Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

TL;DR

The paper demonstrates that certain subgroups of complex hyperbolic lattices have specific finiteness properties and constructs many non-hyperbolic finitely presented subgroups, addressing longstanding questions in geometric group theory.

Contribution

It introduces new finiteness properties of subgroups in complex hyperbolic lattices and constructs non-hyperbolic finitely presented subgroups, answering an open question of Brady.

Findings

Existence of subgroups with specific finiteness properties

Construction of finitely presented non-hyperbolic subgroups

Proof of a special case of Singer's conjecture for aspherical Kähler manifolds

Abstract

We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < {\rm PU}(m,1)$ of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to $\mathbb{Z}$ with kernel of type $\mathscr{F}_{m-1}$ but not of type $\mathscr{F}_{m}$. This provides many finitely presented non-hyperbolic subgroups of hyperbolic groups and answers an old question of Brady. Our method also yields a proof of a special case of Singer's conjecture for aspherical K\"ahler manifolds.