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Rigidity Of Commensurators And Irreducible Lattices
 Invent. Math
"... this paper. We shall address here many ingredients of the well developed linear rigidity theory, such as superrigidity, strongrigidity, arithmeticity, normal subgroup structure and others, in two general situations: lattices in products of (general) topological groups, and commensurators of lattice ..."
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Cited by 75 (3 self)
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this paper. We shall address here many ingredients of the well developed linear rigidity theory, such as superrigidity, strongrigidity, arithmeticity, normal subgroup structure and others, in two general situations: lattices in products of (general) topological groups, and commensurators of lattices in topological groups. The approach we take for this purpose is, naturally, dierent from previous ones. Of course, our results apply also in the conventional linear framework. Some of them are new even in that case, and others provide an alternative, sometimes more natural approach, to several wellknown results (including a new proof of Margulis' arithmeticity and commensuratorarithmeticity theorems for a certain class of lattices).
Deformation and rigidity for group actions and von Neumann algebras
, 2007
"... We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors wit ..."
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Cited by 64 (7 self)
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We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a combination of deformation and rigidity properties. This includes strong rigidity results for factors with calculation of their fundamental group and cocycle superrigidity for actions with applications to orbit equivalence ergodic theory.
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
"... This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Bor ..."
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Cited by 41 (7 self)
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This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Borel if it is a Borel subset of X². Given two
Some computations of 1cohomology groups and construction of non orbit equivalent actions
"... Abstract. For each group G having an infinite normal subgroup with the relative property (T) (e.g. G = H × K, with H infinite with property (T) and K arbitrary) and each countable abelian group Λ we construct free ergodic measurepreserving actions σΛ of G on the probability space such that the 1’st ..."
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Cited by 35 (10 self)
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Abstract. For each group G having an infinite normal subgroup with the relative property (T) (e.g. G = H × K, with H infinite with property (T) and K arbitrary) and each countable abelian group Λ we construct free ergodic measurepreserving actions σΛ of G on the probability space such that the 1’st cohomology group of σΛ, H 1 (σΛ, G), is equal to Char(G) × Λ. We deduce that G has uncountably many non stably orbit equivalent actions. We also calculate 1cohomology groups and show existence of “many ” non stably orbit equivalent actions for free products of groups as above. Let G be a countable discrete group and σ: G → Aut(X, µ) a free measure preserving (m.p.) action of G on the probability space (X, µ), which we also view as an integral preserving action of G on the abelian von Neumann algebra A = L ∞ (X, µ). A 1cocycle for (σ, G) is a map w: G → U(A), satisfying wgσg(wh) = wgh, ∀g, h ∈ G, where
Spectra Of Elements In The Group Ring Of SU(2)
 J. Eur. Math. Soc. (JEMS
, 1999
"... . We present a new method for establishing the "gap" property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S 2 as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the ..."
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Cited by 35 (7 self)
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. We present a new method for establishing the "gap" property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S 2 as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the Nth irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the "unfolded" consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N # #; we establish several results in this direction. For certain special "arithmetic" (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. 1. Introduction The irreducible representations of G = SU(2) are #N = s...
ORBIT INEQUIVALENT ACTIONS OF NONAMENABLE GROUPS
, 2008
"... Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such ..."
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Cited by 28 (3 self)
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Consider two free measure preserving group actions Γ � (X, µ), ∆ � (X, µ), and a measure preserving action ∆ � a (Z, ν) where (X, µ), (Z, ν) are standard probability spaces. We show how to construct free measure preserving actions Γ � c (Y, m), ∆ � d (Y, m) on a standard probability space such that E d ∆ ⊂ E c Γ and d has a as a factor. This generalizes the standard notion of coinduction of actions of groups from actions of subgroups. We then use this construction to show that if Γ is a countable nonamenable group, then Γ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space.
Ergodic Subequivalence Relations Induced by a Bernoulli Action, available at arXiv: 0802.2353
"... Abstract. Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ � [0, 1] Γ, where [0,1] Γ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation R of S, there exists a partition {Xi} i≥0 of [0, 1] Γ with Rinvariant ..."
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Cited by 25 (4 self)
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Abstract. Let Γ be a countable group and denote by S the equivalence relation induced by the Bernoulli action Γ � [0, 1] Γ, where [0,1] Γ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation R of S, there exists a partition {Xi} i≥0 of [0, 1] Γ with Rinvariant measurable sets such that R X0 is hyperfinite and R Xi is strongly ergodic (hence ergodic), for every i ≥ 1. §1. Introduction and statement of results. During the past decade there have been many interesting new directions arising in the field of measurable group theory. One direction came from the deformation/rigidity theory developed recently by S. Popa in order to study group actions and von Neumann algebras ([P5]). Using this theory, Popa obtained striking rigidity
The rank gradient from a combinatorial viewpoint
, 2009
"... This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the ..."
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Cited by 23 (5 self)
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This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby’s trichotomy theorem on finitely presented groups. 1
Actions of F∞ whose II1 factors and orbit equivalence relations have prescribed fundamental group
, 2008
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Orbit Equivalence and Measured Group Theory
 INTERNATIONAL CONGRESS OF MATHEMATICIANS (ICM), HYDERABAD: INDIA
, 2010
"... We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions. ..."
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Cited by 20 (0 self)
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We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions.