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Popa superrigidity and countable Borel equivalence relations
, 2006
"... Abstract. We present some applications of Popa’s Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, we show that the universal countable Borel equivalence relation E ∞ is not essentially free. 1. ..."
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Abstract. We present some applications of Popa’s Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, we show that the universal countable Borel equivalence relation E ∞ is not essentially free. 1.
Lacunary hyperbolic groups
, 2007
"... We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an Rtree. We characterize lacunary hyperbolicgroups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of ..."
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Cited by 8 (2 self)
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We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an Rtree. We characterize lacunary hyperbolicgroups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolicgroups, we solve a problem of Gromov by constructing a group whose asymptotic cone C has countable but nontrivial fundamental group (in fact C is homeomorphic to the direct product of a tree and a circle, so π1(C) = Z). We show that the class of lacunary hyperbolic groups contains elementary amenable groups, groups with all proper subgroups cyclic, and torsion groups. This allows us to solve two problems of Drut¸u and Sapir, and a problem of Kleiner about groups with cutpoints in their asymptotic cones. We also construct a finitely generated group whose divergence function is not linear but is arbitrarily close to being linear. This answers a question of Behrstock. Contents 1
Infinite groups with large balls of torsion elements and small entropy, Archiv der Mathematik 82(2
, 2006
"... Abstract. We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3generator nonvirtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these exam ..."
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Abstract. We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3generator nonvirtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. The Burnside Problem asks whether a finitely generated group all of whose elements have finite order must be finite. We are interested in the following related question: fix n sufficiently large; given a group Γ, with a finite symmetric generating subset S such that every element in the nball is torsion, is Γ finite? Since the Burnside problem has a negative answer, a fortiori the answer to our question is negative in general. However, it is natural to ask for it in some classes of finitely generated groups for which the Burnside Problem has a positive answer, such as linear groups or solvable groups. This motivates the following proposition, which in particular answers a question of Breuillard to the authors.
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
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, 2004
"... We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n 2 log n and other small Dehn functions. Some of ..."
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We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n 2 log n and other small Dehn functions. Some of
NONAMENABLE FINITELY PRESENTED TORSIONBYCYCLIC GROUPS
"... Abstract. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x, y] n =1. 1. ..."
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Abstract. We construct a finitely presented nonamenable group without free noncyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n ≫ 1 by a cyclic group, so it satisfies the identity [x, y] n =1. 1. Short history of the problem Hausdorff [14] proved in 1914 that one can subdivide the 2sphere minus a countable set of points into three parts A, B, C such that each of these three parts can be obtained from each of the other two parts by a rotation, and the union of any two of these parts can be obtained by rotating the third part. This implied that one cannot define a finitely additive measure on the 2sphere which is invariant under the group SO(3). In 1924 Banach and Tarski [3] generalized Hausdorff’s result by proving, in particular, that in R3, every two bounded sets A, B with nonempty interiors can be decomposed A = �n i=1 Ai, B = �n i=1 Bi so that Ai can be rotated to Bi, i =1,...,n (the so called BanachTarski paradox). Von Neumann [20] was first who noticed that the cause of the BanachTarski paradox is not the geometry of R3 but an algebraic property of the group SO(3). He introduced the concept of an amenable group (he called such groups “measurable”) as a group G which has a left invariant finitely additive measure µ, µ(G) = 1, noticed that if a group is amenable, then any set it acts upon freely also has an invariant measure, and proved that a group is not amenable provided it contains a free nonabelian subgroup. He also showed that groups like PSL(2, Z), SL(2, Z) contain free nonabelian subgroups. So analogs of BanachTarski paradox can be found in R2 and even R. Von Neumann showed that the class of amenable groups contains abelian groups, finite groups and is closed under taking subgroups, extensions, and infinite unions of increasing sequences of groups. Day [9] and Specht [31] showed that this Received by the editors January 9, 2001.