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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 $G=\langle H, t; K^t=K^{\prime}\rangle$ 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 $Q$ 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 …

If $\varphi$ is an automorphism of $G = \langle x_1, \ldots, x_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x_1, \ldots, x_n)$, $\overline{\varphi}$, with $$x_i\varphi=_G x_i\ove …

Let $X$ be a topological space, and say that $X$ satisfies the closed fixed point set property if every continuous self-map $f:X\to X$ has fixed point set $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ …