Cavendish - Finite-sheeted covers of 3-manifolds and the Cohomology of Solenoids

The study of finite-sheeted covering spaces of 3-manifolds has been invigorated in recent years by the resolution of several long-standing conjectures by Kahn-Markovic, Agol and Wise.  In this talk, I will discuss how using this work one can reformulate some of the central open questions in the field in terms of objects called solenoids. These objects are formed by taking inverse limits of families of finite-sheeted covering spaces of a compact manifold M, and they can be thought of as pro-finite generalization of finite-sheeted covering spaces of M.  While such an object can in general be quite complicated, I will show in this talk that if M is a compact aspherical 3-manifold, then the solenoid given by taking the inverse limit of the family of all finite-sheeted connected covering spaces of M looks like a disk from the perspective of Cech cohomology with coefficients in any finite module. I will then talk about the relevance of this result to elementary questions about finite-sheeted covers.

Danciger - A geometric transition from hyperbolic to AdS geometry

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. A particular case of interest is that of hyperbolic cone structures that collapse as the cone angle approaches 2π. In this case, the AdS manifolds on the ''other side'' of the transition have so-called tachyon singularities. We will discuss a general theorem about transitions and then construct examples using ideal tetrahedra.

Dumas - Real and complex boundaries in the character variety

The set of holonomy representations of complex projective structures on a compact Riemann surface is a submanifold of the SL(2,C) character variety of the fundamental group. We will discuss the real and complex-analytic geometry of this manifold and its interaction with the Morgan-Shalen compactification of the character variety. In particular we show that the subset consisting of holonomy representations that extend over a given hyperbolic 3-manifold group (of which the surface is an incompressible boundary) is discrete.

Dymarz - Rigidity and enveloping discrete groups by locally compact groups

If a finitely generated group G is a lattice in a locally compact group H then we say that H envelopes G. This terminology was introduced in the sixties by Furstenberg, who also proposed studying the problem of which locally compact groups can envelope which discrete groups. In the case when G is a lattice in a semisimple Lie group and H is a semisimple Lie group, the problem is solved by celebrated rigidity results of Mostow, Prasad and Margulis that show that there is only one possible envelope for each such lattice. In contrast we focus on classes of groups which are not lattices in any Lie group but do sit as lattices in the isometry groups of nice metric complexes. We show how in certain cases techniques from quasi-isometric rigidity can be used to give rigidity results.

Kapovich - On spectrally rigid subsets of free groups

It is well-known that any tree T in the (unprojectivized) Culler-Vogtmann Outer space cvN is uniquely determined by the translation length function (also known as the marked length spectrum) of T, ||.||T : FN → [0,∞). Here for g FN ||g||T = infxT dT (x, gx).

We say that a subset SFN is spectrally rigid in FN if whenever T,T'cvN are such that ||g||T = ||g||T' for every gS then T = T' in cvN. By contrast to similar questions for the Teichmüller space, it is known that for N ≥ 2 there does not exist a finite spectrally rigid subset of FN.

We will discuss known results and open problems about spectral rigidity and nonspectral rigidity of various “natural” infinite subsets of FN.

Young - Higher-order filling functions in solvable groups

Spaces with nonpositive curvature have many geometrical properties which can often be used to control the geometry of their subsets. Such subsets can be used, for example, to construct groups with unusual finiteness properties, like Stallings' group.  Similar constructions result in groups with unusual filling functions.  In this talk, we will describe these constructions, discuss how nonpositive curvature influences the geometry of fillings in solvable groups and sketch new ways of bounding their filling functions.

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