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Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

Claudio Llosa Isenrich, Pierre Py

Abstract

We prove that in a cocompact complex hyperbolic arithmetic lattice $Γ< {\rm PU}(m,1)$ 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 Kähler manifolds.

Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices

Abstract

We prove that in a cocompact complex hyperbolic arithmetic lattice of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to with kernel of type but not of type . 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.
Paper Structure (8 sections, 21 theorems, 40 equations)

This paper contains 8 sections, 21 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Lefschetz theory and finiteness properties
  3. The BNSR invariants
  4. A higher dimensional version of Delzant's theorem
  5. Simpson's theorem
  6. Arithmetic lattices of the simplest type and Albanese maps
  7. Albanese maps are virtually immersions
  8. Conclusion of the proofs

Key Result

Theorem 1

Let $m\ge 2$ and let $\Gamma < {\rm PU}(m,1)$ be a torsion-free cocompact arithmetic lattice of the simplest type. Then $\Gamma$ has a finite index subgroup $\Gamma_0$ with the following property. For every finite index subgroup $\Gamma_1 < \Gamma_0$ the $(m-1)$-th BNSR invariant $\Sigma^{m-1}(\Gamm

Theorems & Definitions (27)

  • Theorem 1
  • Definition 2
  • Corollary 3
  • Theorem 5
  • Theorem 6
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • Definition 11
  • Definition 12
  • ...and 17 more