Surfaces in finite covers of 3-manifolds and aspects of the mapping class groups

PI将研究三维流形的拓扑，重点是与虚拟Haken猜想相关的问题。这一猜想假定每个具有无限基本群的闭3-流形都包含一个曲面，如果一个曲面的基本群是有限覆盖的，则该曲面的基本群注入到这三个流形的曲面中。该提案的最终目的是证明这一猜想。PI将通过两个主题来探讨这个问题：使用随机3-流形来确定最有利可图的攻击途径，以及使用数论来构造有启发性的例子。工作的次要焦点将是曲面的映射类群，特别是研究射影可测分层空间上的随机游动调和测度和勒贝格测度之间的关系。PI打算使用来自包括拓扑学、概率论和数论在内的多个数学学科的方法来研究三维流形拓扑学中的核心问题之一。他还打算开发软件来探索他的研究计划的各个方面。该软件将通过网络向其他研究人员提供。研究生将参与这项研究。

The PI will study the topology of 3-dimensional manifolds focusing on questions related to the Virtual Haken Conjecture. This conjecture posits that every closed 3-manifold with infinite fundamental group contains a surface whose fundamental group injects into that of the three manifold, if one passes to a finite cover. The ultimate goal of the proposal is to prove this conjecture. The PI will approach the question with two main themes: the use of random 3-manifolds to determine the most profitable avenues of attack, and the use of number theory to construct illuminating examples.A secondary focus of the work will be the mapping class groups of surfaces, in particular a study of the relationship between random walk harmonic measures and the Lebesgue measure on the space of projective measured laminations.The PI intends to work on one of the central questions in topology of three-dimensional manifolds using methods coming from several disciplines of mathematics, including topology, probability and number theory. He also intends to develod software to explore aspects of his research program. This software will be made available to other researchers via the web. Graduate students will be involved in this research.