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Approximating simple locally compact groups by their dense subgroups

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NPCW04 - Approximation, deformation, quasification

Co-authors: Pierre-Emmanuel Caprace (Université catholique de Louvain), Colin Reid (University of Newcastle, Australia )  The collection of topologically simple totally disconnected locally compact (t.d.l.c.) groups which are compactly generated and non-discrete, denoted by \ , forms a rich and compelling class of locally compact groups. Members of this class include the simple algebraic groups over non-archimedean local fields, the tree almost automorphism groups, and groups acting on \\\ \\\ cube complexes.

 In this talk, we study the non-discrete t.d.l.c. groups \ which admit a continuous embedding with dense image into some group \\\  ; that is, we study the non-discrete t.d.l.c. groups which approximate  groups \\\  . We consider a class \ which contains all such t.d.l.c. groups and show \ enjoys many of the same properties previously established for \ . Using these more general results, new restrictions on the members of \ are obtained. For any \\\ , we prove that any infinite Sylow pro-\ subgroup of a compact open subgroup of \ is not solvable. We prove further that there is a finite set of primes \ such that every compact subgroup of \\\ \\\ is virtually pro-\ .

This talk is part of the Isaac Newton Institute Seminar Series series.

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