Ends of hyperbolic $3$-manifolds

Stacks of Hyperbolic Spaces and Ends of 3-manifolds

In this paper we introduce the notion of a “stack” of geodesic spaces. Loosely speaking, this consists of a geodesic space decomposed into a sequence of “sheets” indexed by a set of consecutive integers. A stack is said to be “hyperbolic” if it is Gromov hyperbolic and its sheets are uniformly Gromov hyperbolic. We define a Cannon-Thurston map for such a stack, and show that the boundary of a o...

Arithmetic of hyperbolic 3-manifolds

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the “arithmetic topology”. Given 3-dimensional manifold M fibering over the circle we introduce an algebraic number field K = Q( √ d), where d > 0 is an integer number (discriminant) uniquely determined by M . The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that V o...

Tameness of Hyperbolic 3-manifolds

Marden conjectured that a hyperbolic 3-manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3-manifold topologists. We prove this conjecture in theorem 10.2, actually in slightly more generality for PNC ma...

Distribution of resonances for manifolds with hyperbolic ends

Distribution of resonances for manifolds with hyperbolic ends by Kiril Datchev Doctor of Philosophy in Mathematics University of California, Berkeley Professor Maciej Zworski, Chair Quantum decay rates appear as imaginary parts of resonances, or poles of the meromorphic continuation of the resolvent of the Laplacian. As energy grows, decay rates are related to properties of geodesic flow and to...

Horocycles in hyperbolic 3-manifolds