Norm rigidity for arithmetic and profinite groups

TL;DR

This paper investigates the rigidity of conjugation-invariant norms on certain arithmetic groups, establishing a dichotomy that connects to major results in the theory of arithmetic and profinite groups, with implications for group approximation properties.

Contribution

It introduces the dichotomy property for groups with bounded elementary generation, linking it to existing rigidity theorems and exploring consequences for approximations of non residually finite extensions.

Findings

Conjugation-invariant norms are either discrete or precompact under certain conditions.

The dichotomy property relates to key rigidity results like Margulis' normal subgroup theorem.

Constraints are derived for approximations of specific non residually finite arithmetic group extensions.

Abstract

Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.