Surface group representations with maximal Toledo invariant

Details Surface group representations with maximal Toledo invariant Surface group representations with maximal Toledo invariant

2006-05-24

arxiv.org/abs/math/0605656v3

Mathematics

Differential Geometry

56 pages, no figures. The paper has been completely rewritten and will appear in Annals of Math

Scientific paper

We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations. We thus obtain geometric properties of the maximalrepresentations, that is representations whose Toledo invariant achieves the maximum value: we show that maximal representations have discrete image, are faithful and completely reducible and they always preserve a maximal tube type subdomain. This extends to the case of a general Hermitian group some of the properties of the representations in Teichmuller space, as well as results due to Goldman, Toledo, Hernandez, Bradlow--Garcia-Prada--Gothen. An announcement of these results in the case of surfaces without boundary -- where the role of tube type domains had already been emphasized -- appeared in 2003 by the same authors. The congruence relations for the Toledo invariant involve a rotation number function related to a continuous homogeneous quasimorphism which gives an explicit way to compute the Toledo invariant. This rotation number generalizes constructions due to Ghys, Barge--Ghys, and Clerc--Koufany. We establish moreover properties of boundary maps associated to maximal representations which generalize naturally, for the causal structure of the Shilov boundary, monotonicity properties of quasiconjugations of the circle. This, together with the congruence relations leads to the result that the subset of maximal representations is always real semialgebraic.

Affiliated with

Also associated with

No associations

Rating

Surface group representations with maximal Toledo invariant does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Surface group representations with maximal Toledo invariant, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Surface group representations with maximal Toledo invariant will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-119332