On the asymptotic isoperimetric constants for Riemann surfaces and graphs.
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 \(k\) sufficiently large.} The authors' argument gives the explicit bound: \(C\leq .4402\dots\). Theorem 2 is extended to surfaces built out of the Ramanujan graphs. The construction describing how these surfaces are built from \(k\)-regular graphs is sketched. The authors then show the follwoing result. Theorem 3. {There exists a constant \(C < 1/2\) such that, for all \(p\) and for all \(q\) sufficiently large, depending on \(p\), we have that \(h(S_\mathcal O^{p,q})\leq C\), where \(S_\mathcal O^{p,q}\) is a surface constructed from the Ramanujan graph} \(X^{p,q}\). The authors' argument produces a bound: \(C < .467177\dots.\) In contrast to the graph-theoretic results of N. Alon [loc. cit.], the authors get that this constant stays away from 1/2 independent of the regularity \(k\) of the associated graph.
Che\-e\-ger isoperimetric constant of a Riemannian surface
scientific article; zbMATH DE number 2171910