Algebraic & Geometric Topology

Systoles and kissing numbers of finite area hyperbolic surfaces

Federica Fanoni and Hugo Parlier

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We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surface and which grows subquadratically in the genus.

Article information

Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3409-3433.

Received: 1 September 2014
Revised: 3 March 2015
Accepted: 7 April 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15]
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 53C22: Geodesics [See also 58E10]

hyperbolic surfaces kissing numbers systoles


Fanoni, Federica; Parlier, Hugo. Systoles and kissing numbers of finite area hyperbolic surfaces. Algebr. Geom. Topol. 15 (2015), no. 6, 3409--3433. doi:10.2140/agt.2015.15.3409.

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  • C Adams, Maximal cusps, collars, and systoles in hyperbolic surfaces, Indiana Univ. Math. J. 47 (1998) 419–437
  • F Balacheff, E Makover, H Parlier, Systole growth for finite area hyperbolic surfaces, Ann. Fac. Sci. Toulouse Math. 23 (2014) 175–180
  • C Bavard, Systole et invariant d'Hermite, J. Reine Angew. Math. 482 (1997) 93–120
  • C Bavard, Anneaux extrémaux dans les surfaces de Riemann, Manuscripta Math. 117 (2005) 265–271
  • R Brooks, Platonic surfaces, Comment. Math. Helv. 74 (1999) 156–170
  • R Brooks, E Makover, Random construction of Riemann surfaces, J. Differential Geom. 68 (2004) 121–157
  • 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)
  • T Gauglhofer, K-D Semmler, Trace coordinates of Teichmüller space of Riemann surfaces of signature $(0,4)$, Conform. Geom. Dyn. 9 (2005) 46–75
  • M,G Katz, M Schaps, U Vishne, Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups, J. Differential Geom. 76 (2007) 399–422
  • S Makisumi, A note on Riemann surfaces of large systole, J. Ramanujan Math. Soc. 28 (2013) 359–377
  • H Parlier, Kissing numbers for surfaces, J. Topol. 6 (2013) 777–791
  • H Parlier, Simple closed geodesics and the study of Teichmüller spaces, from: “Handbook of Teichmüller theory, IV”, (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc., Zürich (2014) 113–134
  • B Petri, Random regular graphs and the systole of a random surface, preprint (2013)
  • P Przytycki, Arcs intersecting at most once, Geom. Funct. Anal. 25 (2015) 658–670
  • P Schmutz, Congruence subgroups and maximal Riemann surfaces, J. Geom. Anal. 4 (1994) 207–218
  • P Schmutz, Arithmetic groups and the length spectrum of Riemann surfaces, Duke Math. J. 84 (1996) 199–215
  • P Schmutz Schaller, Extremal Riemann surfaces with a large number of systoles, from: “Extremal Riemann surfaces”, (J,R Quine, P Sarnak, editors), Contemp. Math. 201, Amer. Math. Soc. (1997) 9–19