# Torsion subgroups of hyperbolic groups are finite?

Is it true that torsion subgroups of hyperbolic groups are finite?

I have a vague memory that this is true, perhaps due to Ol'shanskii, but have been struggling to find a reference.

(By a torsion subgroup I mean a subgroup $T\leq H$ such that every $t\in T$ has finite order. Although there are finitely many conjugacy classes of finite order elements in $H$, this does not immediately imply that $T$ is finite.)

• See Theorem 1 in here: ams.org/journals/proc/2005-133-02/S0002-9939-04-07578-1/… – Loreno Heer Nov 16 '16 at 11:26
• @LorenoHeer, it's reassuring to see the statement in print, but since the cited paper does not provide a specific reference for the statement, it's not very useful. – HJRW Nov 16 '16 at 14:07
• Not only they are finite but they have bounded order and form finitely many conjugacy classes. It's due to Gromov (in the 1987's original article in English) (the various authors who wrote redigests of Gromov's article reproved it... in this precise case Gromov's proof is reasonably complete). – YCor Nov 16 '16 at 17:20

This is Corollaire 36, Chapitre 8 in the (French) book "Sur les Groupes Hyperboliques d’après Mikhael Gromov", Editors: Etienne Ghys, Pierre de la Harpe, Progress in Mathematics 83. Birkhäuser Boston (1990)

• Great, thanks! That said, I will hold off accepting your answer in case someone comes up with an English-language proof. (Of course this book is a standard reference, but I know no French.) – user101216 Nov 16 '16 at 15:33