Kissing numbers for surfaces

Abstract

The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like $g^{\sfrac{4}{3}-ε}$ for any $\epsilon >0$. The first goal of this article is to give upper bounds on these numbers; in particular the growth is shown to be sub-quadratic. In the second part, a construction of (non hyperbolic) surfaces with roughly $g^{\sfrac{3}{2}}$ systoles is given.

20 pages, 9 figures

Cited by in corpus (4)

• Systoles and kissing numbers of finite area hyperbolic surfaces
• Systolically extremal nonpositively curved surfaces are flat with finitely many singularities
• Systoles on punctured spheres
• Hyperbolic manifolds with a large number of systoles