Maximal representations form unexpected connected components of the character variety of the fundamental group of a hyperbolic surface in a semisimple Lie group, that only consist of injective homomorphsims with discrete image. They thus generalize the Teichmüller space, and can be thought of as parametrizing certain locally symmetric spaces of infinite volume. After a general introduction to character varieties and maximal representations, I will discuss joint work with Andres Sambarino and Anna Wienhard in which we prove a sharp upper bound for the exponential orbit growth rate of the associated actions on the symmetric space.
Maximal representations of complex hyperbolic lattices
Maria Beatrice Pozzetti (Ruprecht-Karls-Universität Heidelberg)
MSRI: Simons Auditorium
There are natural incidence structures on the boundary of the complex hyperbolic space and on some suitable boundary S associated to the group PU(m,n). Such structures have striking rigidity properties: I will prove that a (measurable) map from the boundary of the complex hyperbolic space to S that preserves these incidence structures needs to be algebraic. This implies that, if G is a lattice in SU(1,p) and n is greater than m, there exist Zariski dense maximal representations of G in SU(m,n) only if (m,n) is equal to (1,p). In particular the restriction to G of the diagonal embedding of SU(1,p) in SU(m,pm+k) is locally rigid.
A symmetric space is Hermitian if it admits a complex structure preserved by the isometry group. In this introductory talk I will describe various geometric features of these spaces. I will focus particularly on some natural notion of boundary arising in this context, and emphasize the role played by these objects in studying rigidity questions.