This talk presents the preprint https://arxiv.org/abs/1709.06733.

Introduction

Let \( G \) be a locally compact, second countable group. We introduced the Chabauty space \( \mathrm{Sub}_G \) of closed subgroups of \( G \) and the space \( \mathrm{IRS}(G) \) of invariant random subgroups of \( G \) in the lectures on the Nevo–Stück–Zimmer theorem. A corresponding topological notion is given by the following objects introduced by Eli Glasner and Benjamin Weiss.

Definition: A uniformly reccurent subgroup (URS) of \( G \) is a minimal \( G \)-invariant subset in \( \mathrm{Sub}_G \).

Recall that a closed \( G \)-invariant subset of a topological \[ G \)-space is called minimal if it does not contain a proper closed invariant \( G \) subset. As in the case of IRSs there are immediate examples of URSs:

• The singletons \( \{ {\mathrm{Id}} \}, \{ G \} \) are always URSs;
• More generally, if \( N \) is a closed normal subgroup of \( G \) tnen \( \{ N \} \) is an URS of \( G \).

An URS is said to be nontrivial if it is not one of the two examples in the first item above.

An example which often differs from IRSs (in the sense that it is not the support of any IRS of \( G \)) is given by cocompact subgroups: if \( P \le G \) is cocompact then the orbit \( P^G = \{ gPg^{-1} :\: g \in G \} \) is closed and hence an URS. For an example which is not the support of an URS we can take \( G \) to be semisimple a Lie group (e.g. \( G = \mathrm{SL}_2(\mathbb R) \)) and \( P \) a proper parabolic subgroup (e.g. the upper triangular matrices).

The correspondence between URSs and continuous actions on compact space is looser than in the emasurable category (as stated in this result): it is given by the following result.

Proposition (Glasner–Weiss): If \( G \) acts minimally on a compact space \( X \) then the subset
\[
\overline{ \{ \mathrm{Stab}_G(x) :\: x \in X \} }
\]
contains a unique URS.

Similarly to the case of IRSs the following question is open:

Are there any URSs in the Neretin group \( N_p \) which do not come from the action on \( \partial T_p \)?

We also note that an example of a discrete group which admits (many) IRSs but no non-trivial URSs is the group of finitary premutations of a countable set. On the contrary the Thompson group admits non non-trivial IRSs (Dudko–Medynets) but has URSs coming from the action on the boundary.

The goal of this lecture is to explain the construction of a family of non-dicrete lcsc groups which have no nontrivial URSs. They turn out to have no non-trivial IRSs as well. They are not compactly generated, and no such examples are known.

These groups will occur as subgroups of the group of piecewise affine homeomorphisms of \( {\mathbb Q}_p \). Such maps are by definition invertible maps \( g : {\mathbb Q}_p \to {\mathbb Q}_p \) such that there exists a decomposition into disjoint clopen subsets
\[
{\mathbb Q}_p = X_1 \sqcap \cdots \sqcap X_r
\]
and \( a_1, b_1, \ldots, a_r, b_r \in {\mathbb Q}_p \) such that
\[
\forall x \in X_i :\: gx = a_i x + b_i
\]
(i.e. the restriction to \( X_i \) is affine). It is clear that they form a group, which will be denoted by \( \mathrm{PL}({\mathbb Q}_p) \); it is not closed in \( \mathrm{Homeo}({\mathbb Q}_p) \) but still inherits a group topology from it.

Theorem: Let \( F_n,\, n \ge 1 \) be a family of finite groups. There exists a locally compact subgroup \( G \le \mathrm{PL}({\mathbb Q}_p) \) which contains \( U \) as an open subgroup, and has non nontrivial URSs or IRSs

Biapproximations

In this section \( G \) is a lcsc group.

Definition: A biapproximation of \( G \) is a sequence \( U_n, G_n \) of subgroups satisfying the following properties:

1. For all \( n \), \( U_n \le G_n \);
2. For all \( n \), \( G_n \) is open, \( G_n \le G_n{n+1} \) and
   \[
   G = \bigcup_{n \ge 1} G_n;
   \]
3. The subset \( \bigcup_{n \ge 1} U_n \) is relatively compact in \( G \);
4. \( U_n \) converges to the trivial subgroup \( \{{\mathrm{Id}}\} \) as \( n \to +\infty \).

For example we can have \( U_n \supset U_{n+1} \) and \( bigcap_{n \ge 1} U_n = \{{\mathrm{Id}}\} \).

Definition: Let \( U \le G \) be a compact subgroup and let \( H \le G \). The \( U \)-saturation of \( H \) in \( G \) is the subgroup of elements preserving setwise all orbits of \( H \) in \( G/U \) :
\[
[H]_U^G = \{ g :\: \forall x \in G/U, gx \in Hx \}
\]

It follows immediately from the definition that
\[
[H]_U^G = \bigcap_{x \in G/U} HxUx^{-1}
\]
which implies that

• \( [H]_U^G \) is a closed subgroup of \( G \);
• If \( U \triangleleft G \) then \( [H]_G^U = HU \).

