TITLE: Torsion subgroups of hyperbolic groups are finite?
QUESTION [14 upvotes]: 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.)

REPLY [2 votes]: Alternatively, one can refer to Lemma 3.4 from Osin's article Acylindrically hyperbolic groups.
Proposition: If a group acts on a hyperbolic space uniformly properly$^*$ with unbounded orbits, then it contains a loxodromic element.
This implies directly that torsion subgroups of hyperbolic groups are finite. The result is attributed to Ivanov and Olshanskii, the proof given by Osin being extracted from Lemma 17 in Hyperbolic groups and their quotients of bounded exponents.
$^*$An action $G \curvearrowright (X,d)$ is uniformly proper if, for every $R \geq 0$, there exists $N \geq 0$ such that $\# \{g \in G \mid d(x,gx) \leq R \} \leq N$ for every $x \in X$.