Local Quasiconvexity of Groups acting on Small Cancellation Complexes

TL;DR

This paper establishes a criterion for quasiconvexity of subgroups in groups acting on small cancellation complexes, generalizing the perimeter method and applying it to show coherence of certain one-relator products.

Contribution

It introduces a new criterion for subgroup quasiconvexity in small cancellation complexes, extending the perimeter method and applying it to coherence of high-powered one-relator products.

Findings

All finitely generated subgroups have quasiconvex orbits under the given action.

High-powered one-relator products are coherent if factors are coherent.

The perimeter method is generalized to broader classes of complexes.

Abstract

Given a group acting cellularly and cocompactly on a simply-connected 2-complex, we provide a criterion establishing that all finitely generated subgroups have quasiconvex orbits. This work generalizes the "perimeter method". As an application, we show that high-powered one-relator products $A \ast B / \nclose{r^n}$ are coherent if $A$ and $B$ are coherent.