Extending Properties to Relatively Hyperbolic Groups

TL;DR

This paper develops an axiomatic framework for extending metric properties from subgroups to relatively hyperbolic groups, demonstrating that certain decomposition complexities are extendable and analyzing their equivalences.

Contribution

It introduces an axiomatic approach to extend metric properties in relatively hyperbolic groups and proves the extendability of finite decomposition complexities.

Findings

Finite decomposition complexity is extendable to relatively hyperbolic groups.

Straight finite decomposition complexity is also extendable.

Two notions of straight finite decomposition complexity are shown to be equivalent.

Abstract

Consider a finitely generated group $G$ that is relatively hyperbolic with respect to a family of subgroups $H_1, ..., H_n$. We present an axiomatic approach to the problem of extending metric properties from the subgroups $H_i$ to the full group $G$. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.