Topology and Dynamics of Moduli Spaces of Geometric Structures

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9803518 Goldman A locally homogeneous geometric structure on a manifold M is a system of local coordinates on open subsets of M with values in a homogeneous space G/H. On overlapping patches of M, the coordinates are related by transformations in the Lie group G. Following Cartan and Ehresmann, the coordinate charts globalize to a "development" of the universal covering space U of M into G/H, defining a homomorphism of the fundamental group F of M (realized as the group of deck transformations of U). One forms a moduli space Def(M,G/H) of geometric structures on M modelled on G/H which classifies structures up to equivalence. Def(M,G/H) is itself locally homeomorphic to a moduli space of equivalence classes of homomorphisms Hom(F,G)/G. The prototype of these moduli spaces is the Teichmuller space, classifying hyperbolic structures (here G/H is the real hyperbolic plane and G = PSL(2,R)). Moreover the mapping class group of M acts on both Def(M,G/H) and Hom(F,G)/G, and the holonomy map from Def(M,G/H) - Hom(F,G)/G is equivariant with respect to these actions. This investigation mainly concentrates on the case when F is the fundamental group of a compact surface. When M is a closed surface, these moduli spaces carry invariant symplectic structures, and when G is compact the investigator conjectures that the symplectic measure is ergodic under the action of the mapping class group. When G is SL(2,R), this moduli space is related to Teichmuller space of (possibly singular) hyperbolic structures, and the investigator has used geometric structures to study the dynamics on the moduli space. This dynamical study relates the action of the discrete mapping class group to certain Hamiltonian flows on the moduli space. When G/H is Minkowski 2+1-space, the investigator (in collaboration with Drumm) is studying the deformation spaces of complete flat manifolds, using "crooked planes" developed in Drumm's thesis and ideas of Margulis. When G/H is complex hyperbo lic 2-space, Xia has classified the connected components of Hom(F,G)/G and the investigator (in collaboration with Leeb and Kapovich) has found discrete embeddings in each component. When G/H is the real projective plane, the work of the investigator combined with that of Choi gives a complete description of the deformation space Def(M,G/H). Such geometric structures represent a way of relating topology and geometry. According to Felix Klein's Erlangen program, geometry is the study of relations between objects in a homogeneous space G/H of a Lie group G that are invariant under the transformations in G. They include "rigid" geometric properties such as distance, angle, parallelism and collinearity. On the other hand, topology studies the loose organization of points on a manifold M, where the only relations derive from continuous mapping. A geometric structure on M more rigidly organizes its points through local coordinate charts from open sets in M to G/H. Moduli spaces parametrize equivalence classes of these structures and closely relate to spaces of homomorphisms of the fundamental group of M into G. These moduli spaces themselves enjoy rich geometric structures of their own, and display unusual symmetries arising from their origin as natural geometric objects. A better understanding of such basic objects has time and again repaid the effort through insights in related fields, typically in theoretical physics. ***