Torsion subgroups of small cancellation groups

TL;DR

This paper demonstrates that torsion subgroups in certain small cancellation groups are finite cyclic, and explores fixed points, automatic continuity, and the Tits Alternative in these geometric group theory contexts.

Contribution

It establishes finiteness and cyclicity of torsion subgroups in specific small cancellation groups and links geometric properties to algebraic group actions.

Findings

Torsion subgroups are finite cyclic in C(6), C(4)-T(4), C(3)-T(6) small cancellation groups.

Fixed point existence for locally elliptic actions on small cancellation complexes.

The Tits Alternative holds for groups acting on simply connected C(3)-T(6) complexes with bounded cell stabilizers.

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.