Conjugation-invariant norms on SL2(R) for rings of S-algebraic integers with infinitely many units

Abstract

This paper studies conjugation-invariant norms on for R a ring of S-algebraic integers with infinitely many units in a number field. We will show that barring some differences in the proofs the properties of these norms strongly resemble related properties of conjugation-invariant norms on arithmetic Chevalley groups of higher rank. Specifically, we prove two properties: First, we show that the diameter of each word norm on defined by finitely many conjugacy classes has an upper bound linear in the number of conjugacy-classes. Second, we show that all non-discrete conjugation-invariant norms on have a profinite norm-completion. Both of these properties of conjugation-invariant norms are already known for higher rank arithmetic Chevalley groups.