Noncoherence of arithmetic hyperbolic lattices

Abstract

We prove that all arithmetic lattices in $O\left(n,1\right)$, $n\ge 4$, $n\ne 7$, are noncoherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in $\mathit{SU}\left(n,1\right)$, $n\ge 2$, and of uniform lattices in $\mathit{SU}\left(2,1\right)$ which have infinite abelianization.