Totally disconnected locally compact groups locally of finite rank

Abstract

AbstractWe study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.