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 \cite{Shchur} 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, $\LL^p$-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 $dt^2+\sum_ie^{2\mu_it}dx_i^2$, 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) provided by unipotent locally homogeneous spaces.
Quasi-isometries between hyperbolic metric spaces, quantitative aspects