11/16/2005

A. Goncharov
Brown University

Higher Teichmuller theory

Let S be a topological surface. The classical Teichmuller space parametrises complex structures on S. Thanks to the Poincare uniformization theorem, it can be viewed (barring few trivial cases) as the moduli space of faithful discrete representations of the fundamental group of S to PGL(2, R).

We develope a more explicit approach to the classical
Teichmuller-Thurston  theory. It allows us to show that a big part of this theory admits a generalization where PGL(2, R) is replaced by any semi-simple real split Lie group G with trivial center, e.g. PGL(n, R).

We define the higher Teichmuller space and show that it parametrises faithful discrete representations of the fundamental group of S to G. We find a collection of explicit parametrizations of this space. Using them we define its quantum version, and  construct an infinite dimensional unitary representation of the mapping class group of S. This is a joint work with V. Fock