Stability of the volume growth rate under quasi-isometries

  • Ana Granados
  • Domingo PestanaEmail author
  • Ana Portilla
  • José M. Rodríguez
  • Eva Tourís


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.


Volume growth rate Quasi-isometry Riemann surface Poincaré metric Negative pinched curvature 

Mathematics Subject Classification

30F45 53C20 30F20 31C12 



We would like to thank the referees for their careful reading of the manuscript and several useful comments which have helped us to improve the presentation of the paper.


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Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  1. 1.Saint Louis UniversityMadridSpain
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain
  3. 3.Departamento de Matemáticas, Facultad de CienciasUniversidad Autónoma de MadridMadridSpain

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