Short homotopically independent loops on surfaces

Karam, Steve¹

¹Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, UFR Sciences et Technologie, Parc de Grandmont, 37200 Tours, France

Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1825-1844

Résumé

In this paper, we are interested in short homologically and homotopically independent loops based at the same point on Riemannian surfaces and metric graphs.

First, we show that for every closed Riemannian surface of genus g ≥ 2 and area normalized to g, there are at least ⌈log(2g) + 1⌉ homotopically independent loops based at the same point of length at most Clog(g), where C is a universal constant. On the one hand, this result substantially improves Theorem 5.4.A of M Gromov in [J. Differential Geom. 18 (1983) 1–147]. On the other hand, it recaptures the result of S Sabourau on the separating systole in [Comment. Math. Helv. 83 (2008) 35–54] and refines his proof.

Second, we show that for any two integers b ≥ 2 with 1 ≤ n ≤ b, every connected metric graph Γ of first Betti number b and of length b contains at least n homologically independent loops based at the same point and of length at most 24(log(b) + n). In particular, this result extends Bollobàs, Szemerédi and Thomason’s log(b) bound on the homological systole to at least log(b) homologically independent loops based at the same point. Moreover, we give examples of graphs where this result is optimal.

Détail

DOI : 10.2140/agt.2014.14.1825

Classification : 30F10
Keywords: Riemannian surfaces, homologically independent loops, systole

Bibliographie

[1] F Balacheff, H Parlier, S Sabourau, Short loop decompositions of surfaces and the geometry of Jacobians, Geom. Funct. Anal. 22 (2012) 37

[2] B Bollobás, E Szemerédi, Girth of sparse graphs, J. Graph Theory 39 (2002) 194

[3] B Bollobás, A Thomason, On the girth of Hamiltonian weakly pancyclic graphs, J. Graph Theory 26 (1997) 165

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

[5] M P Do Carmo, Riemannian geometry, Birkhäuser (1992)

[6] R Diestel, Graph theory, Graduate Texts in Mathematics 173, Springer (2010)

[7] M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1

[8] M Gromov, Systoles and intersystolic inequalities, from: "Actes de la Table Ronde de Géométrie Différentielle" (editor A L Besse), Sémin. Congr. 1, Soc. Math. France (1996) 291

[9] W Jaco, On certain subgroups of the fundamental group of a closed surface, Proc. Cambridge Philos. Soc. 67 (1970) 17

[10] S Karam, Uniform growth of balls in the universal cover of graphs and surfaces, to be published in Trans. Amer. Math. Soc.

[11] S B Myers, Connections between differential geometry and topology II: Closed surfaces, Duke Math. J. 2 (1936) 95

[12] S Sabourau, Asymptotic bounds for separating systoles on surfaces, Comment. Math. Helv. 83 (2008) 35