Elementary totally disconnected locally compact groups

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.

Accepted version. To appear in The Proceedings of the London Mathematical Society

References in corpus (3)

• Decomposing locally compact groups into simple pieces
• Abstract commensurators of profinite groups
• Elementary amenable subgroups of R. Thompson's group F

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• Indicability, residual finiteness, and simple subquotients of groups acting on trees
• Totally disconnected locally compact groups with a linear open subgroup
• C*-simplicity has no local obstruction
• Approximating simple locally compact groups by their dense locally compact subgroups