In Summer 2007, the seminar was organised by Javier Aramayona and Samuel Lelièvre.
(note unusual day and time: Friday at 11:30)
Cliff Earle (Cornell)
Inequalities for the Poincaré and Carathéodory metrics in unit balls
Abstract: In Poincaré's unit disk model of the hyperbolic plane, consider a geodesic segment of given (hyperbolic) length. Where should its center be placed in order to maximize the Euclidean distance between its endpoints? The obvious answer to that question can be expressed by an elementary analytic inequality that is a special case of an inequality about the Carathéodory metric in the unit ball of any complex Banach space. Larry Harris and I have examined the general case and studied how the conditions for the inequality to be strict depend on the shape of the unit ball. This talk describes our results.
followed by a Coull Quartet concert in the Common Room at 15:00
(which is why we're starting at 13:45)
Jane Gilman (Rutgers)
Finding parabolic dust and the structure of two-parabolic space
Abstract: Every non-elementary subgroup of PSL(2;C) generated by two parabolic transformations is determined up to conjugacy by a non-zero complex number, lambda, and each nonzero lambda determines a two-parabolic generator group. I will discuss a structure theorem for two-parabolic space, that is, for the full representation space (modulo conjugacy) of non-elementary two-parabolic generator groups. Portions of two-parabolic space have been studied by many authors including Bamberg, Beardon, Lyndon-Ullman, Keen-Series, Minsky, Riley, Wright, and Gilman-Waterman. These include portions corresponding to free and non-free groups, discrete and non-discrete groups, manifold and orbifold groups, Riley groups, NSDC groups and classical and non-classical tangent Schottky groups. I will discuss locating non-free parabolic dust, that is, groups with additional parabolics, by iterating the Gilman-Waterman classical tangent Schottky boundary.
Ken'ichi Ohshika (Osaka)
Divergence and exotic convergence of quasi-Fuchsian groups
One of the goals in the Kleinian group theory is to understand the
topological structure of deformation spaces.
Understanding convergence and divergence of sequence of Kleinian groups is an important topic in this direction.
In this talk, I shall give a sufficient condition for quasi-Fuchsian groups to diverge in the entire deformation space, which seems to have the most general form in terms of the Thurston compactification of the Teichmuller space.
I shall also discuss when the exotic convergence to b-groups can or cannot happen.
Daan Krammer (Warwick)
A Garside type structure on the framed mapping class group
Abstract: I begin by a pictorial introduction to Garside graphs
which I hope is appealing. A Garside graph is a combinatorial version of
(a convex subset of) a real vector space; and an action of a group on it is
a combinatorial version of a linear representation.
I construct a Garside graph on which the framed mapping class group acts faithfully, that is, the subgroup of a surface mapping class group of those elements preserving a nowhere vanishing vector field up to homotopy. It uses laminations on surfaces which I shall recall. Many precise questions remain as to how good the Garside structure is.
(Note change in date and time. Usual room.)
A two-dimensional slice through the parameter space of two-generator Kleinian groups
Abstract: The aim of this talk is to describe all real points of the parameter space of two-generator Kleinian groups with one generator parabolic, that is, to describe a certain two-dimensional slice through this space. I shall also discuss the structure and some interesting properties of the slice.
Caroline Series (Warwick)
Lines of minima are Teichmüller quasi-geodesics
Abstract: We describe Rafi's method of estimating the length of short curves along a Teichmüller geodesic and explain how this can be combined with results of Choi, Rafi and Series to prove that Kerckhoff lines of minima are Teichmüller quasi-geodesics.
Tan Ser Peow (Singapore)
Dynamics of the mapping class group action on the SL(2,C) character variety of the one-holed torus, I
Yasushi Yamashita (Nara Women's University)
Dynamics of the mapping class group action on the SL(2,C) character variety of the one-holed torus, II
Abstract for parts I and II: The SL(2,C) character variety X of a one-holed torus can be identified with complex 3-dimensional space C^3. The mapping class group of the torus acts naturally on the character variety as polynomial automorphisms of C^3. We study the dynamics of this action.
Nikolay Abrosimov (Novosibirsk)
Seidel problem on the volume of hyperbolic tetrahedra
Abstract: In 1986 J.J. Seidel formulated the hypothesis that the volume of an ideal tetrahedron can be expressed by a function of the determinant and the permanent of its Gram matrix and he also supposed that this function should be monotonous in each of its two arguments. Recently more powerful question was raised: is it correct that the volume of a tetrahedron (hyperbolic or spherical) can be expressed by a function in terms of the determinant of its Gram matrix? We give a negative answer for this question in both spherical and hyperbolic cases. We also consider initial Seidel's conjecture and prove it in more precise formulation.
Hugo Akrout (Montpellier)
Voronoi theory in Teichmüller spaces
Abstract:Voronoi theory for lattices is well understood, and gives some important results about SL(n,Z). An equivalent theory in the setting of Teichmuller spaces should improve our understanding of the action of the mapping class group. After recalling some Voronoi theory and essential results in both settings, we will illustrate through an example that things are not as simple as one could imagine in the Teichmuller spaces setting.