Abstract
In this manuscript we study the geometry of some metric spaces called horospherical product. They are constructed out of two Gromov hyperbolic spaces, and contains both discrete or continuous examples such as the Diestel-Leader graphs, the SOL geometry or the treebolic spaces.In the first part of this manuscript, we consider two proper, geodesically complete, Gromov hyperbolic, Busemann spaces X and Y. We construct their horospherical product Xbowtie Y and, after some metric estimations on specific paths in Gromov hyperbolic spaces, we describe a family of distances on Xbowtie Y. More specifically, we show that all these distances produce the same large scale geometry for Xbowtie Y. This description allows us to depict the shape of geodesic segments and geodesic lines. The understanding of the geodesics’ behaviour leads us to the characterization of the visual boundary of Xbowtie Y. For the second part, the two spaces X and Y are endowed with measures. Thanks to these measures, we manage to achieve the geometric rigidity of self quasi-isometries of Xbowtie Y. More specifically, we show that every self quasi-isometry Phi of Xbowtie Y is close to a product map (Phi^X,Phi^Y), where Phi^X:Xto X and Phi^Y:Yto Y are two quasi-isometries. To do so, we first develop several metric and measure theoretic tools regarding a specific family of geodesic called textit{vertical} geodesics. These tools include the textit{coarse differentiation}, introduced by Eskin, Fisher and Whyte for the horospherical product of regular infinite trees and hyperbolic planes. Afterwards, generalising techniques they presented, we obtain geometric rigidity.In the last chapter we present an example on how to use this geometric rigidity on Xbowtie Y in order to get informations on its quasi-isometry group. More precisely, we provide a description of the quasi-isometry group of a family of solvable Lie groups of the form mathbb{R}ltimes _{mathrm{Diag}(A_1,-A_2)}(N_1times N_2), where N_1, N_2 are nilpotent Lie groups and where A_1 and A_2 are matrices whose eigenvalues have all positive real parts.