Distortion of malnormal subgroup of hyperbolic groups

Let $G$ be a countable, Gromov-hyperbolic group.

We say that $H$ is hyperbolically embedded in $G$ if $G$ is relatively hyperbolic to {$H$} (in the strong sense). This definition is due to Osin.

A theorem of Bowditch says that infinite, finitely generated, almost-malnormal and quasi-convex subgroups of $G$ are hyperbolically embedded in $G$. Later Osin has proved that the conditions are necessary (even in the wider context of relatively hyperbolic groups).

Quasi-convex subgroups are not necessary malnormal but they have always finite height by a result of Gitik, Mitra, Rips and Sageev. The height of $H\subset G$ is defined to be the maximal $n$ such that there exist $g_1,\ldots,g_n\in G$ with $g_1Hg_1^{-1}\cap\ldots\cap g_nHg_n^{-1}$ infinite (but all the $g_iHg_i^{-1}$ different).

I would like to know how distorted a malnormal subgroup can be in $G$.

Is there some class of groups for which malnormal implies quasi-convex? Examples?

What about the relatively hyperbolic case?

• I believe this is still an open question (malnormal implies quasiconvex). One could attempt to prove it by induction for cubulated hyperbolic groups, but even that seems tricky. – Ian Agol Jun 18 '13 at 20:54
• Yes, existence of distorted malnormal subgroups in hyperbolic groups is a known open problem. There are classes of hyperbolic groups which are locally quasiconvex, but this does not use malnormality. – Misha Jun 18 '13 at 23:59
• By the way, you should assume that the malnormal subgroup is finitely generated (at least). I think in any hyperbolic group, one can find malnormal infinitely generated (free) subgroups. – Ian Agol Jun 19 '13 at 4:49
• Also, by the technique of combinatorial Dehn filling, the relatively hyperbolic case probably isn't too far from the word-hyperbolic case. – HJRW Jun 19 '13 at 8:19
• Oh, you might want to look at Question 1.8 of Besvina's problem list, which asks whether subgroups of finite width are quasiconvex. Malnormal means width 1. math.utah.edu/~bestvina/eprints/questions-updated.pdf – HJRW Jun 19 '13 at 8:32