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 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 stabilisers.