• Documents
  • Authors
  • Tables
  • Log in
  • Sign up
  • MetaCart
  • DMCA
  • Donate

CiteSeerX logo

Advanced Search Include Citations
Advanced Search Include Citations

Furstenberg entropy realizations for virtually free groups and lamplighter groups. arXiv:1210.5897 (2012)

by Yair Hartman, Omer Tamuz
Add To MetaCart

Tools

Sorted by:
Results 1 - 7 of 7

STABILIZER RIGIDITY IN IRREDUCIBLE GROUP ACTIONS

by Yair Hartman, Omer Tamuz
"... Abstract. 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 hence all irreducible invariant random subgroups, are co-amenable i ..."
Abstract - Cited by 4 (0 self) - Add to MetaCart
Abstract. 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 hence all irreducible invariant random subgroups, are co-amenable in some normal subgroup. As a consequence, we derive rigidity results on irreducible actions that provide generalizations, and new proofs,
(Show Context)

Citation Context

...G, µ)boundary, which is isomorphic to the Poisson boundary of the group G/K, equipped with the projection of µ (see [2, Lemma 2.15]). L. Bowen [3] introduces what we shall call Bowen spaces (see also =-=[11]-=-). Definition 2.4. Let K ≤ G be an IRS in G with distribution λ ∈ IRS(G). Consider the space ˜SubG = {(K; Kg1, Kg2, . . . ) |K ∈ SubG, gn ∈ G}STABILIZER RIGIDITY IN IRREDUCIBLE GROUP ACTIONS 7 with t...

UNIMODULARITY OF INVARIANT RANDOM SUBGROUPS

by Ian Biringer, Omer Tamuz
"... Abstract. An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all el-ements 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 ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
Abstract. An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all el-ements 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 invari-ant random subgroups. 1.
(Show Context)

Citation Context

...e appeared in a number of papers, either as direct subjects of study [12–14,18,26], as probabilistic limits of manifolds with increasing volume [1], or as tools to understand stationary group actions =-=[11,17]-=-. The notion of an IRS is a natural weakening of that of a normal subgroup. As such, it is interesting to understand which properties of normal subgroups hold for IRSs. This is the spirit of [3], and ...

Property (T) and the Furstenberg Entropy of Nonsingular Actions

by Lewis Bowen, Yair Hartman, Omer Tamuz , 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 µ-entropy values of the ergodic, properl ..."
Abstract - Add to MetaCart
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. 1
(Show Context)

Citation Context

... has been a useful tool in the study of stationary actions (e.g., [5, 8]), and the study of the set of entropy values realizable by properly nonsingular stationary actions has attracted some interest =-=[11, 2, 6]-=-. 2 Nevo’s theorem (Theorem A.1) implies that for every (G, µ), where G has property (T), the Furstenberg entropy values of properly nonsingular, ergodic µ-stationary actions are bounded away from zer...

Generic Stationary Measures and Actions

by Lewis Bowen, Yair Hartman, Omer Tamuz , 2015
"... 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 ZG, where Z is any perfect Polish space. We also study the space of µ-stationary, measurable G-actions ..."
Abstract - Add to MetaCart
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 ZG, 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 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 topol-ogy known as the weak topology. Here we show that when G has property (T), the
(Show Context)

Citation Context

...However, there are surprisingly few explicit constructions of stationary actions: aside from measure-preserving actions and Poisson boundaries, there are constructions from invariant random subgroups =-=[10, 31]-=- and methods for combining stationary actions via joinings [22]. There is a general structure theory of stationary actions [22] and a very deep structure theory in the case that G is a higher rank sem...

THE TOPOLOGY OF INVARIANT RANDOM SURFACES

by Ian Biringer, Jean Raimbault
"... Abstract. We study the topological type of a generic surface, with respect to a unimodular measure on the space of pointed hyperbolic surfaces. Uni-modularity is equivalent to saying that the measure comes from an invariant random subgroup of the Lie group of isometries of the hyperbolic plane. 1. ..."
Abstract - Add to MetaCart
Abstract. We study the topological type of a generic surface, with respect to a unimodular measure on the space of pointed hyperbolic surfaces. Uni-modularity is equivalent to saying that the measure comes from an invariant random subgroup of the Lie group of isometries of the hyperbolic plane. 1.
(Show Context)

Citation Context

...iven in [1, Section 12]. IRSs were first studied by Abért-Glasner-Virág [3] for discrete G, see also Vershik [17], were introduced to Lie groups in [1] and have attracted much recent interest, e.g. =-=[7, 8, 9, 10, 14, 16]-=-. Let K = PSO(2), and H2 = G/K be the real hyperbolic plane. If Γ is a discrete, torsion-free subgroup of G, the quotient SΓ = Γ\H2 is a hyperbolic surface. The aim of this brief note is to prove the ...

3 STABILIZER RIGIDITY IN IRREDUCIBLE GROUP ACTIONS

by Yair Hartman, Omer Tamuz
"... ar ..."
Abstract - Add to MetaCart
Abstract not found
(Show Context)

Citation Context

...ver (SubG, λ): Π(G, µ)× (SubG, λ) −→ Bµ(λ) −→ (SubG, λ). We encourage the reader to study the details in Bowen’s original paper [4]. For further discussion and another application of Bowen spaces see =-=[13]-=-. Consider the process on λ\G that first chooses an element according to λ, and then applies a µ-right random walk. Formally, this is the Markov chain {KZn}n∈N where K ∼ λ and Zn ∼ µ n. It follows fro...

4 Property (T) and the Furstenberg Entropy of Nonsingular Actions

by Lewis Bowen, Yair Hartman, Omer Tamuz , 2014
"... ar ..."
Abstract - Add to MetaCart
Abstract not found
(Show Context)

Citation Context

... has been a useful tool in the study of stationary actions (e.g., [5, 8]), and the study of the set of entropy values realizable by properly nonsingular stationary actions has attracted some interest =-=[11, 2, 6]-=-. 2 Nevo’s theorem (Theorem A.1) implies that for every (G, µ), where G has property (T), the Furstenberg entropy values of properly nonsingular, ergodic µ-stationary actions are bounded away from zer...

Powered by: Apache Solr
  • About CiteSeerX
  • Submit and Index Documents
  • Privacy Policy
  • Help
  • Data
  • Source
  • Contact Us

Developed at and hosted by The College of Information Sciences and Technology

© 2007-2019 The Pennsylvania State University