Quasi-isometries between hyperbolic metric spaces, quantitative aspects

Overview & Synopsis

In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ``quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, L^p-cohomology spaces provides useful quasi-isometry invariants and Poincaré constants of balls are their quantitative incarnation. We study how Poincaré constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincaré constants for locally homogeneous metrics of the form dt²+∑_{i}e^{2µ_{i}t}dx²_{i}, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).