36 research outputs found
Stabilizer Rigidity in Irreducible Group Actions
We consider irreducible actions of locally compact product groups, and of
higher rank semi-simple Lie groups. Using the intermediate factor theorems of
Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all
irreducible invariant random subgroups, are co-amenable in their normal
closure. As a consequence, we derive rigidity results on irreducible actions
that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.Comment: 25 page
Generic Stationary Measures and Actions
Let be a countably infinite group, and let be a generating
probability measure on . We study the space of -stationary Borel
probability measures on a topological space, and in particular on ,
where is any perfect Polish space. We also study the space of
-stationary, measurable -actions on a standard, nonatomic probability
space.
Equip the space of stationary measures with the weak* topology. When
has finite entropy, we show that a generic measure is an essentially free
extension of the Poisson boundary of . When is compact, this
implies that the simplex of -stationary measures on is a Poulsen
simplex. We show that this is also the case for the simplex of stationary
measures on .
We furthermore show that if the action of on its Poisson boundary is
essentially free then a generic measure is isomorphic to the Poisson boundary.
Next, we consider the space of stationary actions, equipped with a standard
topology known as the weak topology. Here we show that when has property
(T), the ergodic actions are meager. We also construct a group without
property (T) such that the ergodic actions are not dense, for some .
Finally, for a weaker topology on the set of actions, which we call the very
weak topology, we show that a dynamical property (e.g., ergodicity) is
topologically generic if and only if it is generic in the space of measures.
There we also show a Glasner-King type 0-1 law stating that every dynamical
property is either meager or residual.Comment: To appear in the Transactions of the AMS, 49 page
Subalgebras, subgroups, and singularity
This paper concerns the non-commutative analog of the Normal Subgroup Theorem
for certain groups. Inspired by Kalantar-Panagopoulos, we show that all
-invariant subalgebras of and are (-)
co-amenable. The groups we work with satisfy a singularity phenomenon described
in Bader-Boutonnet-Houdayer-Peterson. The setup of singularity allows us to
obtain a description of -invariant intermediate von Neumann subalgebras
in
terms of the normal subgroups of .Comment: This is the final version. All the comments made by the referee have
been implemented. The paper has been accepted to appear in the Bulletin of
the London Math Society https://doi.org/10.1112/blms.1293
Generic Stationary Measures and Actions
Let G be a countably infinite group, and let μ be a generating probability measure on G. We study the space of μ-stationary Borel probability measures on a topological G space, and in particular on Z^G, where Z is any perfect Polish space. We also study the space of μ-stationary, measurable G-actions on a standard, nonatomic probability space.
Equip the space of stationary measures with the weak* topology. When μ has finite entropy, we show that a generic measure is an essentially free extension of the Poisson boundary of (G, μ). When Z is compact, this implies that the simplex of μ-stationary
measures on Z^G is a Poulsen simplex. We show that this is also the case for the simplex of stationary measures on {0, 1}^G.
We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary.
Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G has property (T), the
ergodic actions are meager. We also construct a group G without property (T) such that the ergodic actions are not dense, for some μ.
Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property (e.g., ergodicity) is topologically generic if and only if it is generic in the space of measures. There we also show a Glasner-King type 0-1 law stating that every dynamical property is either meager or residual
Property (T) and the Furstenberg Entropy of Nonsingular Actions
We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condition
An Abramov formula for stationary spaces of discrete groups
Let (G,mu) be a discrete group equipped with a generating probability
measure, and let Gamma be a finite index subgroup of G. A mu-random walk on G,
starting from the identity, returns to Gamma with probability one. Let theta be
the hitting measure, or the distribution of the position in which the random
walk first hits Gamma.
We prove that the Furstenberg entropy of a (G,mu)-stationary space, with
respect to the induced action of (Gamma,theta), is equal to the Furstenberg
entropy with respect to the action of (G,mu), times the index of Gamma in G.
The index is shown to be equal to the expected return time to Gamma.
As a corollary, when applied to the Furstenberg-Poisson boundary of (G,mu),
we prove that the random walk entropy of (Gamma,theta) is equal to the random
walk entropy of (G,mu), times the index of Gamma in G.Comment: 16 page