# Approximating simple locally compact groups by their dense subgroups

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.