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We consider a quantitative form of the quasi-isometry problem. We discussseveral arguments which lead us to different results and bounds ofquasi-isometric distortion: comparison of volumes, connectivity etc. Then westudy the transport of Poincar\'e constants by quasi-isometries and we givesharp lower and upper bounds for the homotopy distortion growth for aninteresting class of hyperbolic metric spaces.

On the quantitative quasi-isometry problem: transport of Poincar\'e inequalities and different types of quasi-isometric distortion growth