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 $dt^2+\sum_ie^{2\mu_it}dx_i^2$, 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).
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00867709 |
| Date | 08 July 2013 |
| Creators | Shchur, Vladimir |
| Publisher | Université Paris Sud - Paris XI |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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