韩国裸舞

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Seminar Geometric Group Theory
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New bounds for `homotopical Ramsey theory' on surfaces

Farb and Leininger asked: How many distinct (isotopy classes of) simple closed curves on a finite-type surface S may pairwise intersect at most k times? Przytycki has shown that this number grows at most as a polynomial in |蠂(S)| of degree k^2+k+1. We present narrowed bounds by showing that the above quantity grows slower than |蠂(S)|^{3k}. The most interesting case is that of k=1, in which case the size of a `maximal 1-system' grows sub-cubically in |蠂(S)|. Following Przytycki, the proof uses the hyperbolic geometry of a surface of negative Euler characteristic essentially. In particular, we require bounds for the maximum size of a collection of curves of length at most L on a hyperbolic surface homeomorphic to S of the form F(L)路|蠂|, a point of view that yields intriguing questions in its own right. This is joint work with Tarik Aougab and Ian Biringer.

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