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.
