Non-coherence of arithmetic hyperbolic lattices

TL;DR

This paper demonstrates that, assuming the virtual fibration conjecture, most arithmetic hyperbolic lattices in higher dimensions are non-coherent, and it also establishes noncoherence results for certain lattices in complex hyperbolic spaces.

Contribution

It proves non-coherence of arithmetic lattices in O(n,1) for n>4 (except 7) and in SU(n,1), extending understanding of their algebraic properties.

Findings

Arithmetic lattices in O(n,1), n>4, are non-coherent under the conjecture.

Uniform arithmetic lattices of the simplest type in SU(n,1) are non-coherent.

Certain lattices in SU(2,1) with infinite abelianization are non-coherent.

Abstract

We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in SU(2,1) which have infinite abelianization.