Kissing numbers for surfaces

Abstract. The so-called 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 g4/3−ε for any ε> 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 g3/2 systoles is given. 1