Subgroups of word hyperbolic groups in dimension 2 over arbitrary rings

TL;DR

This paper extends Gersten's 1996 result by using isoperimetric inequalities over arbitrary rings to characterize subgroups of hyperbolic groups, establishing new criteria for higher rank hyperbolicity.

Contribution

It introduces a ring-independent approach to hyperbolicity via isoperimetric inequalities and provides elementary definitions for higher rank hyperbolic groups.

Findings

Linearity of the discrete isoperimetric function characterizes hyperbolicity.

Conditions are established for subgroups of higher rank hyperbolic groups to retain hyperbolicity.

Equivalence between isoperimetric and coning inequalities is demonstrated in simplicial complexes.

Abstract

In 1996, Gersten proved that finitely presented subgroups of a word hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use isoperimetric inequalities over arbitrary rings to extend this result to any ring. In particular, we study the discrete isoperimetric function and show that its linearity is equivalent to hyperbolicity, which is also equivalent to it being subquadratic. We further use these ideas to obtain conditions for subgroups of higher rank hyperbolic groups to be again higher rank hyperbolic of the same rank. The appendix discusses the equivalence between isoperimetric inequalities and coning inequalities in the simplicial setting and the general setting, leading to combinatorial definitions of higher rank hyperbolicity in the setting of simplicial complexes and allowing us to give elementary definitions of higher rank hyperbolic groups.