Short Loop Decompositions of Surfaces and the Geometry of Jacobians

Florent Balacheff, Hugo Parlier, Stéphane Sabourau*

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

14 Citations (Scopus)

Abstract

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 λ ∈ (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 [λ g] 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 4π(g + n/2 - 1) admits a pants decomposition whose total length (sum of the lengths) does not exceed C gn 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 ℤ in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal.

Original languageEnglish
Pages (from-to)37-73
Number of pages37
JournalGeometric and Functional Analysis
Volume22
Issue number1
DOIs
Publication statusPublished - Feb 2012

Keywords

  • Jacobian
  • pants decomposition
  • period lattice
  • Riemann surfaces
  • Schottky problem Bers' constant
  • short homology basis
  • simple closed geodesics
  • systole
  • systolic area of groups
  • Teichmüller and moduli spaces

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