Totally disconnected locally compact groups locally of finite rank

TL;DR

This paper investigates the structure of totally disconnected locally compact second countable groups with finite rank, revealing their decomposition into simpler components and extending understanding of p-adic Lie groups.

Contribution

It provides new structural decomposition theorems for t.d.l.c.s.c. groups with finite rank, including those with pro- compact open subgroups, and applies these to p-adic Lie groups.

Findings

Groups are virtually extensions of simple groups by elementary groups.

Decomposition results for groups with finite rank and compact open pro-nilpotent subgroups.

Structural insights into topologically simple p-adic Lie groups.

Abstract

We 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-$\pi$ compact open subgroup for some finite set of primes $\pi$ 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 prove 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.