Geodesic
Systoles and kissing numbers of finite area hyperbolic surfaces
Fanoni, Federica1 ; Parlier, Hugo2
1Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
2Department of Mathematics, University of Fribourg, Ch. du Musée 23, CH-1700 Fribourg, Switzerland
Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3409-3433
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2015.15.3409
Classification : 30F10, 32G15, 53C22
Keywords: hyperbolic surfaces, kissing numbers, systoles

Fanoni, Federica  1   ; Parlier, Hugo  2

1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
2 Department of Mathematics, University of Fribourg, Ch. du Musée 23, CH-1700 Fribourg, Switzerland 
@article{10_2140_agt_2015_15_3409,
     author = {Fanoni, Federica and Parlier, Hugo},
     title = {Systoles and kissing numbers of finite area hyperbolic surfaces},
     journal = {Algebraic and Geometric Topology},
     pages = {3409--3433},
     year = {2015},
     volume = {15},
     number = {6},
     doi = {10.2140/agt.2015.15.3409},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.3409/}
}
TY  - JOUR
AU  - Fanoni, Federica
AU  - Parlier, Hugo
TI  - Systoles and kissing numbers of finite area hyperbolic surfaces
JO  - Algebraic and Geometric Topology
PY  - 2015
SP  - 3409
EP  - 3433
VL  - 15
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.3409/
DO  - 10.2140/agt.2015.15.3409
ID  - 10_2140_agt_2015_15_3409
ER  - 
%0 Journal Article
%A Fanoni, Federica
%A Parlier, Hugo
%T Systoles and kissing numbers of finite area hyperbolic surfaces
%J Algebraic and Geometric Topology
%D 2015
%P 3409-3433
%V 15
%N 6
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.3409/
%R 10.2140/agt.2015.15.3409
%F 10_2140_agt_2015_15_3409
Fanoni, Federica; Parlier, Hugo. Systoles and kissing numbers of finite area hyperbolic surfaces. Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3409-3433. doi: 10.2140/agt.2015.15.3409

[1] C Adams, Maximal cusps, collars, and systoles in hyperbolic surfaces, Indiana Univ. Math. J. 47 (1998) 419

[2] F Balacheff, E Makover, H Parlier, Systole growth for finite area hyperbolic surfaces, Ann. Fac. Sci. Toulouse Math. 23 (2014) 175

[3] C Bavard, Systole et invariant d'Hermite, J. Reine Angew. Math. 482 (1997) 93

[4] C Bavard, Anneaux extrémaux dans les surfaces de Riemann, Manuscripta Math. 117 (2005) 265

[5] R Brooks, Platonic surfaces, Comment. Math. Helv. 74 (1999) 156

[6] R Brooks, E Makover, Random construction of Riemann surfaces, J. Differential Geom. 68 (2004) 121

[7] P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser (1992)

[8] P Buser, P Sarnak, On the period matrix of a Riemann surface of large genus, Invent. Math. 117 (1994) 27

[9] B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)

[10] T Gauglhofer, K D Semmler, Trace coordinates of Teichmüller space of Riemann surfaces of signature $(0,4)$, Conform. Geom. Dyn. 9 (2005) 46

[11] M G Katz, M Schaps, U Vishne, Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups, J. Differential Geom. 76 (2007) 399

[12] S Makisumi, A note on Riemann surfaces of large systole, J. Ramanujan Math. Soc. 28 (2013) 359

[13] H Parlier, Kissing numbers for surfaces, J. Topol. 6 (2013) 777

[14] H Parlier, Simple closed geodesics and the study of Teichmüller spaces, from: "Handbook of Teichmüller theory, IV" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc., Zürich (2014) 113

[15] B Petri, Random regular graphs and the systole of a random surface, preprint (2013)

[16] P Przytycki, Arcs intersecting at most once, Geom. Funct. Anal. 25 (2015) 658

[17] P Schmutz, Congruence subgroups and maximal Riemann surfaces, J. Geom. Anal. 4 (1994) 207

[18] P Schmutz, Arithmetic groups and the length spectrum of Riemann surfaces, Duke Math. J. 84 (1996) 199

[19] P Schmutz Schaller, Extremal Riemann surfaces with a large number of systoles, from: "Extremal Riemann surfaces" (editors J R Quine, P Sarnak), Contemp. Math. 201, Amer. Math. Soc. (1997) 9

Cité par Sources :