Complexity volumes of splittable groups

TL;DR

This paper introduces Euler characteristic type invariants for splittable groups using graph of groups decompositions, applicable to various classes of finitely presented hyperbolic and limit groups, providing new tools for understanding their structure.

Contribution

It defines new Euler characteristic type invariants for splittable groups, extending their applicability to a broad class of finitely presented hyperbolic and limit groups.

Findings

Invariants are non-zero for many interesting classes of groups.

Applicable to elementarily free groups and certain hyperbolic groups.

Provides new insights into the structure of splittable groups.

Abstract

Using graph of groups decompositions of finitely generated groups, we define Euler characteristic type invariants which are non-zero in many interesting classes of finitely presented, hyperbolic, limit and CSA groups, including elementarily free groups and one-ended torsion-free hyperbolic groups whose JSJ decomposition contains a maximal hanging Fuchsian vertex group.