Geometry and topology of curves and surfaces in closed hyperbolic manifolds

California Institute Of Technology, Pasadena CA

Investigators

Abstract

The PI will study questions about geometry and topology of closed hyperbolic manifolds. In connection with the Virtual Haken Conjecture, the question of whether a hyperbolic 3-manifold contains an abundance of equidistributed essential 3-manifolds with incompressible boundary will be addressed. Higher dimensional hyperbolic manifolds will be studied and the aim is to prove that every closed hyperbolic manifold of dimension at least 4 contains an essential 3-manifold group. Also, it will be shown that the Simple Loop Conjecture fails in every dimension greater than or equal to 4. Related problems (in particular the Surface Subgroup Conjecture) will be address for other hyperbolic spaces, like complex hyperbolic spaces or hyperbolic groups. Pure mathematics is a breeding ground for ideas that are later utilized in natural sciences like physics and biology. In physics, the universe is described as a 3-dimensional space, thus studying geometry and topology of 3-manifolds may prove very important in the real life physical questions that we face.