Decomposing groups by codimension-1 subgroups

TL;DR

This paper proves a conjecture about splitting finitely generated groups over certain subgroups by introducing the concept of finite splitting height, extending classical theorems in geometric group theory.

Contribution

It introduces finite splitting height for subgroups and proves the Kropholler-Roller conjecture for subgroups with this property, extending existing decomposition theorems.

Findings

Kropholler-Roller conjecture holds for subgroups with finite splitting height

Finite splitting height generalizes finite-height property in group theory

Results extend Stallings' theorem and generalize Sageev's theorem

Abstract

The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include relatively quasiconvex subgroups of relatively hyperbolic groups with virtually polycyclic peripheral subgroups. In particular, our results extend Stallings' theorem and generalise a theorem of Sageev on decomposing a hyperbolic group by quasiconvex subgroups.