Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

Abstract

We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma <\mathrm{PU}(m,1)$ of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to $\mathbb{Z}$ with kernel of type ${\mathcal{F}}_{m-1}$ but not of type ${\mathcal{F}}_m$. 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 Kähler manifolds.

22 pages

References in corpus (4)

• Residually finite rationally solvable groups and virtual fibring
• Kähler groups and subdirect products of surface groups
• Virtual algebraic fibrations of Kähler groups
• Irrational pencils and Betti numbers

Cited by in corpus (2)

• Holomorphic functions on geometrically finite quotients of the ball
• Groups with exotic finiteness properties from complex Morse theory