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 symplectic 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 symmetric 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

Noncommutative coordinates for symplectic representations

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with boundary into the symplectic group Sp(2n,R). These coordinates provide a noncommutative generalization of the parameterizations of the spaces of representations into SL(2,R) or PSL(2,R) given by Thurston, Penner, Kashaev and Fock-Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group O(n). This allows us to describe the homotopy type and, when n=2, to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.Comment: 80 page