Isoperimetric inequalities in Riemann surfaces and graphs

TL;DR

This paper extends the understanding of isoperimetric inequalities in Riemann surfaces, including those with zero injectivity radius, by relating them to graphs and exploring implications for Gromov boundary.

Contribution

It introduces a new graph-based method to study isoperimetric inequalities in Riemann surfaces with zero injectivity radius, expanding Kanai's theorem.

Findings

Allows analysis of isoperimetric inequalities in surfaces with zero injectivity radius

Establishes a connection between Gromov boundary and isoperimetric inequalities

Provides a new graph construction inspired by Kanai's graph

Abstract

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.