Journal article

Short loop decompositions of surfaces and the geometry of jacobians

  • Balacheff, Florent Laboratoire Paul Painlevé, Université Lille 1, Villeneuve d’Ascq, France
  • Parlier, Hugo Department of Mathematics, University of Fribourg, Switzerland
  • Sabourau, Stéphane Laboratoire d’Analyse et de Mathématiques, Appliquées, Université Paris-Est Créteil, Créteil, France
Published in:
  • Geometric and functional analysis. - 2012, vol. 22, no. 1, p. 37-73
English Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions.First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any l Î (0, 1) there exists a constant C λ such that every closed Riemannian surface of genus g whose area is normalized at 4π (g – 1) has at least [λg] homologically independent loops of length at most C λ log(g). This result extends Gromov’s asymptotic log(g) bound on the homological systole of genus g surfaces. We construct hyperbolic surfaces showing that our general result is sharp. We also extend the upper bound obtained by P. Buser and P. Sarnak on the minimal norm of nonzero period lattice vectors of Riemann surfaces in their geometric approach of the Schottky problem to almost g homologically independent vectors.Then, we consider the lengths of pants decompositions on complete Riemannian surfaces in connexion with Bers’ constant and its generalizations. In particular, we show that a complete noncompact Riemannian surface of genus g with n ends and area normalized to 4p(g + \fracn2-1) admits a pants decomposition whose total length (sum of the lengths) does not exceed C g n log(n + 1) for some constant C g depending only on the genus.Finally, we obtain a lower bound on the systolic area of finitely presentable nontrivial groups with no free factor isomorphic to \mathbbZ in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal.
Faculté des sciences et de médecine
Département de Mathématiques
  • English
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