Arithmetic Trialitarian Hyperbolic Lattices Are Not Locally Extended Residually Finite

Abstract

A group is LERF (locally extended residually finite) if all its finitely generated subgroups are separable. We prove that the trialitarian arithmetic lattices in $\mathbf{PSO}_{7,1}(\mathbb{R})$ are not LERF. This result, together with previous work by the third author, implies that no arithmetic lattice in $\mathbf{PO}_{n,1}(\mathbb{R})$, $n>3$, is LERF.