A Survey of Elementary Totally Disconnected Locally Compact Groups
The class of elementary totally disconnected locally compact (t.d.l.c.) groups is the smallest class of t.d.l.c. second countable (s.c.) groups which contains the second countable profinite groups and the countable discrete groups and is closed under taking closed subgroups, Hausdorff quotients, group extensions, and countable directed unions of open subgroups. This class appears to be fundamental to the study of t.d.l.c. groups. In these notes, we give a complete account of the basic properties of the class of elementary groups. The approach taken here is more streamlined than previous works, and new examples are sketched.
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This survey originated from a mini-course given at the Mathematical Research Institute MATRIX. The author thanks the institute for its hospitality. He also thanks Colin Reid and Simon M. Smith for their many suggestions for improvements to these notes and for reading an initial draft. The author is supported by ERC grant #278469.
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