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Let $G_1$ be the wreath product of $\mathbb{Z}$ with $\mathbb{Z}$. There is a surjective homomorphism $G_1\rightarrow \mathbb{Z}$ whose kernel $G_2$ is isomorphic to $G_1$. This gives a sequence of …

Is there an algorithm to distinguish the elements of the set $S$? If so, here is a word problem algorithm. This doesn't seem to use any transitivity properties, just faithfulness. Start with the pos …

It is true that a group $G$ is $FP_2$ if and only if $G$ acts freely cellularly on a connected CW-complex with trivial first homology group. I see from the comments that you are worried about finite …

This does not answer Greg's question, but it is related. You can realize any $\mathbb{Z}G$-module you that like as $H_1$ of a based 2-complex, or as $H_2$ of a 3-complex if you insist that the comple …

Consider the following two transformations of $\mathbb{R}^3$: $a:(x,y,z)\mapsto (x+1,y,z)$ and $b:(x,y,z)\mapsto (-x,-y,z+1)$. The translation axis for $a^nb$ is the line $x=n/2$, $y=0$ and $a^nb$ t …

This question for a product of one Gromov hyperbolic space is a famous open problem (see for example Bestvina's problem list) and as far as I am aware the general case is also open.

If you add the relations that the standard generators have order two, then instead of getting the braid group on $n$ strands, you get the symmetric group on $n$ symbols. The generator that correspond …

This answers the other part of your question, not answered by Thompson's group. For each $i\geq 3$ there is a finitely presented group $G_i$ with the property that $H_i(G_i\mathbb{Q})$ is infinite di …

There has been a lot of work on cases of this conjecture connected to Coxeter groups. M. Davis and R. Charney made a conjecture that comes from these cases in 1995 in The Euler characteristic of a no …

The Bestvina-Brady construction of non-finitely presented groups of type FP produces groups of cohomological dimension two. Bestvina-Brady groups are parametrized by finite flag simplicial complexes. …