Elementary totally disconnected locally compact groups

By Phillip R. Wesolek. 2014

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1aa67392dcc9ab23631e9963d8cdb78dd5af98f6

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In Algebra and Number Theory

1. P. Caprace

In Geometry and Topology

1. Colin D. Reid
2. George A. Willis
3. P. Caprace
4. Denis V. Osin
5. Daan Krammer
6. Anthony Genevois
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In Mathematical Physics

1. Colin D. Reid
2. George A. Willis
3. P. Caprace
4. Denis V. Osin
5. Anthony Genevois
6. K. Fujiwara
7. Michah Sageev
8. Mark F. Hagen

Citations in the corpus

• LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART II: COMPACTLY GENERATED SIMPLE GROUPS (2014, background): “heory of connected locally compact groups. (A complementary role, analogous to the role of soluble Lie groups in Lie theory, could be played by the class of elementary groups introduced by P. Wesolek [65].) We shall exclude discrete simple groups from consideration, since their known behaviour precl”