The dichotomy property of ${\rm SL}_2(R)$-A short note

TL;DR

This paper extends the understanding of topological properties of certain algebraic groups, showing that ${ m SL}_2(R)$ with specific rings of algebraic integers also exhibits restricted topologies similar to higher rank Chevalley groups.

Contribution

It demonstrates that the dichotomy property applies to ${ m SL}_2(R)$ over rings of algebraic integers with infinitely many units, expanding previous results beyond higher rank groups.

Findings

Non-discrete, conjugation invariant norms lead to profinite norm-completions.

The dichotomy property applies to ${ m SL}_2(R)$ with rings of algebraic integers.

The argument extends the known topological restrictions to a broader class of groups.

Abstract

A recent paper by Polterovich, Shalom and Shem-Tov has shown that non-discrete, conjugation invariant norms on arithmetic Chevalley groups of higher rank give rise to very restricted topologies. Namely, such topologies always have profinite norm-completions. In this note, we sketch an argument showing that this also holds for ${\rm SL}_2(R)$ for $R$ a ring of algebraic integers with infinitely many units.