Aritmethic lattices of $\SO(1,n)$ and units of group rings

TL;DR

This paper proves conjugacy separability for arithmetic subgroups of SO(1,n) and applies this to unit groups of specific integer group rings, showing finite quotients determine the original group rings.

Contribution

It establishes conjugacy separability for certain arithmetic groups and demonstrates that finite quotients of unit groups determine the original group rings, a novel connection.

Findings

Arithmetic subgroups of SO(1,n) are conjugacy separable.

Finite quotients of unit groups determine the original group rings.

Application to units of integer group rings.

Abstract

We establish that standard arithmetic subgroups of a special orthogonal group ${\rm SO}(1,n)$ are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.