Phillip Wesolek

Totally disconnected locally compact groups locally of finite rank

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-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 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. There are a number of theorems that point to a deep relationship between the compact open subgroups and the global structure of a totally disconnected locally compact (t.d.l.c.) group; cf. [1], [5], [8], [9], [22]. These global structural consequences of local properties are, following M. Burger and S. Mozes [5], often called local-to-global structure theorems. In the work at hand, we contribute to the body of local-to-global structure theorems by proving results for t.d.l.c. groups that have a compact open subgroup of finite rank where Definition 1·1. A profinite group has rank 0 < r 6 ∞ if every closed subgroup contains a dense r-generated subgroup. When a profinite group has rank r < ∞, we say it has finite rank. These t.d.l.c. groups are of wide interest as locally compact p-adic Lie groups have a compact open subgroup of finite rank. Furthermore, our results supply a compelling application of the theory of elementary groups developed in [21]. Remark 1·2. We study t.d.l.c. groups that are also second countable (s.c.). The second countability assumption is quite mild. Indeed, most natural examples of t.d.l.c. groups are second countable. On the other hand, t.d.l.c. groups may always be written as a directed union of open compactly generated subgroups, and these subgroups are second countable modulo a compact normal subgroup [12, (8.7)]. Therefore, in a sense, the study of t.d.l.c. groups reduces to the study of second countable groups. Of course, one may be temped to say the study of t.d.l.c. groups reduces to the study of compactly generated