Splittings and C-complexes

TL;DR

This paper explores how the intersection patterns of limit sets in hyperbolic groups relate to group splittings, providing new insights and reproofs of existing results on group decompositions over malnormal subgroups.

Contribution

It establishes a connection between the disconnectedness of incidence graphs and group splittings, extending to subgroups of any group and rederiving known results.

Findings

Disconnected incidence graphs imply group splittings.

Reproves Kropholler's results on splittings over malnormal subgroups.

Provides a new coding of intersection patterns in hyperbolic groups.

Abstract

The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of this graph for codimension one subgroups leads to splittings. We also reprove some results of Peter Kropholler on splittings of groups over malnormal subgroups and variants of them.