Kissing numbers for surfaces

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 $g^{\sfrac{4}{3}-\epsilon}$ 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.