## Notes of Jean Lécureux’s lecture

Amenable invariant random subgroups

Joint with Bader, Duchesne, Glasner, Lazarovich.

1. IRS

Let ${G}$ be a locally compact second counable group. Let ${S(G)}$ be the topological space of closed subgroups of ${G}$: ${H_n}$ converges to ${H}$ if

• Every ${h\in H}$ is a limit of elements ${h_n\in H_n}$.
• Every converging subsequence of a sequence ${h_n\in H_n}$ converges to an element of ${H}$.

Definition 1 (Abert-Glazner-Virag) An invariant random subgroup (IRS) is a ${Ad_G}$-invariant probability measure on ${S(G)}$.

Examples.

• ${\delta_{\{e\}}}$, ${\delta_{G}}$.
• If ${N}$ is normal in ${G}$, ${\delta_N}$ is an IRS.
• Let ${\Gamma be a lattice. Then ${\Gamma\setminus G\rightarrow S(G)}$ mapsto
• Every IRS arises as follows. Let ${X}$ be a probability space with measure preserving ${G}$-action. Then ${x\mapsto Stab_G(x)}$ maps the measure to an IRS.

The space of IRS is compact. This raises the question of describing limits of IRS associated with lattices. Also, which sequences of lattices have the property that the corresponding IRS co,verge to ${\delta_{\{e\}}}$ ?

Lück’s theorem on ${\ell^2}$ Betti numbers can be interpreted in terms of convergence of IRS.

2. Amenable IRS

Definition 2 An IRS is amenable if it is supported on amenable subgroups.

The following is a generalization of a theorem of Kesten about normal subgroups.

Theorem 3 (Abert-Glasner-Virag) Let ${\mu}$ be an IRS. Then ${\mu}$ is amenable iff for ${\mu}$ a.e. subgroup ${H, the spectral radius of he simple random walk does not change when passing from ${G}$ to ${G/H}$.

2.1. Structure theorem

Theorem 4 (Bader-Duchesne-Lecureux) Every amenable IRS is supported on ${S(R_a(G))\subset S(G)}$, where ${R_a(G)}$ is the amenable radical, i.e. the largest normal amenable subgroup of ${G}$.

2.2. Proof

Can assume IRS is not supported on the space of subgroups of an any proper subgroup. Then one must show that ${G}$ is amenable. Use the fixed point property. Let ${G}$ act on a weakly compact convex set ${C}$. Assume ${C}$ is minimal (no proper compact convex invariant set). Almost every ${H}$ fixes a point in ${C}$. It suffices to prove that there is a common fixed point for a.e. ${H}$.

Baby case. ${H}$ has a unique fixed point. The map ${H\mapsto}$ fixed point of ${H}$ pushes forward ${\mu}$ to a ${G}$-invariant measure on ${C}$. Its barycenter is a fixed point in ${C}$.

General case. The fixed point set is a convex set ${C_H}$.

Lemma 5 The space ${C(E)}$ of compact convex subsets of ${E}$ is a convex cone in a a topological locally convex vector space. The subspace of convex subsets of a fixed convex set ${C}$ is convex and comapct in that space.

Indeed, compact convex sets can be added and dilated, and this corresponds to the linear structure on ${{\mathbb R}^{E^*}}$ when a convex set ${C}$ is mapped the function which to a linear form associates its max on ${C}$.

Since ${G}$ acts on ${{\mathbb R}^{E^*}}$

3. IRS in groups acting on ${CAT(0)}$ spaces

Let ${X}$ be a ${CAT(0)}$ space. Assume that either ${X}$ is proper and ${\partial X}$ is finite dimensional, or that ${X}$ has finite telescopic dimension. Assume too that ${X}$ is irreducible, and not a line.

Definition 6 Say ${G}$ acts geometrically densely on ${X}$ if there is no invariant convex subspace and no fixed point at the boundary.

Theorem 7 Assume that ${G}$ acts geometrically densely on ${X}$. Let ${\mu}$ be an IRS that does not charge the trivial subgroup. Then ${\mu}$-a.e. subgroup ${H}$ acts geometrically densely.

This is an elaboration on a result by Caprace and Monod.

3.1. Proof

Use Adams-Ballmann fixed point theorem for amenable groups acting on ${CAT(0)}$ spaces.

Let ${C_H}$ denote the fixed point set of subgroup ${H}$. Average distance to ${C_H}$ over ${\mu}$. Get concex, ${G}$-quasi-invariant function. If inf is achieved, it is on a convex subset. Otherwise, it is achieved on ${\partial X}$. Then ${[H,H]}$ stabilizes horoballs.