3-manifold geometry and topology

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

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.