# Subgroups of torsion-free hyperbolic groups versus subgroups of hyperbolic groups

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 that $$G\hookrightarrow H$$). Define $$\mathcal{S}_{\mathrm{tf}}$$ to be the class of finitely presented torsion-free groups which occur as the subgroup of some torsion-free hyperbolic group. Clearly $$\mathcal{S}_{\mathrm{tf}}\subseteq \mathcal{S}$$.

My question is: Are $$\mathcal{S}_{\mathrm{tf}}$$ and $$\mathcal{S}$$ equal?

That is, when studying the class of finitely presented torsion-free subgroups of hyperbolic groups $$\mathcal{S}$$, is anything lost by restricting to torsion-free hyperbolic groups?

• Nice question! it also makes sense for finitely generated subgroups, and even arbitrary subgroups. I'm not even sure to be able to answer in the probably easier case of a torsion-free subgroup of a virtually torsion-free hyperbolic groups. At least if this case is doable, then no counterexample to your example is known, since it's unknown whether every hyperbolic f.g. group is virtually torsion-free. – YCor Nov 8 at 13:57
• @YCor Thanks. Yes, the other questions make sense and I would be interested to know their answer also. I only asked one question as I didn't want to make the question too convoluted (and this specific question cropped up in my work). – ADL Nov 8 at 14:05
• I insist but it would be nice to settle the virtually torsion-free case. Indeed, there exist a hyperbolic (indeed virtually free) group $G$ with a torsion-free (normal) subgroup $N$ not contained in any torsion-free finite index subgroup of $G$. Namely, choose a $k$-generated infinite simple group with an element of order 2 (there are many); surject $G=F_k\ast C_2$ onto it with $C_2$ mapped injectively, and let $N$ be the kernel. Then $N$ is torsion-free and the only finite index subgroup of $G$ containing $N$ is $G$ itself, which is not torsion-free. – YCor Nov 8 at 16:14
• The answer is yes for small cancelation groups (Gersten 96): every f.p. subgroup of a small cancelation group is itself hyperbolic. – Mark Sapir Nov 8 at 19:04