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Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and rela …

In the 1970s Ol'shanskii constructed a non-cyclic finitely generated group $G$ with the following properties: Every proper, non-trivial subgroup of $G$ is infinite cyclic. If $X^m=Y^n$ for $X, Y\in G …

The group defined by $\langle x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx\rangle$ has order 27, exponent 3 and is non-abelian. (Checking exponent 3 basically comes down to ensuring that $( …

A two-generator, one-relator group with torsion is a group with presentation of the form $\langle a, b\mid R^n\rangle$, $R\in F(a, b)$ and $n>1$. Their isomorphism problem is decidable in quadratic ti …

The group $$G=\langle a, b, x, y\mid [a, b]^2=[x, y]^2\rangle$$ is a torsion-free group which is not free by cyclic. However, $G$ is free-by-$D_{\infty}$ and so virtually free-by-cyclic (containing an …

Let $F$ be the free group on (say) two generators, $a$ and $b$. Let $A$ and $B$ be (freely reduced) elements of $F$. Let $W(X, Y)$ denote a word on the words $X, Y$. -Is it ever true that the equatio …

Let $G=\langle H, t; K^t=K^{\prime}\rangle$ be an HNN-extension of $H$, with $t$ inducing the isomorphism $\phi: K\rightarrow K^{\prime}$. I was wondering if the following question can be answered, an …

I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite. However, …

Let $X=(x_1, \ldots, x_n)$ be an $n$-tuple of elements of a given group $G$. Then two $n$-tuples $X$ and $Y$ are Nielsen equivalent if there exists an automorphism of the free group on $n$-generators, …

Let $Q$ be a recursively presented group. Is it possible to embed $Q$ into a finitely presented group $G$ such that the image of $Q$ is malnormal in $G$? Note that a subgroup $H$ of $G$ is malnormal …

Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such …

A triangle group has a presentation of the form, $G=\langle a, b; a^{\alpha}, b^{\beta}, c^{\gamma}, abc\rangle, \alpha, \beta, \gamma \geq 2$ (I believe that these are also called von Dyke groups, …

There is a classic result of Baumslag which states, Thm: If $G$ is residually finite then so is $\operatorname{Aut}(G)$. While Grossman proved the (essentially) analogous result for $\operatorname{O …

By a mapping tori of $G$, I mean a semidirect product $G\rtimes\mathbb{Z}$, and by a trivial mapping tori I mean one isomorphic to $G\times\mathbb{Z}$. If $G$ is finitely generated but not finitely pr …

Is it possible for a finitely generated, recursively presented group to have a non-recursively presented outer automorphism group? Or is the following true, $G$ finitely generated, recursively pre …