Topology and Dynamics of Moduli Spaces of Geometric Structures

9803518流形M上的局部齐次几何结构是M的开子集上具有齐次空间G/H上的值的局部坐标系。在M的重叠面片上，坐标通过李群G中的变换联系在一起。在Cartan和Ehresmann之后，坐标图全球化为M的泛覆盖空间U到G/H的“发展”，定义了M的基本群F的同态(实现为U的甲板变换群)。一种是以G/H为模型，在M上形成几何结构的模空间Def(M，G/H)，它将结构分类为等价的。Def(M，G/H)本身局部同胚到同态等价类的模空间Hom(F，G)/G。这些模空间的原型是TeichMuller空间，对双曲结构进行分类(这里G/H是实双曲平面，G=PSL(2，R))。此外，M的映射类群同时作用于Def(M，G/H)和Hom(F，G)/G，且Def(M，G/H)-Hom(F，G)/G的完整映射关于这两个作用是等变的。本文主要研究当F是紧致曲面的基本群时的情形。当M是闭曲面时，这些模空间具有不变的辛结构，当G是紧的时，研究者猜想在映射类群的作用下辛测度是遍历的。当G是SL(2，R)时，这个模空间与(可能是奇异的)双曲结构的TeichMuller空间有关，并且研究者利用几何结构研究了模空间上的动力学。这一动力学研究将离散映射类群的作用与模空间上的某些哈密顿流联系起来。当G/H是Minkowski 2+1-空间时，研究者(与Drumm合作)利用Drumm的命题和Marguis的思想研究了完备平坦流形的形变空间。当G/H是复超2-空间时，夏对Hom(F，G)/G的连通分支进行了分类，研究者(与Leeb和Kapovich合作)在每个分支中发现了离散嵌入。当G/H为实射影平面时，研究者的工作结合Choi的工作给出了形变空间Def(M，G/H)的完整描述。这样的几何结构代表了一种将拓扑和几何联系起来的方式。根据Felix Klein的Erlangen程序，几何学是研究李群G的齐次空间G/H中的对象之间的关系，这些对象在G中的变换下是不变的。它们包括诸如距离、角度、平行度和共线等“刚性”几何性质。另一方面，拓扑学研究流形M上的点的松散组织，其中唯一的关系来自连续映射。M上的一种几何结构更严格地通过从M中的开集到G/H的局部坐标图来组织它的点.模空间将这些结构的等价类参数化，并与M到G的基本群同态空间密切相关.这些模空间本身具有丰富的几何结构，并表现出作为自然几何对象起源而产生的不寻常的对称性.对这些基本天体的更好理解一次又一次地通过在相关领域的洞察来回报这一努力，通常是在理论物理方面。***

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. ***