The dichotomy property of \rm SL₂(R)-A short note

2022/02/16 by Trost, Alexander Alois
#20-xx #51Fxx #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2202.08319

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 SL₂(R) for R a ring of algebraic integers with infinitely many units.

Related

• Norm rigidity for arithmetic and profinite groups
• Strong boundedness of \rm SL₂(R) for rings of S-algebraic integers with infinitely many units
• The second homology of SL₂ of S-integers
• Defining \mathbb Z using unit groups
• Branching rules for the restriction of regular representations of GL₂(\mathfrako/\mathfrakp^r) to SL₂(\mathfrako/\mathfrakp^r).