Finiteness properties and Relatively Hyperbolic Groups

TL;DR

This paper establishes that certain finiteness properties in relatively hyperbolic groups depend on their peripheral subgroups, and demonstrates the existence of infinitely many quasi-isometry classes with specific finiteness properties.

Contribution

It proves that properties $F_n$ and $FP_n$ for relatively hyperbolic groups are equivalent to these properties holding for all peripheral subgroups, and constructs infinitely many groups with distinct finiteness properties.

Findings

Finiteness properties $F_n$ and $FP_n$ depend on peripheral subgroups.

Existence of countably many quasi-isometry classes with specific finiteness properties.

Characterization of these properties in relatively hyperbolic groups.

Abstract

We show that properties $F_n$ and $FP_n$ hold for a relatively hyperbolic group if and only if they hold for all the peripheral subgroups. As an application we show that there are at least countably many distinct quasi-isometry classes of one-ended groups that are type $F_n$ but not $F_{n+1}$ and similarly of type $FP_n$ and not $FP_{n+1}$ for all positive integers $n$.