Back to WTC Winter 2005

Daryl Cooper
Covers of 3-manifolds and the group determinant
We will discuss the question of when a compact 3-manifold with torus boundary has a Dehn-filling with large first betti number, and related questions. We will apply a new tool from the representation theory of finite groups: a symmetrized form of the group-determinant investigated by Dedekind and Frobenius. This is joint work with Genevieve Walsh.

Cameron Gordon
Toroidal Dehn fillings
We classify all hyperbolic 3-manifolds admitting a pair of Dehn fillings with intersection number at least 4 that yield toroidal manifolds. As a corollary we determine all hyperbolic knots in S^3 whose exteriors have such a pair of fillings. This is joint work with Ying-Qing Wu.

Cyril Lecuire

Sequences of Kleinian groups
We consider a hyperbolic 3-manifold with boundary and we study sequences of faithful discrete representations of its fundamental group in the group of isometry of the hyperbolic space. We will discuss a joint work with J.Anderson. The object of this work is to study relations between the behaviour of a sequence of representations and the behaviour of some invariants associated to each representation, as its bending measured geodesic lamination.

Misha Kapovich
Real projective Gromov-Thurston examples
In 1987 Gromov and Thurston constructed examples of negatively curved compact n-manifolds M (n > 3) for which the curvature pinching constant is abritrarily small, but M do not admit metrics of constant negative curvature. Recently, Yves Benoist proved that if M is a compact (real) projective manifold with convex universal cover then the fundamental group of M is Gromov-hyperbolic if and only if the universal cover is strictly convex. In this talk I will show that some of the Gromov-Thurston examples admit projective structure with convex (and therefore strictly convex) universal cover.

Steve Kerckhoff
Degenerating geometric structures via projective geometry
This will be a discussion of the process of rescaling degenerating geometric structures in dimension 3, particularly the transition between hyperbolic, euclidean, and spherical geometry, from the point of view of projective geometry.  Applications to the Orbifold Theorem and global and local rigidity of cone manifolds. This is joint work with Daryl Cooper.

Bruce Kleiner
Higher dimensional analogs of Gromov hyperbolicity

Ben McReynolds
Constructing isospectral manifolds
I will discuss a new construction of isospectral manifolds which combines work of Spatzier and Brooks-Gornet-Gustafson. The result of this venture is the production of large numbers of isospectral, nonisometric, locally symmetric manifolds and pairs of infinite isospectral towers of finite covers of such manifolds. This provides the first examples of isospectral nonisometric manifolds in many settings and new examples in every setting.

Saul Schleimer
A metric survey of curve complexes
I will review the work of Masur and Minsky on the geometric structure of the curve complex.  Their ideas generalize to many other "combinatorial moduli spaces" such as the arc complex, the disk complex, and others. We'll end with a large collection of conjectures, some proven, some not proven, detailing the coarse metric structure of such generalized curve complexes. (joint work with H. Masur)

Jennifer Schultens
Destabilizing amalgamated Heegard splittings
We investigate the behavior of Heegaard splittings under gluings of manifolds along boundary components.  We discuss inequalities relating the Heegaard genera involved and how strict inequalities arise as a result of destabilizations after amalgamation.

Karen Vogtmann
Tethers and Homology stability
The homology of many natural sequences of groups {G_n} is stable, in the sense that H_i(G_n) is independent of n for n sufficiently large with respect to i.  This has been established for braid groups, mapping class groups, and automorphism groups of free groups, among many others.  The standard way of proving homology stability theorems is to find a highly-connected complex on which the group G_n acts so that the lower rank groups G_i appear as cell stabilizers,  then to work with the equivariant homology spectral sequence arising from this action.

We will explain "tethered" variations of standard complexes which result in simpler proofs of homology stability for the groups mentioned above.  Similar complexes can also be used to establish homology stability for new classes of groups. This is joint work with Allen Hatcher.