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

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 ${\mathrm{SL}}_2(R)$ for $R$ a ring of algebraic integers with infinitely many units.