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48
On the superrigidity of malleable actions with spectral gap
 J. Amer. Math. Soc
"... Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certa ..."
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Cited by 79 (7 self)
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Abstract. We prove that if a countable group Γ contains a nonamenable subgroup with centralizer infinite and “weakly normal ” in Γ (e.g. if Γ is nonamenable and has infinite center or is a product of infinite groups) then any measure preserving Γaction on a probability space which satisfies certain malleability, spectral gap and weak mixing conditions is cocycle superrigid. We also show that if Γ � X is an arbitrary free ergodic action of such a group Γ and Λ � Y = T Λ is a Bernoulli action of an arbitrary infinite conjugacy class group, then any isomorphism of the associated II1 factors L ∞ X ⋊Γ ≃ L ∞ Y ⋊Λ comes from a conjugacy of the actions. 1.
The algebraization of Kazhdan’s property (T)
 IN PROC. INTERNAT. CONGR. OF MATHEMATICIANS, VOL. II, EMS PUBL
, 2006
"... We present the surge of activity since 2005, around what we call the algebraic (as contrasted with the geometric) approach to Kazhdan’s property (T). The discussion includes also an announcement of a recent result (March 2006) regarding property (T) for linear groups over arbitrary finitely generate ..."
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Cited by 19 (0 self)
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We present the surge of activity since 2005, around what we call the algebraic (as contrasted with the geometric) approach to Kazhdan’s property (T). The discussion includes also an announcement of a recent result (March 2006) regarding property (T) for linear groups over arbitrary finitely generated rings.
Isometry groups of nonpositively curved spaces: structure theory
 J. Topol
"... Abstract. We develop the structure theory of full isometry groups of locally compact nonpositively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and BruhatTits buildings. Applications to di ..."
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Cited by 15 (1 self)
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Abstract. We develop the structure theory of full isometry groups of locally compact nonpositively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and BruhatTits buildings. Applications to discrete groups and further developments on nonpositively curved lattices are exposed in a companion paper [CM08b]. 1.
Local rigidity of group actions: past, present, future
, 2007
"... This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to inte ..."
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Cited by 9 (1 self)
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This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research.
Linear representations and arithmeticity of lattices in product of trees, preprint, p
"... In this paper we continue our study of lattices in the automorphisms groups of products of trees initiated in [BM97], [BM00a], [BM00b], [Moz98] (see also [Gla03], [BG02], [Rat04]). We concentrate here on the interplay between the linear representation theory and the structure of these lattices. Befo ..."
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Cited by 8 (1 self)
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In this paper we continue our study of lattices in the automorphisms groups of products of trees initiated in [BM97], [BM00a], [BM00b], [Moz98] (see also [Gla03], [BG02], [Rat04]). We concentrate here on the interplay between the linear representation theory and the structure of these lattices. Before turning to the main results of this paper it may be worthwhile to put certain concepts and results from [BM00a], [BM00b] in perspective, and explain the motivations for our approach. A lattice Γ in a locally compact group G is a discrete subgroup such that the quotient space G/Γ carries a finite Ginvariant measure. If in addition G/Γ is compact the lattice is called cocompact or uniform. Consider the following special setting: let Qp be the field of padic numbers and let G = PSL(2,Qp) × PSL(2,Qq). Lattices in products fall into two natural classes: reducible and irreducible. A lattice Γ < G is called reducible if it has a subgroup of finite index which is a direct product Γp × Γq where Γp < PSL(2,Qp) and Γq < PSL(2,Qq) are
Arithmeticity vs. nonlinearity for irreducible lattices
 Geom. Dedicata
"... Abstract. We establish an arithmeticity vs. nonlinearity alternative for irreducible lattices in suitable product groups, for instance products of topologically simple groups. This applies notably to a (large class of) KacMoody groups. The alternative relies heavily on the superrigidity theorem we ..."
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Abstract. We establish an arithmeticity vs. nonlinearity alternative for irreducible lattices in suitable product groups, for instance products of topologically simple groups. This applies notably to a (large class of) KacMoody groups. The alternative relies heavily on the superrigidity theorem we propose in [Md], since we follow Margulis ’ reduction of arithmeticity to superrigidity. 1.
Actions of higherrank lattices on free groups
"... If G is a semisimple Lie group of real rank at least 2 and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image. ..."
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Cited by 8 (2 self)
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If G is a semisimple Lie group of real rank at least 2 and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.