Hyperbolic groups containing subgroups of type $\mathscr{F}_{3}$ not $\mathscr{F}_{4}$

TL;DR

This paper constructs examples of hyperbolic groups with subgroups of type  F_3  but not  F_4 , using Dehn filling on a non-uniform lattice in PO(8,1), expanding understanding of subgroup structures.

Contribution

It provides explicit examples of hyperbolic groups with specific subgroup finiteness properties, previously unknown in the literature.

Findings

Hyperbolic groups with subgroups of type  F_3  but not  F_4  are constructed.

Uses Dehn filling on a non-uniform lattice in PO(8,1) to obtain these groups.

Extends the class of known subgroup configurations in hyperbolic groups.

Abstract

We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_{3}$ but not of type $\mathscr{F}_{4}$. These groups are obtained by Dehn filling starting from a non-uniform lattice in ${\rm PO}(8,1)$ which was previously studied by Italiano, Martelli and Migliorini.