A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

TL;DR

This paper establishes a new combination theorem for hyperbolic groups acting on complexes with combinatorial non-positive curvature features, expanding understanding of hyperbolicity in complex group actions.

Contribution

It introduces a combinatorial property that ensures hyperbolicity for groups acting on certain complexes, including weakly systolic and small cancellation complexes.

Findings

Constructed a potential Gromov boundary for the groups

Analyzed boundary dynamics to characterize hyperbolicity

Applied results to small cancellation over graphs of hyperbolic groups

Abstract

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch's characterization of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes. As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.