Groups with exotic finiteness properties from complex Morse theory

TL;DR

This paper constructs new hyperbolic groups and subgroups of Kähler groups with unusual finiteness properties, advancing understanding of the complex relationships between group theory and geometric structures.

Contribution

It introduces hyperbolic groups with specific finiteness properties and constructs nonnormal subgroups of Kähler groups exhibiting exotic finiteness behaviors.

Findings

Existence of hyperbolic groups with kernels of type _k but not _{k+1}.

Construction of nonnormal subgroups of Kähler groups with unique finiteness properties.

Extension of previous results on finiteness properties in geometric group theory.

Abstract

Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of K\"ahler groups and their subgroups. In this work, we push this further and exhibit, for each integer $k$, new hyperbolic groups admiting surjective homomorphisms to $\mathbb{Z}$ and to $\mathbb{Z}^{2}$, whose kernel is of type $\mathscr{F}_{k}$ but not of type $\mathscr{F}_{k+1}$. By a fibre product construction, we also find examples of nonnormal subgroups of K\"ahler groups with exotic finiteness properties.