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Greendlinger's Lemma is about the geometry of disk diagrams. If we instead consider annular diagrams, the assumption that $\lambda<1/8$ gives an improved structural result. Annular diagrams encode the …

A group $G$ is locally indicable if every nontrivial finitely generated subgroup surjects onto the infinite cyclic group $\mathbb{Z}$. The following theorem was proven by Higman in 1940 [1]: Theorem. …

Bass-Serre theory is the classical theory of groups acting on trees. Trees are hyperbolic, so this theory fits nicely into the context of this question. Standard examples of groups acting on trees co- …

This partial answer just covers the case of $\psi$ non-surjective, i.e. not an automorphism. Just to set some notation: Define $A=\langle w_1, w_2\rangle=\operatorname{im}(\phi)\leq F_2$. We will show …

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 …

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 …

$\DeclareMathOperator\rank{rank}$Let $F$ be a non-cyclic free group. For which finitely generated subgroups $H< F$ such that $H$ is not of finite index in a free factor of $F$ does there exist a two- …

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 …

That there exist finitely presentable non-Hopfian groups. [I still remember my shock when I was first learned this result!] A group G is Hopfian if every surjective homomorphism $\phi:G\to G$ is in fa …

Here is a partial answer: The Kervaire-Laudenbach Conjecture states that, for any group $A$, $(A\ast\mathbb{Z})/\operatorname{ncl}(w)$ is non-trivial. This was proven by Klyachko for torsion free grou …

$\DeclareMathOperator\BS{BS}$The linearity of the Baumslag-Solitar groups $\BS(m, n)=\langle a, t\mid t^{-1}a^mt=a^n\rangle$ is completely understood, and it may be phrased as: $\BS(m, n)$ is linear i …

Let $G$ be a finitely generated group, and let $F_1, F_2$ be two subgroups of $G$ which are free of finite rank at least 2. I am wondering what conditions can be placed on $G$ so that $F_1\cap F_2$ is …

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 …

I wanted to answer this question to give a solid journal citation (as requested in the comments to HJRW's answer), but also I feel there is some historical interest here. It seems that the result you …

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 …