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.
Groups acting on the circle
Kathryn Mann (University of California, Berkeley)
MSRI: Simons Auditorium
This talk will introduce you to the study of groups acting on the circle and the moduli spaces of such actions. We'll discuss the role of these "moduli spaces" in geometry and topology, and survey key techniques, results, and open problems.