Kanai proved the stability under quasi-isometries of numerous global properties (including isoperimetric inequalities) between Riemannian manifolds of bounded geometry. Even though quasi-isometries highly distort local properties, recently it was shown that the injectivity radius is preserved (in some appropriate sense) under these maps between genus zero Riemann surfaces. In the present work, results along these lines are obtained even for infinite genus. As a consequence, the stability of the isoperimetric inequality in this context (without the hypothesis of bounded geometry) is also obtained.
Injectivity radius; Linear isoperimetric inequality; Quasi-isometry; Riemann surface; Poincare metric