Geometric structures on manifolds and varieties of representations. (English) Zbl 0659.57004

Goldman, William M.

Geometry of group representations. Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo. 1987, Contemp. Math. 74, 169-198 (1988).

[For the entire collection see Zbl 0651.00007.]
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 \(\pi\) of M in G. The deformation space for such structures is “locally modelled” on the space Hom(\(\pi\),G)/G of equivalence classes of representations of \(\pi\) \(\to G\). 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 structure 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.

MSC:

57N05	Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57N10	Topology of general \(3\)-manifolds (MSC2010)
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
53C99	Global differential geometry
30F20	Classification theory of Riemann surfaces

Keywords:

representation of the fundamental group; deformation space for structures; geometric structures on manifolds; structures locally modelled on homogeneous spaces; variety of representations; representations of surface groups in SL(2; R); hyperbolic structure on surfaces; Fenchel-Nielsen coordinates on Teichmüller space; convex real projective structures on surfaces; representations of Schwarz triangle groups in SL(3; C)

Citations:

Zbl 0651.00007