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https://doi.org/10.1142/S1793525318500309Cited by:4 (Source: Crossref)

Let Fg denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of Fg if its complement is a disjoint union of discs. The mapping class group Mod(Fg) of genus g acts on the set of fillings of Fg. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of Fg are in the same Mod(Fg)-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of F2 whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of Mod(F2).

We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of F2 is two. Finally, given positive integers g and k with (g,k)(2,1), we construct a filling pair of Fg such that the complement is a union of k topological discs.

References

  • 1. J. W. Anderson, H. Parlier and A. Pettet , Relative shapes of thick subsets of moduli space, Amer. J. Math. 138 (2016) 473–498. CrossrefGoogle Scholar
  • 2. J. W. Anderson, H. Parlier and A. Pettet , Small filling sets of curves on a surface, Topol. Appl. 158 (2011) 84–92. CrossrefGoogle Scholar
  • 3. T. Aougab, Local geometry of the k-curve graph, arXiv:1508.00502. Google Scholar
  • 4. T. Aougab and S. Huang , Minimally intersecting filling pairs on surface, Algebr. Geom. Topol. 15 (2015) 903–932. CrossrefGoogle Scholar
  • 5. L. Chekhov and M. Shapiro , Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables, Int. Math. Res. Not. IMRN 2014 (2014) 2746–2772. CrossrefGoogle Scholar
  • 6. B. Eynard , Recursion between Mumford volumes of moduli spaces, Ann. Henri Poincaré 12 (2011) 1431–1447. CrossrefGoogle Scholar
  • 7. F. Fanoni and H. Parlier , Filling sets of curves on punctured surfaces, New York J. Math. 22 (2016) 653–666. Google Scholar
  • 8. B. Farb and D. Margalit , A Primer on Mapping Class Groups, Princeton Mathematical Series, Vol. 49 (Princeton Univ. Press, 2012). Google Scholar
  • 9. R. J. Marsh and S. Schroll , The geometry of Brauer graph algebras and cluster mutations, J. Algebra 419 (2014) 141–166. CrossrefGoogle Scholar
  • 10. P. S. Schaller , Systoles and topological Morse functions for Riemann surfaces, J. Differential Geom. 52 (1999) 407–452. CrossrefGoogle Scholar
  • 11. W. Thurston, A spine for Teichmüller space, preprint (1986). Google Scholar