# 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. Jun 18, 2013 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. Jun 18, 2013 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. Jun 19, 2013 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, 2013 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, 2013 at 8:32