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15
Asymptotic geometry of the mapping class group and Teichmüller space
 GEOMETRY & TOPOLOGY
, 2006
"... In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. ..."
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Cited by 33 (6 self)
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In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is treegraded in the sense of Dru¸tu and Sapir; this treegrading has several consequences including answering a question of Drutu and Sapir concerning relatively hyperbolic groups. Another application is a generalization of the result of Brock and Farb that for low complexity surfaces Teichmüller space, with the Weil–Petersson metric, is ı–hyperbolic. Although for higher complexity surfaces these spaces are not ı–hyperbolic, we establish the presence of previously unknown negative curvature phenomena in the mapping class group and Teichmüller space for arbitrary surfaces.
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
FUNDAMENTAL GROUPS OF ASYMPTOTIC CONES
, 2004
"... We show that for any metric space M satisfying certain natural conditions, there is a finitely generated group G, an ultrafilter ω, and an isometric embedding ι of M to the asymptotic cone Coneω(G) such that the induced homomorphism ι ∗ : π1(M) → π1(Coneω(G)) is injective. In particular, we prove ..."
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Cited by 4 (1 self)
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We show that for any metric space M satisfying certain natural conditions, there is a finitely generated group G, an ultrafilter ω, and an isometric embedding ι of M to the asymptotic cone Coneω(G) such that the induced homomorphism ι ∗ : π1(M) → π1(Coneω(G)) is injective. In particular, we prove that any countable group can be embedded into a fundamental group of an asymptotic cone of a finitely generated group.
ULTRAPRODUCTS OF FINITE ALTERNATING GROUPS
"... Abstract. We prove that if U is a nonprincipal ultrafilter over ω, then the set of normal subgroups of the ultraproduct Q U Alt(n) is linearly ordered by inclusion. We also prove that the number of such ultraproducts up to isomorphism is either 2 ℵ0 ..."
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Cited by 2 (2 self)
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Abstract. We prove that if U is a nonprincipal ultrafilter over ω, then the set of normal subgroups of the ultraproduct Q U Alt(n) is linearly ordered by inclusion. We also prove that the number of such ultraproducts up to isomorphism is either 2 ℵ0
DIMENSION OF ASYMPTOTIC CONES OF LIE GROUPS
, 2007
"... Abstract. We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of connected Lie groups that quasiisometrically embed in ..."
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Cited by 2 (1 self)
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Abstract. We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of connected Lie groups that quasiisometrically embed into a nonpositively curved metric space. 1.
On sequences of finitely generated discrete groups
, 2007
"... We consider sequences of discrete subgroups Γi = ρi(Γ) of a rank 1 Lie group G, with Γ finitely generated. We show that, for algebraically convergent sequences (Γi), unless Γi’s are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discr ..."
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Cited by 2 (0 self)
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We consider sequences of discrete subgroups Γi = ρi(Γ) of a rank 1 Lie group G, with Γ finitely generated. We show that, for algebraically convergent sequences (Γi), unless Γi’s are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Γi) we show that the resulting action Γ � T on a real tree satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Γ splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in Isom(T). 1
Geometry of quasiplanes
, 2004
"... Abstract. In this paper we discuss metric cell complexes satisfying a coarse form of 2dimensional Poincaré duality. We prove that such spaces are either Gromovhyperbolic or have polynomial growth. As an application we prove that 2dimensional Poincaré duality groups over commutative rings are comm ..."
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Cited by 2 (1 self)
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Abstract. In this paper we discuss metric cell complexes satisfying a coarse form of 2dimensional Poincaré duality. We prove that such spaces are either Gromovhyperbolic or have polynomial growth. As an application we prove that 2dimensional Poincaré duality groups over commutative rings are commensurable with surface groups. 1.
Asymptotic cones, biLipschitz ultraflats, and the geometric rank of geodesics. Available at the arXiv:0801.3636
"... Abstract. Let M be a closed nonpositively curved Riemannian (NPCR) manifold, ˜ M its universal cover, and X an ultralimit of ˜ M. For γ ⊂ ˜ M a geodesic, let γω be a geodesic in X obtained as an ultralimit of γ. We show that if γω is contained in a flat in X, then the original geodesic γ supports ..."
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Cited by 1 (1 self)
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Abstract. Let M be a closed nonpositively curved Riemannian (NPCR) manifold, ˜ M its universal cover, and X an ultralimit of ˜ M. For γ ⊂ ˜ M a geodesic, let γω be a geodesic in X obtained as an ultralimit of γ. We show that if γω is contained in a flat in X, then the original geodesic γ supports a nontrivial, normal, parallel Jacobi field. In particular, the rank of a geodesic can be detected from the ultralimit of the universal cover. We strengthen this result by allowing for biLipschitz flats satisfying certain additional hypotheses. As applications we obtain (1) constraints on the behavior of quasiisometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splittings of complete, simply connected NPCR manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the BallmannBurnsSpatzier rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov’s rigidity theorem for higher rank locally symmetric spaces.
ON THE NUMBER OF UNIVERSAL SOFIC GROUPS
"... (Communicated by Julia Knight) Abstract. If CH fails, then there exist 2 2 ℵ 0 isomorphism. universal sofic groups up to 1. ..."
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(Communicated by Julia Knight) Abstract. If CH fails, then there exist 2 2 ℵ 0 isomorphism. universal sofic groups up to 1.
Supported by the Austrian Federal Ministry of Education, Science and Culture
"... Abstract. If CH fails, then there exist 2 2 ℵ 0 universal sofic groups up to isomorphism. 1. ..."
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Abstract. If CH fails, then there exist 2 2 ℵ 0 universal sofic groups up to isomorphism. 1.