Symplectic groups over noncommutative rings and maximal representations.

Maximal representations into Lie groups of Hermitian type have been introduced in [7], and further studied in [2,6,26]. All maximal representation are discrete embeddings, and spaces of maximal representations are unions of connected components of the character varieties, hence they provide examples of so-called higher Teichmüller spaces. Connected components of spaces of maximal representations have complicated topology which is not well understood.

In this thesis, we study classical Hermitian Lie groups of tube type and give a parametrization of spaces of decorated (maximal) representations of the fundamental group of a punctured surface into a Hermitian Lie group of tube type. Using this parametrization, we describe the topology and the structure of the spaces of maximal representations.

In the first chapter, we introduce coordinates on the space of Lagrangian decorated representations of the fundamental group of a surface with punctures into the sym- plectic group Sp(2n, R). These coordinates provide a noncommutative generalization of the parametrization of the space of representations into SL(2, R) given by V. Fock and A. Goncharov. The locus of positive coordinates maps to the space of decorated maximal representations. We use this to determine the homotopy type and the homeomorphism type of the space of decorated maximal representations, and when n = 2, to describe its finer structure as a smooth locus and kind of singularities.

In the second chapter, we study Hermitian Lie groups of tube type and their complexifications uniformly as Sp2(A) over some special real algebra A. We use this approach to describe the flag variety of such groups corresponding to a maximal parabolic subgroup, a maximal compact subgroup and different models of the sym- metric space. For complexified groups this construction is new. Further, we introduce in these terms coordinates on the space of decorated maximal representations of the fundamental group of a punctured surface into a Hermitian Lie group of tube type and use them to determine the homotopy type and the homeomorphism type of the space of decorated maximal representations.