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STABILIZER RIGIDITY IN IRREDUCIBLE GROUP ACTIONS
"... Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semisimple Lie groups. Using the intermediate factor theorems of BaderShalom and NevoZimmer, we show that the action stabilizers, and hence all irreducible invariant random subgroups, are coamenable i ..."
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Abstract. We consider irreducible actions of locally compact product groups, and of higher rank semisimple Lie groups. Using the intermediate factor theorems of BaderShalom and NevoZimmer, we show that the action stabilizers, and hence all irreducible invariant random subgroups, are coamenable in some normal subgroup. As a consequence, we derive rigidity results on irreducible actions that provide generalizations, and new proofs,
UNIMODULARITY OF INVARIANT RANDOM SUBGROUPS
"... Abstract. An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G there almost surely exists an invariant measure on G/H. Equival ..."
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Abstract. An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G there almost surely exists an invariant measure on G/H. Equivalently, the modular function of H is almost surely equal to the modular function of G, restricted to H. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups. 1.
Property (T) and the Furstenberg Entropy of Nonsingular Actions
, 2014
"... 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 µentropy values of the ergodic, properl ..."
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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 µentropy values of the ergodic, properly nonsingular Gactions are bounded away from zero. We show that this is also a sufficient condition. 1
Generic Stationary Measures and Actions
, 2015
"... Let G be a countably infinite group, and let µ be a generating probability measure on G. We study the space of µstationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of µstationary, measurable Gactions ..."
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Let G be a countably infinite group, and let µ be a generating probability measure on G. We study the space of µstationary Borel probability measures on a topological G space, and in particular on ZG, where Z is any perfect Polish space. We also study the space of µstationary, measurable Gactions 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 µstationary measures on ZG 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
THE TOPOLOGY OF INVARIANT RANDOM SURFACES
"... Abstract. We study the topological type of a generic surface, with respect to a unimodular measure on the space of pointed hyperbolic surfaces. Unimodularity is equivalent to saying that the measure comes from an invariant random subgroup of the Lie group of isometries of the hyperbolic plane. 1. ..."
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Abstract. We study the topological type of a generic surface, with respect to a unimodular measure on the space of pointed hyperbolic surfaces. Unimodularity is equivalent to saying that the measure comes from an invariant random subgroup of the Lie group of isometries of the hyperbolic plane. 1.
4 Property (T) and the Furstenberg Entropy of Nonsingular Actions
, 2014
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