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Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including the volume growth rate) between non-bordered Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses are not usually satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we try to fill that gap and prove the stability of the volume growth rate by quasi-isometries, under hypotheses that many bordered or non-bordered Riemann surfaces (and even Riemannian surfaces with pinched negative curvature) satisfy. In order to get our results, it is shown that many bordered Riemannian surfaces with pinched negative curvature are bilipschitz equivalent to bordered surfaces with constant negative curvature.