| Abstract: When a manifold is homotopy equivalent to a connected sum, it may or may not admit a corresponding decomposition. This talk will describe work with Frank Connolly which gives a complete description of when such a decomposition exists, at least when the manifolds involved have dimension greater than four. Here is how the story goes - surgery theory converts the topological question to an algebraic question involving the fundamental groups of the two manifolds in the connected sum. The algebraic question then can be reduced to an algebraic question involving the quadratic forms over the integral group ring of PSL_2(Z) and the infinite dihedral group. This algebraic question is then attacked by techniques of controlled topology and dynamics, following the lead of Farrell and Jones. In some sense, the quadratic forms can be controlled except in the neighborhood of closed geodesics in the orbifold H^2/PSL_2(Z). The talk will attempt to survey the above, with a concentration on the geometric aspects. |
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| Abstract: We study the asymptotic cones of the universal covering spaces of closed 4-dimensional nonpositively curved real analytic manifolds. We show that the existence of nonstandard components in the Tits boundary, discovered by Hummel and Schroeder, depends only on the quasi-isometry type of the fundamental group. |
| Abstract:One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose boundaries are ``rigged'' with analytic parameterizations. The fundamental operation is the sewing of such surfaces using the parameterizations, which was shown to be holomorphic by D. Radnell. We generalize this picture to quasisymmetric boundary parameterizations, which results in a greatly simplified picture. In particular we prove that the universal Teichmueller space induces complex manifold structures on the Riemann and Teichmueller moduli spaces of rigged surfaces and that the border and puncture pictures of rigged moduli/Teichmueller spaces are biholomorphically equivalent. This model also provides a straightforward proof of the holomorphicity of the sewing operation. Joint work with D. Radnell. |
| Abstract: I will talk about the following theorem of T. Napier and myself. If the filtered ends of a Kahler group is at least three then a finite index subgroup maps onto a co-compact Fuchsian group. Applications of this result to thompson's groups will be discussed. |
| Abstract: The mapping class group is the group of isotopy classes of orientation-preserving homeomorphisms of a surface. We will discuss some new 3-dimensional bordism invariants of certain subgroups of the mapping class group. In particular, these are invariants of the Johnson filtration of the mapping class group. We will discuss a new representation in terms of spin bordism which combines into a single homomorphism all of the information given by many of the well-known representations, including the Johnson homomorphism, Birman-Craggs homomorphism, and Morita homomorphism. |
| Abstract: I'll describe a joint work with E. Breuillard, proving a uniform version of Tits' alternative and various applications of it to the theory of amenability, Riemannian foliations and growth of groups. |
| Abstract: A Poincare space has a ``failure-of-local-duality'' chain defining an element in homology with L coefficients. A homology manifold in this homotopy type is determined by a trivialization of this chain. Versions of this are due to Bryant-Ferry-Mio-Weinberger and Ranicki, and this is the space level of Sullivan's characterization of topological bundles as L-oriented spherical fibrations. We describe a new approach to the proof, and speculate about a version for smooth manifolds. |
| Abstract: We will discuss how a bounded cohomological invariant can be used to study group actions on Hermitian symmetric spaces of noncompact type. The main focus will lie on surface groups, where the invariant singles out certain connected components of the moduli space of representations. We will explain how this connected components may be viewed as generalizations of Teichmueller space. |
| Abstract: I will discuss two metric geometric notions: horoballs, which are limits of balls, and stars which are limits of halfspaces. Horoballs play for example a role in a conjectural law of large numbers for random walks on groups and the dynamics of semicontractions. Stars associate a generalized Tits geometry at infinity of metric spaces. I will discuss their relevance for the dynamics of isometries with consequences for the existence of free subgroups, a metric analog of Furstenberg's lemma, the Dirichlet problem at infinity etc. This gives in particular new results for CAT(0)-geometry, holomorphic maps, and Teichmuller theory. |