Torsion subgroups of small cancellation groups

Abstract

We prove that torsion subgroups of groups defined by $C(6)$, $C(4)$–$T(4)$, or $C(3)$–$T(6)$ small cancellation presentations are finite cyclic groups. This follows from a more general result on the existence of fixed points for locally elliptic (every element fixes a point) actions of groups on simply connected small cancellation complexes. We present an application concerning automatic continuity. We observe that simply connected $C(3)$–$T(6)$ complexes may be equipped with a $\mathrm{CAT}(0)$ metric. This allows us to get stronger results on locally elliptic actions in that case. It also implies that the Tits alternative holds for groups acting on simply connected $C(3)$–$T(6)$ small cancellation complexes with a bound on the order of cell stabilizers.