Norm rigidity for arithmetic and profinite groups

Let (Formula presented.) be a commutative ring, and assume that every non-trivial ideal of (Formula presented.) has finite index. We show that if (Formula presented.) has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If (Formula presented.) is any group satisfying this dichotomy, we say that (Formula presented.) has the 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.