Amenable invariant random subgroups
Joint with Bader, Duchesne, Glasner, Lazarovich.
Let be a locally compact second counable group. Let be the topological space of closed subgroups of : converges to if
- Every is a limit of elements .
- Every converging subsequence of a sequence converges to an element of .
Definition 1 (Abert-Glazner-Virag) An invariant random subgroup (IRS) is a -invariant probability measure on .
- , .
- If is normal in , is an IRS.
- Let be a lattice. Then mapsto
- Every IRS arises as follows. Let be a probability space with measure preserving -action. Then 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 ?
Lück’s theorem on 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 be an IRS. Then is amenable iff for a.e. subgroup , the spectral radius of he simple random walk does not change when passing from to .
2.1. Structure theorem
Theorem 4 (Bader-Duchesne-Lecureux) Every amenable IRS is supported on , where is the amenable radical, i.e. the largest normal amenable subgroup of .
Can assume IRS is not supported on the space of subgroups of an any proper subgroup. Then one must show that is amenable. Use the fixed point property. Let act on a weakly compact convex set . Assume is minimal (no proper compact convex invariant set). Almost every fixes a point in . It suffices to prove that there is a common fixed point for a.e. .
Baby case. has a unique fixed point. The map fixed point of pushes forward to a -invariant measure on . Its barycenter is a fixed point in .
General case. The fixed point set is a convex set .
Lemma 5 The space of compact convex subsets of is a convex cone in a a topological locally convex vector space. The subspace of convex subsets of a fixed convex set is convex and comapct in that space.
Indeed, compact convex sets can be added and dilated, and this corresponds to the linear structure on when a convex set is mapped the function which to a linear form associates its max on .
Since acts on
3. IRS in groups acting on spaces
Let be a space. Assume that either is proper and is finite dimensional, or that has finite telescopic dimension. Assume too that is irreducible, and not a line.
Definition 6 Say acts geometrically densely on if there is no invariant convex subspace and no fixed point at the boundary.
Theorem 7 Assume that acts geometrically densely on . Let be an IRS that does not charge the trivial subgroup. Then -a.e. subgroup acts geometrically densely.
This is an elaboration on a result by Caprace and Monod.
Use Adams-Ballmann fixed point theorem for amenable groups acting on spaces.
Let denote the fixed point set of subgroup . Average distance to over . Get concex, -quasi-invariant function. If inf is achieved, it is on a convex subset. Otherwise, it is achieved on . Then stabilizes horoballs.