Definition: The subgroup \( H \) is said to be \( U \)-saturated if \( [H]_U^G = H \).

Note that in case \( U \triangleleft G \) saturated subgroups are exactly subgroups containg \( U \) by the above, so that there is a bijection between the \( U \)-saturated subgroups and \( \mathrm{Sub}_{G/U} \). In general we have the following fact.

Lemma: The map \( \mathrm{Sub}_G \to \mathrm{Sub}_G \) defined by \( H \mapsto [H]_U^G \) is semi-continuous, in particular Borel.

Groups with biapproximations

Let \( U_n, G_n \) be a biapproximation of a lcsc group \( G \). Then for all \( n \) it follows from the above lemma the self-map \( \lambda_n \) of \( \mathrm{Sub}_G \) defined by
\[
\lambda_n(H) = [H \cap G_n]_{U_n}^{G_n}
\]
is semi-continuous. Let \( \mathcal F(G) \) be the space of closed subsets of \( \mathrm{Sub}_G \), with the Hausdorff topology, and define the map \( \lambda_n^* \) by
\[
\bar\lambda_n(\mathcal H) = \overline{\lambda_n(\mathcal H)}.
\]
The crucial result is then the following.

Proposition:

• The maps \( \lambda_n^* \) converge uniformly to the identity on ths space \( \mathrm{Prob}(\mathrm{Sub}_G) \) of IRS of \( G \);
• If \( \mathcal H_n \to \mathcal H \) in \( \mathcal F(G) \) then \( \bar\lambda_n(\mathcal H_n) \to \mathcal H \) as well.

An immediate consequence for later use is that the following properties are equivalent for the group \( G \):

1. \( G \) has no non-trivial IRSs;
2. If \( \mu_n \) is an IRS of \( G_n \) which is almost surely \( U_n \)-saturated then every accumulation point of \( \mu_n \) in \( \mathrm{Prob}(\mathrm{Sub}_G) \) is a convex combination of \( \delta_{\mathrm{Id}}, \delta_G \).

Application

Let \( \Gamma_p \) be the subgroup of \( \mathrm{PL}({\mathbb Q}_p) \) consisting of elements which are compactly supported (i.e. equal to the identity outside of a compact subset) piecewise affine maps whose coefficients belong to \( p^{\mathbb Z} \) (for the multiplicative part) and \( {\mathbb Z}[1/p] \) (for the additive part).

For \( n \ge 1 \) le \( \mathcal X_n \) be the union of nontrivial cosets of \( p^{-n}{\mathbb Z}_p \) in \( p^{-n-1}{\mathbb Z}_p \):
\[
\mathcal X_n = p^{-n-1}{\mathbb Z}_p \setminus p^{-n}{\mathbb Z}_p.
\]
Let \( \mathcal F = (F_m)_{m \ge 1} \) be a family of finite subgroups such that any elements of \( F_m \) is supported on \( \mathcal X_m \), and define the following subgroup: \( G_{\mathcal F} \) is the subgroup consisting of all \( g \in \mathrm{PL}({\mathbb Q}_p) \) such that:

1. \( g(p^{-N}{\mathbb Z}_p) \mathrm{Sub}set p^{-N}{\mathbb Z}_p \) for some \( N \ge 1 \);
2. \( g|_{p^{-N}{\mathbb Z}_p} \in \Gamma_p|_{p^{-N}{\mathbb Z}_p} \);
3. \( g|_{{\mathbb Q}_p \setminus p^{-N}{\mathbb Z}_p} \in \prod_{m \ge N+1} F_m \).

This subgroup contains \( \Gamma_p \) and \( \prod_{m \ge 1} F_m \); it is not closed in the compact-open topology but we give it a locally compact group topology by decreting the commensurated subgroup \( \prod_n F_n \) to be open.

This group \( G_{\mathcal F} \) admits a biaproximation by the following subgroups:

• \( G_n \) is defined by the conditions 1, 2, 3 above but with \( N = n \) fixed;
• \( U_n \) is defined by 1, 3 (with \( N = n \)) and \( g|_{p^{-n}{\mathbb Z}_p} = {\mathrm{Id}} \) for all \( g \in U_n \).

As \( U_n \triangleleft G_n \), the application of the criterion in the previous section shows that \( G_{\mathcal F} \) has nontrivial IRSs if and only if the locally finite group
\[
\bigcup_{n \ge 1} G_n/U_n \cong \Gamma_p
\]
does. As \( \Gamma_p \) is isomorphic to a Thompson group and those were proven to not have nontrivial IRSs by Dudko–Medynets we conclude that \( G_{\mathcal F} \) does not as well.

A similar reasoning gives the result for URSs.