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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 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 . 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.Comment: 20 pages, 9 figure
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