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 ${S}$ ?