On the quantitative quasi-isometry problem: transport of Poincaré inequalities and different types of quasi-isometric distortion growth

We consider a quantitative form of the quasi-isometry problem. We discuss several arguments which lead us to different results and bounds of quasi-isometric distortion: comparison of volumes, connectivity etc. Then we study the transport of Poincaré constants by quasi-isometries and we give sharp lower and upper bounds for the homotopy distortion growth for an interesting class of hyperbolic metric spaces.