Finitely presented kernels of homomorphisms from hyperbolic groups onto free abelian groups

TL;DR

This paper constructs specific non-hyperbolic, finitely presented subgroups within hyperbolic groups as kernels of surjective homomorphisms onto free abelian groups, highlighting new examples of such structures with particular finiteness properties.

Contribution

It provides explicit examples of non-hyperbolic, finitely presented kernels of homomorphisms from hyperbolic groups onto free abelian groups, with specific finiteness types.

Findings

Existence of non-hyperbolic finitely presented subgroups as kernels

Construction of examples with finiteness type F_2 but not F_3

Subgroups are kernels of surjective homomorphisms onto Z^m

Abstract

For every $m\geq 2$ we produce an example of a non-hyperbolic finitely presented subgroup $H < G$ of a hyperbolic group $G$, which is the kernel of a surjective homomorphism $\phi: G\to \mathbb{Z}^m$. The examples we produce are of finiteness type $F_2$ and not $F_3$.