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.
- Simple Lie groups.
- Simple algebraic groups over non-archimedean fields.
- Automorphism groups of trees.
- Subgroups like Burger Mozes group 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 cube complexes, buildings.
We shall focus on the class 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
van Dantzig: Every totally discnnected locally compact groups has a basis of compact open subgroups which are profinite.
Profinite groups have a Sylow theory. If is a pro- Sylow subgroup of . Its commensurator is dense (Reid 2011). It is nondiscrete if is infinite. If has infinite index and is second countable, .
What does this look like for simple groups ?
We call a dense locally compact subgroup a locally compact group which has a continuous homomorphism to 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 . Nevertheless, they are very different from . For instance, for ,
- They need not be compactly generated.
- They need not be simple.
Say that a property 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 .
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 of robustly monolithic groups contains .
Theorem 1 1. is stable under taking non-discrete dense locally compact subgroups.
2. Groups in do not admit open solvable subgroups.
3. Every group in is locally pro- for a finite set of primes .
4. is a regional property.
Whereas is a global property.
(2) implies that, for , Sylow subgroup is not virtually solvable (apply (2) to ).
2.3. New examples
The 2-localization of belongs to -by-compact.
The 2-localization of , , belongs to -by-finite.
Both groups belong to Le Boudec’s family but with an illegal colouring . More generally, we have criteria for to virtually belong to .
Lubotzky: consider the abstract commensurability group of the free pro-finite group. Does this belong to ?