** Uniformly recurrent subgroups and rigidity of non-free minimal actions **

Joint work with A. Le Boudec and T. Tsankov.

**1. The Chabauty space of a group **

Let be a locally compact group. The Chabauty space of is the set of subgroups of , where two subgroups are nearby if their intersections with every compact set are. If is countable, the topology coincides with that induced from .

The action by conjugation on is usually interesting. Glasner and Weiss suggested to study *uniformly recurrent subgroups*, aka URS, i.e. minimal invariant subsets of this action.

**Examples**.

- Normal subgroups.
- Conjigacy classes of cocompact subgroups.
- Stabilizers of actions on compact spaces.

Indeed, the closure of the set of stabilizers of a -action on contains a unique minimal subset, called the stabilizer URS of the action, (Glasner-Weiss).

Theorem 1 (Matte Bon-Tsankov, Elek)Conversely, every URS of is the stabilizer URS of some action of on a compact space.

Our initial motivation was to study simplicity of countable groups, following this theorem.

Theorem 2 (Kalantar-Kennedy, Kennedy)If is countable, the followng are equivalent.

- is -simple.
- acts freely on its universal Furstenberg boundary.
- has no non-trivial URS consisting of amenable subgroups.

Our interest has shifted to examples.

**2. Examples: Thomson’s groups **

These are three groups acting respectvely on , on the circle and on the Cantor set . Each of them consists of all homeomorphisms acting locally like . and act minimally (but does not), with large stabilizers.

Theorem 3 (Le Boudec-Matte Bon)Let act on a compact space . Assume action is minimal and not topologically free. Then there is a continuous surjective equivariant map . Moreover, for all but countably many, the action of the stabilizer of on the fiber is trivial.

Topologically free means that the set of points with trivial stabilizer is a dense .

Free actions on the circle can be blown up: certain points are replaced with intervals. This gives mny examples.

Bounded cohomology method allow to show that certain group actions on the circle cannot be topologically free.

A similar statement holds fro , and for , it says that there are no such actions at all.

** 2.1. Proof **

- Classify all URS of . There are only three: trivial, and the stabilizer URS of the action on the circle.
- Construct a map to the circle in the third case.

The first step has a more general character. Let be a countable group acting faithfully on a compact Hausdorff space . We shall focus on subgroups fixing pointwise the complement of open sets , , called rigid stabilizers.

Proposition 4Let be a subgroup. Either a sequence of conjugates of converges to the trivial subgroup, or there exists an open subset and a subgroup stabilizing and whose action on coincides with that of a finite index subgroup of . Moreover, if the the action on is extremely proximal (every proper closed set can be shrinked to a point), then there exists a finite index subgroup of whose commutator subgroup is contained in .

**3. Non-discrete locally compact groups without URS nor IRS **

This is work in progress.

** 3.1. Invariant random subgroups **

An IRS is an invariant probability measure on Chabauty space. IRS are stabilizers of probability measure preserving actions.

Can a group have no IRS, or no URS, at all ? For instance, Neretin’s group is simple and has no lattices, this rules out the most obvious examples of IRS. Does it have others?

** 3.2. Answers in the discrete case **

Tarski monsters have only countably many subgroups, so no URS nor IRS.

Finitary alternating group has no URS but plenty of IRS (Vershik).

Thompson’s groups and have 1 URS and no IRS (Dudko-Medynets).

The commutator subgroup of Thompson’s group has no URS, no IRS.

** 3.3. The non-discrete case **

Compactly generated examples can be obtained, using dense subgroups in them obtained by slightly enlarging a known group. Neretin’s group is of this sort, starting from Thompson’s group .

We describe non compactly generated examples. Let be a countable group action faithfully on a set . Let be finite subsets of with pairwise dsjoint supports, and such that for every , the support of has only finitely many points in common with any support. Consider the closed subgroup generated by and the product of , in the topology where is open.

Such constructions arise in Willis, Akim-Glasner-Weiss, Caprace-Cornulier to produce various counterexamples.

We take , and for even, and trivial if s odd. Then the resulting group has no URS, but many IRS.

Let us take , the group of homeos locally modelled on . It is a non-discrete variant of Thompson’s Then the resulting group has no IRS, but many URS.