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Question 1: As mentioned in comments, presentations with the same number of generators and relations are called balanced. The triviality problem for balanced presentations appears to be a question of …

This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them. Recall that a subgroup $K$ of a g …

If you are willing to expand your final question slightly to include general graphs of groups then the answer to Question 2 is certainly “no”. In their famous 2008 GAFA paper Special cube complexes, H …

In this 2024 preprint, your question is attributed as a conjecture of Bridson. See Conjecture 1.2, and note that a subdirect product of limit groups is the same thing as a residually free group. There …

Finitely presented $\mathbb{Z}$-linear groups with unsolvable conjugacy problem are known to exist, although writing them down explicitly will be extremely painful! I'm not sure (or have perhaps forgo …

Higman’s embedding theorem can indeed be made to preserve the property of being torsion-free. See, for instance, this paper of Chiodo—Vyas, who prove the stronger fact that torsion length can be prese …

As mentioned in comments, if $\mathcal{H}$ is a subgraph of a graph of groups $\mathcal{G}$, with the natural induced structure, then the map $H=\pi_1(\mathcal{H})\to G=\pi_1(\mathcal{G})$ induced by …

As mentioned in comments, the answer is "yes" and there are many ways to see it. I would refer you to §2 of Peter Scott's survey paper "The geometries of 3-manifolds", in which he discusses 2-dimensio …

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}$If I have chased through the literature correctly, I think the answer to your question is "yes". Specifically: Dyer–Formanek–Grossman showed …

You need to prove the following folklore lemma, which is well known to researchers in the field but perhaps not written down anywhere. The proof is a nice exercise. Folklore lemma: Let $S$ be a finit …

I think the subgroup $H=\langle a,b^2,ba^2b^{-1}\rangle$ of $F=\langle a,b\rangle$ provides a counterexample. To see why, first note that $H$ has infinite index in $F$. Now consider the Stallings core …

In comments, the OP indicates that what they really want is a uniqueness result for reduced words in arbitrary graphs of groups. (Indeed, what the question actually asks for, that any word can be tran …

Many more examples, including ones where the free kernel is finitely generated, arise by looking at knot complements. Let $K$ be any non-trivial knot with Alexander polynomial $\Delta_K(t)=1$, (appare …

Let $F$ be a non-abelian free group and let $G=\prod_\omega F$ be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharla …

As suggested in the comments, what you are asking for is essentially the presentation complex of a finitely presented, infinite, simple group. Thus it suffices to exhibit a presentation for such a gro …