Abstract

This is the first of four Leverhulme lectures on ``Higher Teichmueller theory”, whose aim is to single out connected components of the G-representation variety of the fundamental group of a compact surface S which are formed of representations with geometric significance.

For G = PSl(2,R) the component of interest, Teichmueller space, is formed by all holonomy representations of hyperbolic structures on S. We’ll describe two characterizations of Teichmueller space, one by Fenchel Nielsen coordinates which for G = PSl(n,R) leads to the Hitchin component, and the other by the maximality of the Euler number which for G = Sp(2n,R) leads to the components formed by maximal representations. From the point of view of geometric group theory, in all cases these representations give rise to quasi-isometric embeddings and this connection is provided by the concept of Anosov representation.