Geometric Structures and Varieties of Representations

Many interesting geometric structures on manifolds can be interpreted as structures locally modelled on homogeneous spaces. Given a homogeneous space (X,G) and a manifold M , there is a deformation space of structures on M locally modelled on the geometry of X invariant under G. Such a geometric structure on a manifold M determines a representation (unique up to inner automorphism) of the fundamental group π of M in G. The deformation space for such structures is “locally modelled” on the space Hom(π,G)/G of equivalence classes of representations of π → G. A strong interplay exists between the local and global structure of the variety of representations and the corresponding geometric structures. The lecture in Boulder surveyed some aspects of this correspondence, focusing on: (1) the “Deformation Theorem” relating deformation spaces of geometric structures to the space of representations; (2) representations of surface groups in SL(2;R), hyperbolic structures on surfaces (with singularities), Fenchel-Nielsen coordinates on Teichmüller space; (3) convex real projective structures on surfaces; (4) representations of Schwarz triangle groups in SL(3;C). This paper represents an expanded version of the lecture.