A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric inequality in a Riemann surface by using a graph related to it, even if the surface has injectivity radius zero (this graph is inspired in Kanai’s graph, but it is different from it). We also present an application relating Gromov boundary and isoperimetric inequality.

中文翻译：

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黎曼曲面和图形中的等距不等式

Kanai的一个著名定理指出，准等距保留了一致的黎曼流形（具有正注入半径）和图之间的等距不等式。我们的主要结果表明，即使表面的注入半径为零，我们也可以通过使用与之相关的图来研究黎曼曲面上的（Cheeger）等距不等式（此图的灵感来自Kanai的图，但与它不同）。 。我们还提出了与Gromov边界和等参不等式有关的应用程序。