Minimally intersecting filling pairs on surfaces

Aougab, Tarik; Huang, Shinnyih

doi:10.2140/agt.2015.15.903

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Minimally intersecting filling pairs on surfaces

Tarik Aougab and Shinnyih Huang

Algebraic & Geometric Topology 15 (2015) 903–932

DOI: 10.2140/agt.2015.15.903

Abstract

Let $S_{g}$ denote the closed orientable surface of genus $g$ . We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_{g}$ 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 $ℱ_{g}$ 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 $ℱ_{g}$ 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 $g$ .

Keywords

mapping class group, filling pairs

Mathematical Subject Classification 2010

Primary: 57M20, 57M50

References

Publication

Received: 26 January 2014

Revised: 4 July 2014

Accepted: 7 July 2014

Published: 22 April 2015

Authors

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

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

Volume 15, issue 2 (2015)