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I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a …

Ian Agol's answer is complete and achieves more than is required. It is perhaps worth pointing out that if you just want an example that meets the Title Criteria you can take $G$ to be the free produc …

In answer to the question "Is there a flat manifold with trivial first homology?" I proposed choosing a finite perfect group $P$ and a surjection $\phi:F\to P$ where $F$ is a free group of finite rank …

Many years ago I had the idea to use non-standard analysis to prove that a group acting freely on an $\mathbb R$-tree must be linear. The heuristic went like this: A non-standard model $G^*$ of the g …

It's puzzled me for a long time why two arguments in group cohomology look connected but no immediate visible connection is available. First, it is a theorem that if a group $G$ is the union of a chai …

Here is an idea for making examples. Let $F$ be a free group and let $N$ be a normal subgroup of $F$. Then $F/[N,N]$ is torsion-free. To see this, suppose $w\in F$ has finite order modulo $[N,N]$, and …

A solvable group is, by definition, a group with a finite series of normal subgroups such that the successive factor groups are abelian. It is the content of the Fundamental Theorem of Galois Theory t …

There is an account of the general theory of group extensions with non-abelian kernel in Gruenberg's Springer Lecture Notes 143, Cohomological Topics in Group Theory: here he refers to his own paper ' …

If $N$ is a normal subgroup of $G$ then there is a coupling: that is, a representation of $G/N$ in $\operatorname{Out}(N)$. In that case, the extensions of $N$ by $G/N$ affording the same coupling are …

Does anyone know an example of an amenable group with all subgroups finitely generated that is not elementary amenable?