## Algebraic & Geometric Topology

### Minimally intersecting filling pairs on surfaces

#### Abstract

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 $g$.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 2 (2015), 903-932.

Dates
Revised: 4 July 2014
Accepted: 7 July 2014
First available in Project Euclid: 28 November 2017

https://projecteuclid.org/euclid.agt/1511895793

Digital Object Identifier
doi:10.2140/agt.2015.15.903

Mathematical Reviews number (MathSciNet)
MR3342680

Zentralblatt MATH identifier
1334.57013

Keywords
mapping class group filling pairs

#### Citation

Aougab, Tarik; Huang, Shinnyih. Minimally intersecting filling pairs on surfaces. Algebr. Geom. Topol. 15 (2015), no. 2, 903--932. doi:10.2140/agt.2015.15.903. https://projecteuclid.org/euclid.agt/1511895793

#### References

• H Akrout, Singularités topologiques des systoles généralisées, Topology 42 (2003) 291–308
• J,W Anderson, H Parlier, A Pettet, Small filling sets of curves on a surface, Topology Appl. 158 (2011) 84–92
• F Balacheff, H Parlier, Bers' constants for punctured spheres and hyperelliptic surfaces, J. Topol. Anal. 4 (2012) 271–296
• K Bezdek, Ein elementarer Beweis für die isoperimetrische Ungleichung in der Euklidischen und hyperbolischen Ebene, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27 (1984) 107–112
• P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser, Boston (1992)
• P Buser, P Sarnak, On the period matrix of a Riemann surface of large genus, Invent. Math. 117 (1994) 27–56
• B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
• D,D Long, C Maclachlan, A,W Reid, Arithmetic Fuchsian groups of genus zero, Pure Appl. Math. Q. 2 (2006) 569–599
• G,A Margulis, Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. 17, Springer, Berlin (1991)
• D Mumford, A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc. 28 (1971) 289–294
• H Parlier, The homology systole of hyperbolic Riemann surfaces, Geom. Dedicata 157 (2012) 331–338
• P Schmutz, Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal. 3 (1993) 564–631
• P Schmutz Schaller, Geometry of Riemann surfaces based on closed geodesics, Bull. Amer. Math. Soc. 35 (1998) 193–214
• P Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces, J. Differential Geom. 52 (1999) 407–452
• K Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) 201–212