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 tree-graded 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.

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. 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 non-trivial 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 cut-points 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

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

Abstract. In this paper we discuss metric cell complexes satisfying a coarse form of 2-dimensional Poincaré duality. We prove that such spaces are either Gromov-hyperbolic or have polynomial growth. As an application we prove that 2-dimensional Poincaré duality groups over commutative rings are commensurable with surface groups. 1.

Abstract. Divergence functions of a metric space estimate the length of a path connecting two points A, B at distance ≤ n avoiding a large enough ball around a third point C. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. That property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has “many ” periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditch’s properties that are weaker than local compactness. This gives a new proof of Behrstock’s result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the Q-rank is 1 and when the lattice is SLn(OS) where

Abstract. Let M be a closed non-positively 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 non-trivial, 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 bi-Lipschitz flats satisfying certain additional hypotheses. As applications we obtain (1) constraints on the behavior of quasi-isometries 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 Ballmann-Burns-Spatzier 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.

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

(Communicated by Julia Knight) Abstract. If CH fails, then there exist 2 2 ℵ 0 isomorphism. universal sofic groups up to 1.

Abstract. If CH fails, then there exist 2 2 ℵ 0 universal sofic groups up to isomorphism. 1.

ABSTRACT. Let M be a complete locally compact CAT(0)-space, and X an asymptotic cone of M. Forγ⊂Mak-dimensional flat, let γω be the k-dimensional flat in X obtained as the ultralimit of γ. In this paper, we identify various conditions on γω that are sufficient to ensure that γ bounds a (k+1)-dimensional half-flat. As applications we obtain: (1) constraints on the behavior of quasi-isometries between locally compact CAT(0)-spaces; (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds; (3) a correspondence between metric splittings of a complete, simply connected non-positively curved Riemannian manifolds, and metric splittings of its asymptotic cones; and (4) an elementary derivation of Gromov’s rigidity theorem from the combination of the Ballmann, Burns-Spatzier rank rigidity theorem and the classic Mostow rigidity theorem. 1.