On the asymptotic isoperimetric constants for Riemann surfaces and graphs.
type: Item
instance of: scholarly article
title: On the asymptotic isoperimetric constants for Riemann surfaces and graphs.
zbMATH Open document ID: 1065.05091
author: Robert Brooks
author: Andrzej Zuk
published in: Journal of Differential Geometry
publication date: 2005-05-26T00:00:00Z
review text: Let \(X\) be a finite graph of degree \(k\). For a finite subset of vertices \(U\subset X\), its boundary \(\partial U\) is defined as the set of edges with one extremity in \(U\) and the other in \(X \setminus U\). The Cheeger isoperimetric constant \(h(X)\) is defined as \(h(X) = \min\left.\left\{\,\frac{| \partial U| }{| U| }\,\right|\, U\subset X\,\, \text{and}\,\, 1\leq | U| \leq \frac{1}{2}| X| \,\right\}\). Let inj\((X)\) denote the injectivity radius of the graph \(X\), i.e., the maximal number \(r\) such that in every ball of radius \(r\) in \(X\) there is no cycle. Theorem 1. {Let \(X\) be a finite \(k\)-regular graph. Then} \(h(X)\leq\frac{k-2}{2}+\varepsilon(\text{inj}(X)). \) A result stronger than Theorem 1 is proved in \textit{N.~Alon} [Comb. Probab. Comput. 6, 145--152 (1997; Zbl 0881.05077)]. The authors present Theorem 1 with the weaker assumption and weaker conclusion as the dependence on injectivity radius and universal covering will reappear in the geometric setting, and generalize this theorem to the setting of manifolds, and in particular to the case of Riemann surfaces. For a Riemannian manifold \(M^n\) of dimension \(n\) its Cheeger isoperimetric constant \(h(M^n)\) is defined as \(h(M^n)=\inf_A\left\{ \text{vol}_{n-1}(\partial A) / \text{vol}_{n}(A) \right\},\) where \(A\) runs over subdomains of \(M^n\) of volume less than (1/2)vol(\(M^n\)), and where vol\(_{n-1}(\partial M)\) and vol\(_n(A)\) are the measures with respect to the Riemannian metric. The authors prove the following result. Theorem 2. {There is a constant \(C < 1/2\) with the following property: Let \(S_k\) be the modular surface \(S_k=\mathbb H^2/\Gamma_k\), where \(\Gamma_k\) is the subgroup of \(\text{ PSL}(2,\mathbb Z\)) given by \[ \Gamma_k=\left\{\left( \begin{matrix} a&b\\ c&d \end{matrix}\right)\left|\,\left( \begin{matrix} a&b\\ c&d \end{matrix}\right) \right. \equiv \left( \begin{matrix} 1&0\\ 0&1 \end{matrix} \right) \pmod {k}\right\}. \] Then \(h(S_k)\leq C\) for …
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 2171910
zbMATH Keywords: probability
zbMATH Keywords: Che\-e\-ger isoperimetric constant of a Riemannian surface
zbMATH Keywords: Ramanujan graphs
zbMATH Keywords: modular surfaces
zbMATH Keywords: Ramanujan surfaces
Wikidata QID: Q115201378
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090425529
OpenAlex ID: W1512248353
DOI: 10.4310/JDG/1090425529
description: scientific article; zbMATH DE number 2171910