Kissing numbers for surfaces

Hugo Parlier

doi:10.1112/jtopol/jtt012

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arXiv · 1111.3573

Kissing numbers for surfaces

Hugo Parlier

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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 $ε>0$. 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 $g^{\sfrac{3}{2}}$ systoles is given.

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Hugo Parlier. 2011-11-15. Kissing numbers for surfaces. https://doi.org/10.1112/jtopol%2Fjtt012

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