Notes of Phillip Wesolek’s Cambridge lecture 11-05-2017

Approximating simple locally compact groups by dense subgroups

Joint with Caprace and Reid.

1. Simple locally compact groups

Here, simple means topologically simple. Does there exist a locally compact compactly generated group which is topologically simple but not simple? Still unknown.

Examples.

  • Simple Lie groups.
  • Simple algebraic groups over non-archimedean fields.
  • Automorphism groups of trees.
  • Subgroups like Burger Mozes group {U^+(Alt(7))} where link subgroups are preescribed.
  • Tree almost automorphism groups: homeos of the boundary which coincide clopenwise with automorphisms.
  • Certain Kac-Moody groups.
  • Certain groups acting on {CAT(0)} cube complexes, buildings.

We shall focus on the class {S} of topologically simple totally disconnected locally compact compactly generated groups which are not discrete. There are continuously many non-isomorphic such groups (Smith 2014). Smith’s examples act on biregular trees.

Cartan 1936: properties have been first observed on examples and only later has one found general explanations.

2. Dense locally compact subgroups

2.1. {p}-localization

van Dantzig: Every totally discnnected locally compact groups {G} has a basis of compact open subgroups {U} which are profinite.

Profinite groups have a Sylow theory. If {U_p} is a pro-{p} Sylow subgroup of {U}. Its commensurator {G_p} is dense (Reid 2011). It is nondiscrete if {U_p} is infinite. If {U} has infinite index and {G} is second countable, {G_p\not=G}.

What does this look like for simple groups ?

2.2. Results

We call a dense locally compact subgroup a locally compact group which has a continuous homomorphism to {G} with dense image.

Discrete dense locally compact subgroups always exist and do not seem to tell much. However, non-discrete dense locally compact subgroups can be thought of as approximations of {G}. Nevertheless, they are very different from {G}. For instance, for {G\in S},

  • They need not be compactly generated.
  • They need not be simple.

Say that a property {P} is

  • local if holds for arbitraril small compact open subgroups.
  • regional if it holds for arbitrarily large compactly generated open subgroups
  • global if it holds for {G}.

Say a group is robustly monolithic if it is monolithic and the monolith (intersection of all normal subgroups) is regionally faithful (a weak form of being compactly generated). The class {R} of robustly monolithic groups contains {S}.

Theorem 1 1. {R} is stable under taking non-discrete dense locally compact subgroups.

2. Groups in {R} do not admit open solvable subgroups.

3. Every group in {R} is locally pro-{\pi} for a finite set of primes {\pi}.

4. {R} is a regional property.

Whereas {S} is a global property.

(2) implies that, for {G\in S}, Sylow subgroup {U_p} is not virtually solvable (apply (2) to {G_p}).

2.3. New examples

The 2-localization of {Aut^+(T_5)} belongs to {S}-by-compact.

The 2-localization of {U^+(Alt(d))}, {d\geq 7}, belongs to {S}-by-finite.

Both groups belong to Le Boudec’s family {G_k(F,F')} but with an illegal colouring {k}. More generally, we have criteria for {G(F,F')} to virtually belong to {S}.

3. Question

Lubotzky: consider the abstract commensurability group of the free pro-finite group. Does this belong to {S} ?

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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