Hyperbolic Groups with Finitely Presented Subgroups not of Type $F_3$

TL;DR

This paper constructs infinite families of hyperbolic groups with finitely presented subgroups not of type F3, demonstrating their diversity and limitations in higher dimensions, advancing understanding of hyperbolic group structures.

Contribution

It generalizes Brady and Lodha's constructions to produce new hyperbolic groups with specific subgroup properties, and analyzes their isomorphism classes and dimensional limitations.

Findings

Infinitely many non-isomorphic hyperbolic groups constructed.

Existence of finitely presented subgroups not of type F3 in these groups.

Higher-dimensional generalizations are not possible beyond dimension 3.

Abstract

We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than $3$.