Strict C(6) complexes

TL;DR

This paper introduces strict C(6) small-cancellation complexes, proving that groups acting properly and cocompactly on such complexes are relatively hyperbolic, and establishes a convex cocompact core theorem for their quasiconvex subgroups.

Contribution

It defines strict C(6) complexes, proves relative hyperbolicity of groups acting on them, and demonstrates a convex cocompact core theorem specific to strict C(6) groups.

Findings

Groups acting on strict C(6) complexes are relatively hyperbolic.

A convex cocompact core theorem holds for strict C(6) groups.

Counterexamples show the theorem fails without strict C(6) condition.

Abstract

We define strict C(n) small-cancellation complexes, intermediate to C(n) and C(n+1), and we prove groups acting properly cocompactly on a simply-connected strict C(6) complex are hyperbolic relative to a collection of maximal virtually free abelian subgroups of rank 2. We study geometric walls in a simply-connected strict C(6) complex, and we use them to prove a convex cocompact (cosparse) core theorem for (relatively) quasiconvex subgroups of strict C(6) groups. We provide an examples showing the convex cocompact core theorem is false without the strict C(6) assumption.