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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
Popa Superrigidity and Countable Borel Equivalence Relations
 In preparation
"... 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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Cited by 7 (5 self)
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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.
Review of: Subgroup lattices of groups, by Roland Schmidt, Expositions in Math., vol. 14, de Gruyter, 1994, xv+572 pp.
, 1996
"... ical considerations to solve the classification problem of finite simple groups." However this hope was not realized; much more powerful techniques, primarily character theory and "local analysis", were used. Similarly in Abelian group theory Baer's lattice theory techniques are no longer used. (See ..."
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ical considerations to solve the classification problem of finite simple groups." However this hope was not realized; much more powerful techniques, primarily character theory and "local analysis", were used. Similarly in Abelian group theory Baer's lattice theory techniques are no longer used. (See page 86 of Kaplansky's monograph [13].) So group theory and lattice theory went their separate ways. (For that matter, group theory nowadays has little in common with Abelian group theory.) Group theory had other techniques and lattice theory had its own deep problems to work on, and most of the applications of lattice theory to algebra were in the field of universal algebra. In the last several years some connections between lattice theory and group theory have resurfaced. One problem of interest in general algebra: is every finite lattice isomorphic to the congruence lattice of some algebraic system?P.Palfy and 1991 Mathematics Subject Classification. Primary 20E15; Secondary 06B15
UNIVERSAL BOREL ACTIONS OF COUNTABLE GROUPS
"... Abstract. If the countable group G has a nonabelian free subgroup, then there exists a standard Borel Gspace such that the corresponding orbit equivalence relation is countable universal. In this paper, we will consider the question of whether the converse also holds. 1. ..."
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Abstract. If the countable group G has a nonabelian free subgroup, then there exists a standard Borel Gspace such that the corresponding orbit equivalence relation is countable universal. In this paper, we will consider the question of whether the converse also holds. 1.
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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