3-manifold geometry and topology

这个项目提出了几种研究双曲三维流形的拓扑学和几何学的方法。主要内容是研究三维流形中的浸没曲面、双曲三维流形之间的映射以及双曲三维流形的体积。该项目最雄心勃勃的方面涉及虚拟Haken猜想，这相当于证明了双曲3-流形有一个有限片覆盖，其中包含一个嵌入的pi_1-内射曲面。PI想把这与瑟斯顿的虚拟纤化猜想联系起来，该猜想指出双曲三维流形有一个有限薄片覆盖，它纤化在圆上，并暗示了虚拟哈肯猜想。PI证明了Haken 3-流形同伦等价于紧致CAT(0)立方体复形。接下来，PI想要证明双曲3-流形的基本群是LERF。如果格罗莫夫-双曲群是剩余有限的，那么这将是必然的，尽管许多几何群论者认为这是错误的。根据Haglund和Wise的程序，这些猜想将意味着Haken双曲流形有基本群，它实际上嵌入到直角反射群中。这反过来又意味着这些流形实际上是PI先前工作的纤维。PI还计划调查双曲Haken流形的体积，以及流形上的各种拓扑限制如何引起体积的下界。例如，PI希望了解具有n个顶点的可定向双曲3-流形的体积的渐近行为。对3维空间的数学研究可以追溯到Poincare的工作。因为经典物理学将我们的宇宙描述为三维空间，所以对三维空间的分类是一项重要的数学努力，因为它可能会对我们宇宙的全球结构产生影响。最近，佩雷尔曼惊人地实现了这种分类，他使用汉密尔顿的利玛奇流解决了瑟斯顿的几何化猜想，并由此得出了100年前的庞加莱猜想。几何化猜想意味着任何有限的三维空间都有一个几何片的标准分解，其中最有趣的是双曲三维流形。研究3-流形的数学家们仍在理清佩雷尔曼的工作和几何化猜想的含义。这个项目将探索几何化猜想的各个方面的分支。主要的也是最雄心勃勃的项目将研究三维空间中的二维对象，特别是满足某些强拓扑限制的双曲对象。研究这些2维物体对于有限的3维空间的整体结构具有重要意义。该项目还旨在研究双曲三维流形的几何，特别是了解最简单的此类流形及其几何性质(如体积)与其拓扑性质(如边界的多少个分量)之间的关系。

This project proposes several avenues of study relating the topology and geometry of hyperbolic 3-manifolds. The principal topics are the study of immersed surfaces in 3-manifolds, maps between hyperbolic 3-manifolds, and volumes of hyperbolic 3-manifolds. The most ambitious aspect of the project involves the virtual Haken conjecture, which is equivalent to showing that hyperbolic 3-manifolds have a finite-sheeted cover which contains an embedded pi_1-injective surface. The PI would like to relate this to the virtual fibering conjecture of Thurston, which states that a hyperbolic 3-manifold has a finite-sheeted cover which fibers over the circle, and implies the virtual Haken conjecture. The PI would like to show that Haken 3-manifolds are homotopy equivalent to a compact CAT(0) cube complex. Next, the PI would like to show that fundamental groups of hyperbolic 3-manifolds are LERF. This would follow if Gromov-hyperbolic groups are residually finite, although this is believed to be false by many geometric group theorists. Following a program of Haglund and Wise, these conjectures would imply that a Haken hyperbolic manifold has fundamental group which virtually embeds in a right-angled reflection group. This in turn would imply that these manifolds virtually fiber by previous work of the PI. The PI also plans to investigate the volumes of hyperbolic Haken manifolds, and how various topological restrictions on the manifold give rise to lower bounds on volume. For example, the PI would like to understand the asymptotic behavior of the volumes of orientable hyperbolic 3-manifolds with n cusps.The mathematical study of 3-dimensional spaces goes back to the work of Poincare. Because classical physics describes our universe to be a 3-dimensional space, the classification of 3-dimensional spaces is an important mathematical endeavor, since it may have ramifications for the global structure of our universe. Recently, this classification was spectacularly achieved by Perelman, using Hamilton's Ricci flow, who resolved Thurston's geometrization conjecture, and as a consequence the Poincare conjecture, which goes back 100 years. The geometrization conjecture implies that any finite 3-dimensional space has a canonical decomposition into geometric pieces, the most interesting of which are hyperbolic 3-manifolds. Mathematicians studying 3-manifolds are still sorting out the implications of Perelman's work and of the geometrization conjecture. This project will pursue various aspects of the ramifications of the geometrization conjecture. The principal and most ambitious project will study 2-dimensional objects inside of 3-dimensional spaces, especially the hyperbolic ones, which satisfy certain strong topological restrictions. Studying these 2-dimensional objects has implications for the global structure of finite 3-dimensional spaces. The project also aims to study the geometry of hyperbolic 3-manifolds, and in particular understand the simplest such manifolds and how their geometric properties (such as volume) relate to their topological properties, such as how many components of the boundary.