Hyperbolic Geometry and Combinatorial Surface Topology

PI: Yair Minsky题目：双曲几何和组合曲面提案：9971596拓扑摘要：Minsky将研究双曲3-流形（Kleinian群）和曲面的映射类群的相关理论中的刚性、变形和分类现象。在Kleinian群中，Minsky计划继续研究Thurston的end LaminationConjecture，该猜想指出双曲3流形是由其拓扑类型和描述其端点的渐近几何的不变量列表唯一决定的。这一猜想的结果是球上Kleinian群作用的刚性定理，以及同构Kleinian群的参数空间的描述。在与H. Masur和B. Farb的合作中，Minsky计划考虑曲面的映射类群的准等距刚性问题，这表明该群的所有准等距实际上必须尊重其群结构直至有界误差。这两组问题的通用工具是曲面上的“曲线复合体”，它是对曲面上简单曲线的同位素类集合的组合结构进行编码的简单复合体。Masur和Minsky先前已经建立了这个复合体的双曲性，并将其应用于映射类群的共轭问题。Minsky已经发现了这种复合体的几何性质与表面群的Kleinian表示之间的明确联系，这项工作应该可以应用于结束层压问题。他还将与R. Canary和J. Brock共同研究Kleinian群的几何极限结构，这可能非常复杂。对所有几何极限集合的良好描述是获得单个群的一致估计的重要工具。一百多年来，几何、拓扑和动力学之间的相互作用一直是数学和物理的一个美丽而强大的特征。动力学是对数学或物理系统的时间演化的研究，而几何学和拓扑学涉及“静态”对象，如表面或高维类似物，通常是动态过程的背景。亨利·庞加莱已经知道，标准的圆球，经典分析和几何的背景，也可以作为一种奇异的非欧几里得几何的“无限视界”，我们现在称之为双曲空间。球体变换的动力学性质转化为该空间刚性运动的几何性质，并产生对称瓷砖族，其结构可以用几何和拓扑方法研究。这些系统的复杂性对它们的约束如此之大，以至于组合（或拓扑）描述足以唯一地确定它们，这就是我们所说的刚性。这种现象在整个几何和动力学中以多种形式出现，并且与系统分类、绘制稳定和不稳定区域、系统族的变形和分岔以及遍历等概率性质等问题相关，所有这些在纯数学和应用数学中都具有重要意义。本项目所研究的特定方面在某些方面具有典型性，在另一些方面又具有特殊性。通过举例和组织原则的制定，更好地理解它们有望为该领域的其他部分提供启示。

PI: Yair MinskyTitle: Hyperbolic Geometry and Combinatorial Surface Proposal: 9971596TopologyAbstract: Minsky will investigate phenomena of rigidity, deformation and classification in the related theories of hyperbolic 3-manifolds (Kleinian groups) and mapping class groups of surfaces. In Kleinian groups Minsky plans to continue work on Thurston's Ending LaminationConjecture, which states that a hyperbolic 3-manifold is uniquely determined by its topological type and a list of invariants that describe the asymptotic geometry of its ends. Consequences of this conjecture are a rigidity theorem for Kleinian group actions on the sphere, and a description of parameter spaces of isomorphic Kleinian groups. In collaboration with H. Masur and B. Farb, Minsky plans to consider the question of quasi-isometric rigidity of the mapping class group of a surface, which says that all quasi-isometries of this group must in fact respect its group structure up to a bounded error. A common tool to both sets of questions is the "complex of curves" on a surface, which is a simplicial complex encoding the combinatorial structure of the set of isotopy classes of simple curves on the surface. Masur and Minsky have previously established a hyperbolicity property for this complex, with applications to the conjugacy problem for mapping class groups. Minsky has found explicit connections between geometric properties of this complex and those of Kleinian representations of surface groups, and this work should have applications to the ending lamination problem. Jointly with R. Canary and J. Brock, he will also investigate the structure of geometric limits of Kleinian groups, which can be quite intricate. A good description of the set of all geometric limits is an important tool for obtaining uniform estimates on individual groups.The interactions between geometry, topology and dynamics have been a beautiful and powerful feature of mathematics and physics for more than a hundred years. Dynamics is the study of time-evolution of mathematical or physical systems, whereas geometry and topology involve "static" objects such as surfaces or higher-dimensional analogues, often the background for a dynamical process. Henri Poincare already knew that the standard round sphere, the setting of classical analysis and geometry, functioned also as a "horizon at infinity" for an exotic non-Euclidean geometry that we now call Hyperbolic space. Dynamical properties of transformations of the sphere translate to geometric properties of rigid motions of this space, and give rise to families of symmetric tilings whose structure we can study by geometric and topological methods. The complexity of these systems can constrain them so much that a combinatorial (or topological) description suffices to determine them uniquely, and this is what we call rigidity. This phenomenon occurs in many guises throughout geometry and dynamics, and is relevant to issues such as classification of systems, mapping out regions of stability and instability, deformation and bifurcation of families of systems, and probabilistic properties such as ergodicity, all of which have significance in both pure and applied mathematics. The particular aspects studied in this project are typical in some ways and special in others. A better understanding of them promises to shed light on other parts of the field, both by way of examples and by the formulation of organizing principles.