Coherence and Negative Sectional Curvature in Complexes of Groups

TL;DR

This paper extends the understanding of group coherence by analyzing complexes of groups with negative sectional curvature, generalizing curvature notions, and employing L^2-Betti numbers, also establishing local quasiconvexity for CAT(0) spaces.

Contribution

It introduces a generalized curvature condition for complexes of groups, extends the combinatorial Gauss-Bonnet theorem, and links these to group coherence and quasiconvexity.

Findings

Groups acting on complexes with negative sectional curvature are coherent.

The generalized curvature condition applies to broader classes of complexes.

Local quasiconvexity holds for groups acting on CAT(0) complexes under certain conditions.

Abstract

We examine a condition on a simply connected 2-complex X ensuring that groups acting properly on X are coherent. This extends earlier work on 2-complexes with negative sectional curvature which covers the case that G acts freely. Our extension of these results involves a generalization of the notion of sectional curvature, an extension of the combinatorial Gauss-Bonnet theorem to complexes of groups, and surprisingly requires the use of L^2-Betti numbers. We also prove local quasiconvexity of G under the additional assumption that X is CAT(0) space.