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

TL;DR

The paper shows that standard arithmetic lattices of $SO(1,n)$ are conjugacy separable by leveraging their embedding into virtually compact special groups, and uses this to deduce conjugacy separability for unit groups of integral group rings of certain finite groups. It develops a framework linking profinite properties, Wedderburn components, and exceptional components to deduce goodness, hereditary conjugacy separability, and determinant-by-finite-quotients results for unit groups $U(ZG)$. A detailed classification of the relevant simple components guides the proofs, and the authors relate these properties to the Congruence Subgroup Problem, providing bounds on the CSP kernel's cohomological dimension. The results contribute to understanding how finite quotients constrain algebraic structures such as group rings and their unit groups, with implications for rigidity and identifiability in arithmetic and group-ring settings.

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.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Proposition 3.1
• Definition 3.2
• Definition 3.3