3-manifold geometry and topology

这个计画提出了几种研究双曲三维流形的拓扑与几何的方法。主要研究三维流形中的浸入曲面、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哈肯猜想，这相当于证明双曲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希望将此与Thurston的虚覆盖猜想联系起来，该猜想指出双曲三维流形有一个在圆上纤维化的有限片覆盖，并暗示虚哈肯猜想。PI想证明哈肯三维流形同伦等价于一个紧的CAT（0）立方复形。接下来，PI想证明双曲三维流形的基本群是LERF。如果Gromov双曲群是剩余有限的，这将遵循，尽管许多几何群理论家认为这是错误的。根据Haglund和Wise的一个程序，这些图将意味着一个哈肯双曲流形有一个基本群，它实际上嵌入一个直角反射群。这反过来又意味着这些流形实际上是由PI以前的工作纤维。PI还计划研究双曲哈肯流形的体积，以及流形上的各种拓扑限制如何引起体积的下限。例如，PI想了解具有n个尖点的可定向双曲3-流形的体积的渐近行为。3维空间的数学研究可以追溯到庞加莱的工作。由于经典物理学将我们的宇宙描述为三维空间，因此三维空间的分类是一项重要的数学奋进，因为它可能对我们宇宙的整体结构产生影响。最近，Perelman使用汉密尔顿的Ricci流，解决了Thurston的几何化猜想，并因此解决了Poincare猜想，这可以追溯到100年前。几何化猜想意味着任何有限的三维空间都有一个典型的分解成几何片段，其中最有趣的是双曲三维流形。研究三维流形的数学家仍在整理佩雷尔曼的工作和几何化猜想的含义。这个项目将追求几何化猜想的各个方面的分支。主要和最雄心勃勃的项目将研究三维空间内的二维物体，特别是满足某些强拓扑限制的双曲物体。研究这些二维物体对有限三维空间的整体结构有着重要意义。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以及它们的几何属性（如体积）如何与它们的拓扑属性（如边界的多少个分量）相关。

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.