  Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has
finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary
generation then every conjugation-invariant norm on it is either discrete or
precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has
the \emph{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.
