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*

**Abstract:**
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.)

**Natasha Kopteva**

*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:**

Programme of the seminar in past years: 2005-2006; 2004-2005; 2003-2004.