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The easiest trick, if it happens to work, is to construct a surjection onto another group that you already know is not finitely generated. A canonical quotient to try is the abelianization.

Good questions! To bump the discussion of torsion out of the comments: the group of units of $\mathbb{F}_2[P]$ is torsion-free. Suppose we have $q$-torsion and factor $0 = x^q - 1 = (x - 1)(x^{q-1} + …

No, there is no such dichotomy. If $G$ is an infinite group with Property (T) and $H$ is any non-trivial group, then $G*H$ has neither of the two properties. This is because groups with Property (T) h …

A locally hyperbolic group is in particular coherent (i.e. locally finitely presented), which is already a special property. To add to the examples already given by Sam Nead: ascending HNN extensions …

To answer the vaguer question: I think there is no known bound on the Dehn functions of finitely presented simple groups. Recall: Boone–Higman Embedding Theorem. A finitely presented group has solvabl …

There is no computable upper bound on the hyperbolicity constant of hyperbolic groups in terms of maximal relator length. Hyperbolicity is not a decidable property of finitely presented groups by the …

As another example, the problem of computing the order of an element of the finitely presented simple Brin–Thompson group $2V$ is undecidable by Belk, James; Bleak, Collin, Some undecidability results …

Regarding the question of spaces $X$ for which $QI(X)$ is known, a good keyword is quasi-isometric rigidity. One reference would be the survey in Chapter 25 of the Druţu–Kapovich book "Geometric group …

Yes. More generally, for any field $K$ we have an embedding of $\operatorname{Aff}(K^n)$ in $\operatorname{GL}_{n+1}(K)$, and so if $K$ has characteristic zero we can apply Selberg's lemma to conclude …

Property (T) implies Property FA: every action on a tree has a global fixpoint. The Magnus–Moldavanskii hierarchy expresses every one-relator group as (a subgroup of) an HNN extension of a "simpler" o …

A recent preprint of Friedl, Park, Petri, Raimbault, and Ray classifies the compact 3-manifolds with empty or toroidal boundary that have the topological analogue of this property. The authors do not …

Like Misha and Henry, I'll also add to the list of constraints on IF-groups rather than giving any examples. In The Surface Group Conjectures for groups with two generators we prove: Theorem. Let $G$ …