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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 $A\subset \mathbb{N}$ …

Yes, The Grigorchuk group embeds into a 2 generated amenable group which is also finitely presented. For a reference you can look at the paper of Grigorchuk titled "Solved and unsolved problems around …

In this paper, Anna Erschler constructs uncountably many non residually finite central extensions of the first Grigorchuk Group.

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 …

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?

Assume that we have two residually finite groups $G$ and $H$. Which properties of $G$ and $H$ could be used to show that their pro-finite (or pro-p) completions are different? I asked a while ago in …

It is known that all amenable groups do not contain free subgroups (of rank>1). But there are amenable groups containing free semigroups. Which amenable groups cannot contain free semigroups?

I asked the following question in math.stackexchange and did not get any answer, so I am asking it here: Let $G$ be a group and let $G_n$ be its series of dimension subgroups defined as follows: $$ …

Let G be a finitely generated group. Suppose we have two families F1 and F2 of finite index subgroups of G, and each family has trivial intersection and is filtered from below (i.e. for any two elemen …

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 …

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

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 …

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 …

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

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 …