A hyperbolic group with a finitely presented subgroup that is not of type $FP_3$

TL;DR

This paper presents a new construction demonstrating that certain hyperbolic groups can have finitely presented subgroups that are not of type $FP_3$, using Bestvina-Brady Morse theory without branched coverings.

Contribution

It provides an alternative proof of Brady's theorem with a novel construction leveraging Morse theory, avoiding branched coverings.

Findings

Existence of hyperbolic groups with finitely presented subgroups not of type $FP_3$

New construction method using Bestvina-Brady Morse theory

Avoids branched coverings in the proof

Abstract

Brady proved that there are hyperbolic groups with finitely presented subgroups that are not of type $FP_3$ (and hence not hyperbolic). We reprove Brady's theorem by presenting a new construction. Our construction uses Bestvina-Brady Morse theory, but does not involve branched coverings.