Norm rigidity for arithmetic and profinite groups

Let A $A$ be a commutative ring, and assume that every non-trivial ideal of has finite index. We show if SL n ( ) ${\rm {SL}}_n(A)$ bounded elementary generation then conjugation-invariant norm on it is either discrete or precompact. If G $G$ any group satisfying this dichotomy, we say the dichotomy property. relate property, as well some natural variants it, to other rigidity results in theory arithmetic profinite groups such celebrated normal subgroup theorem Margulis seminal work Nikolov Segal. As consequence derive constraints possible approximations certain non-residually central extensions groups, which hope might have further applications study sofic groups. In last section provide several open problems for research.