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
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.