic (number- or ring-theoretical) justification for residual finiteness (and implicitly LERF, though in fact the correct statement is that …

Thompson's group F is an example. It's finitely presented and, according to this paper of Ken Brown, the integral homology is free abelian of rank 2 in every positive dimension.

Many examples can be exhibited using a theorem of Gersten: Theorem (Gersten): Let $G$ be a hyperbolic group of cohomological dimension 2. Every finitely presented subgroup $H$ of $G$ is hyperbolic. Th …

Finitely presented $\mathbb{Z}$-linear groups with unsolvable conjugacy problem are known to exist, although writing them down explicitly will be extremely painful! I'm not sure (or have perhaps forgo …

The following is an elaboration of the last paragraph of Misha's answer. For me, the thing that makes non-linear (discrete) groups interesting is that we are not very good at constructing them! Line …

There are plenty of other possibilities. Here are a few examples: The boundary of the fundamental group of an acylindrical hyperbolic 3-manifold with totally geodesic boundary is homeomorphic to a S …

At the risk of being subjective (and possibly even argumentative), I feel like I should offer an answer to the implicit question, answered piquantly by Derek Holt: 'a large proportion of research …

It is an open problem to find a coherent hyperbolic group with a finitely generated, non-hyperbolic subgroup. See Wise's survey article for the state of the art on coherent groups. Wise, Daniel T. (3- …

$\DeclareMathOperator\BS{BS}$Since this question goes in several directions, I hope a discursive answer is appropriate. The question fits into an important family of questions in geometric group theor …

This might help. Lemma If $A$ does not split freely and $C$ is a non-trivial subgroup of $A$ then the HNN extension $G=A*_C$ does not split freely. The proof uses Bass--Serre theory---see Serre's bo …