Minimally intersecting filling pairs on surfaces

Aougab, Tarik¹ ; Huang, Shinnyih¹

¹Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06510, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 903-932

Résumé

Let Sg denote the closed orientable surface of genus g. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill Sg and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group. Next, we demonstrate that minimally intersecting filling pairs are combinatorially optimal, in the sense that there are many simple closed curves intersecting the pair exactly once. We conclude by initiating the study of a topological Morse function ℱg over the moduli space of Riemann surfaces of genus g, which, given a hyperbolic metric σ, outputs the length of the shortest minimally intersecting filling pair for the metric σ. We completely characterize the global minima of ℱg and, using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper, we show that the number of such minima grows at least exponentially in g.

Détail

DOI : 10.2140/agt.2015.15.903

Classification : 57M20, 57M50
Keywords: mapping class group, filling pairs

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