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Let $G$ be a finitely presented group and $H$ a finitely generated normal subgroup. Is it always true that the Schur Multiplier $H_2(H,\mathbb{Z})$ is a direct product of finitely generated abelian gr …

Chou asked in this paper whether The Higman group $H$ has a maximal normal subgroup $N$ such that $H/N$ has no (non-abelian) free subgroups (or is amenable). Is it known now if such subgroups exist …

Is there a group with a presentation $\left< X \mid r_i, i \in \mathbb{N} \right>$ (where $X$ is finite) with $\left< X \mid r_i, i \in A \right>$ is amenable if and only if $A\subset \mathbb{N}$ …

Suppose that $G$ is a finitely presented group and $H$ is a finitely generated normal subgroup such that $G/H$ is infinite cyclic. Is it true that $H$ is finitely presented?

Let $G$ be the automorphism group of the binary rooted tree. The downward Löwenheim-Skolem theorem states that G has a countable elementary sub-structure. My question is whether such sub-structure i …

The Higman Embedding theorem says that any finitely generated and recursively presented group can be embedded in a finitely presented group. My question is if one can embed such a group as a normal s …

This maybe a very general question. If we have a group given by its presentation only, what kind of properties could be proven about it? I know examples about non-amenability of some Burnside groups …

A $k$-marked groups is a pair $(G,S)$ where $G$ is a group and $S$ is an ordered set of $k$ generators of $G$. Each such pair can be identified with a normal subgroup of the free group $F_k$ of rank $ …

Is there a function $f$ such that for any presentation $$G=\langle x_1,\ldots,x_n \mid r_1,\ldots,r_k\rangle\quad \text{with}\quad |r_i|\leq 3$$ $k\leq f(n)$ implies that $G$ has non-abelian free …

If you have a finitely generated linear group whose growth is sub-exponential, then by Tits alternative (mentioned above) the group has to be virtually solvable and by a theorem of Milnor the group h …

What are some examples of finitely generated (finitely presented) elementary amenable groups which are not virtually solvable?

The first answer is incomplete and moreover I suspect that it is incorrect! Here is an answer submitted to me via email by Andrew Brunner, who asked me to post his answer for him since he is not sig …

A theorem in Geoghean's book is the following (theorem 18.3.18): Let $G$ be a finitely presented group and let the rank of $G/G'$ (as a $\mathbb{Z}$-module) be at least 2. If $G$ has no non-abelian …

If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect product …

Can you give me an example of a finitely generated infinitely presented amenable group which is a quotient of a finitely presented amenable group?