Approximation by p-adic Lie Groups

Abstract

We discuss classes of topological groups which can be approximated by p-adic Lie groups, and varieties of Hausdorff groups generated by classes of \hboxp-adic Lie groups (for a single or multiple p). We give several characterizations of locally compact pro-p-adic Lie groups and locally compact pro-discrete groups, and prove a “pro-version” of Cartan's Theorem: whenever a locally compact group is a pro-p-adic Lie group and a pro-q-adic Lie group for distinct primes p and q, it is pro-discrete. If a locally compact group can be approximated by p-adic Lie groups for variable primes p, then it is a pro-p-adic Lie group for some prime p.