The isoperimetric profile of infinite genus surfaces
The isoperimetric profile of a surface at \(t\) is the infimum of the lengths of the boundaries of subsets whose area is \(t\). \textit{P. Papasoglou} [Trans. Am. Math. Soc. 361, No.~10, 5139--5162 (2009; Zbl 1183.53030)] showed that for a surface of finite genus, equipped with a Riemannian metric and a simplicial triangulation, the isoperimetric profile satisfies a gap condition with respect to \(t\), i.e., when it grows faster than \(\sqrt t\), it grows as a linear function. The main result of the paper under review is that the result does not hold for surfaces of infinite genus. The construction of such surfaces is based on the existence of infinite sequences of expander graphs with fixed regularity. More precisely, starting with such a sequence, a new sequence is constructed by adding terminal vertices to the graphs. Then, for each graph in the sequence, a surface is constructed by thickening the edges. That is done by replacing the edges by cylinders and gluing, with the appropriate triangulation. The choice of triangulation allows the gluing of the cylinders at the vertices. The construction of the surfaces with the required property starts by making holes in the hyperbolic plane and gluing the surfaces that were constructed above. That produces a surface of infinite genus whose isoperimetric profile does not have a gap with respect to \(t\). It can grow with any rate between \(\sqrt t\) and \(t\).
scientific article; zbMATH DE number 5818360
scientific article