Geometric Structures and Character Varieties

December 28 ~ 30, 2015, Jeju National University

Main       Schedule and Abstracts       Travel       Registration

The ends of convex real projective manifolds and orbifolds

Suhyoung Choi (KAIST)

A real projective structure on a manifold or an orbifold is given by locally modeling the space by pieces of a real projective space and gluing with projective patching maps. Hyperbolic manifolds are examples. We will give some introductory survey (but not including the deformation spaces). Then we will study convex real projective structures with radial or totally geodesic ends. We will discuss how to classify some reasonable collections of these types of ends. (Ballas, Cooper, Danciger, Long, and Tillman also are studying ends with analogous conditions.)

Primitive stable representations in higher rank semi-simple Lie groups

Sungwoon Kim (Jeju National University)

We define primitive stable representations of free groups into higher rank semisimple Lie groups and study their properties. Then we show that the holonomies of convex projective structures of finite Hilbert volume on a compact surface with one boundary component are primitive stable. Moreover, we will discuss about the regularity of positive representations introduced by Fock and Goncharov.

On Schottky groups of rank two

Jaejeong Lee (KIAS)

The first explicit example of non-classical Schottky group was found by Yamamoto in 1991. I will exhibit his example and explore some ideas to study two-generator subgroups of $PSL(2,\mathbb{C})$ in general.

Differential geometry of convex projective structures

Xin Nie (KIAS)

We will survey the differential-geometric aspect of convex real projective projective structures and cyclic Higgs bundles, with emphasis on Wang's equation and more generally, affine Toda equations, which are particular cases of Hitchin's self-duality equation. We will present some open problems, whose difficulty lies in the analysis of these equations.

On the $SL(2,\mathbb{C})$ character variety of the rank two free group (I and II)

Ser Peow Tan (National University of Singapore)

The $SL(2,{\mathbb C})$ character variety of the rank 2 free group $F_2$ can be identified with ${\mathbb C}^3$ on which $Out(F_2)$ acts by polynomial automorphisms generated by the Vieta moves. There are two important subsets of the character variety: the first is defined geometrically and corresponds to discrete, faithful, geometrically finite representations, and the second, defined dynamically corresponds to representations which satisfy some conditions defined by Bowditch, and on which $Out(F_2)$ acts properly discontinuously. We will discuss various aspects of the geometry and dynamics of these subsets, and various related subsets like the diagonal slice. In particular, we will discuss the relations between these two sets under certain restrictions, a description of these sets in the diagonal slice, and identities satisfied by representations satisfying the Bowditch conditions.

Pressure metric on the Teichmuller space of surfaces with boundary

Binbin Xu (KIAS)

Let $S$ be an oriented compact surface with negative Euler characteristic. Teichmuller space of $S$ is the space of isotopy classes of marked hyperbolic structures on $S$. A well-known Riemannian metric on the Teichmuller space is the Weil-Petersson metric. It has many interesting properties, for example: Kahler, negatively curved, incomplete, geodesically convex, etc. When $S$ is closed, this metric can be interpreted as the pressure metric by using thermordynamic formalism. When $S$ has boundary, both of these two metrics are still well-defined, but the relation between them are not known yet. By studying the incompleteness of the pressure metric in the second case, we are capable to answer that they are not equivalent to each other.