Elementary totally disconnected locally compact groups

TL;DR

This paper defines and characterizes elementary totally disconnected locally compact groups, demonstrating their structural properties, permanence under various operations, and applications to understanding the broader class of such groups.

Contribution

It introduces the class of elementary t.d.l.c.s.c. groups, provides their characterization via descriptive set theory, and explores their structural and local-to-global properties.

Findings

Elementary groups are closed under extensions, subgroups, quotients, and inverse limits.

A compactly generated t.d.l.c.s.c. group decomposes into elementary and simple groups.

Locally solvable and [A]-regular t.d.l.c.s.c. groups are elementary.

Abstract

We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.d.l.c.s.c. groups are elementary and [A]-regular t.d.l.c.s.c. groups are elementary.