QUASI-ISOMETRIES BETWEEN HYPERBOLIC METRIC SPACES, QUANTITATIVE ASPECTS

Abstract. This is a preliminary version of my PhD thesis. In this text 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 [30] where an optimal upper-bound for Morse Lemma is given, together with the symmetric 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, Lp-cohomology spaces provides useful quasi-isometry invari-ants and Poincare ́ constants of balls are their quantitative incarnation. We study how Poincare ́ constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincare ́ constants for locally homogeneous metrics of the form dt2 +