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 a number of 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 a certain class of hyperbolic metric spaces, a quotient of a Heintze group R⋉RnR⋉Rn by ZnZn. We also prove the linear distortion growth between hyperbolic space Hn,n≥3Hn,n≥3 and a tree.