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29
Stable group theory and approximate subgroups
 J. Amer. Math. Soc
"... Abstract. We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sumproduct phenomenon. For a simple linear group G, we show that a finite subset X with XX −1 X/X  bounded is close to a finite subgroup, or else to a subset of a pro ..."
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Cited by 48 (0 self)
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Abstract. We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sumproduct phenomenon. For a simple linear group G, we show that a finite subset X with XX −1 X/X  bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Modeltheoretically we prove the independence theorem and the stabilizer theorem in a general firstorder setting. 1.
A NEW PROOF OF GROMOV’S THEOREM ON GROUPS OF POLYNOMIAL GROWTH
"... Abstract. We give a proof of Gromov’s theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. The proof does not rely on the MontgomeryZippinYamabe structure theory of locally compact groups. ..."
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Abstract. We give a proof of Gromov’s theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. The proof does not rely on the MontgomeryZippinYamabe structure theory of locally compact groups.
Dimension and rank for mapping class groups
, 2007
"... We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of BrockFarb’s Rank Conjecture wh ..."
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Cited by 32 (5 self)
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We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of BrockFarb’s Rank Conjecture which asserts that MCG has quasiflats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasiflats in Teichmüller space with the WeilPetersson metric.
On Random Walks on Wreath Products
 Ann. Probab
, 2001
"... Wreath products are a type of semidirect products. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to th ..."
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Cited by 29 (3 self)
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Wreath products are a type of semidirect products. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group.
G.A.Margulis, Almost isometric actions, property (T), and local rigidity
 Invent. Math
"... Abstract. Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions ..."
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Cited by 28 (4 self)
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Abstract. Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2]. One definition of property (T) is that a group Γ has property (T) if every isometric Γ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with property (T). Some of these are used in the proofs of our local rigidity theorems. 1.
A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples
, 2001
"... ..."
Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
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Cited by 22 (2 self)
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this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field
Spectral asymptotics of percolation Hamiltoninas on amenable Cayley graphs
 In Methods of Spectral Analysis in Mathematical Physics (Lund, 2006), Volume 186 of Oper. Theory Adv. Appl
, 2008
"... Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these r ..."
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Cited by 11 (2 self)
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Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem. 1.
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
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Cited by 10 (1 self)
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Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1