https://hal.archives-ouvertes.fr/hal-00795807Balacheff, FlorentFlorentBalacheffLPP - Laboratoire Paul Painlevé - UMR 8524 - Université de Lille - CNRS - Centre National de la Recherche ScientifiqueParlier, HugoHugoParlierDepartment of Mathematics - Albert-Ludwigs-Universität FreiburgSabourau, StéphaneStéphaneSabourauLAMA - Laboratoire d'Analyse et de Mathématiques Appliquées - UPEM - Université Paris-Est Marne-la-Vallée - Fédération de Recherche Bézout - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - CNRS - Centre National de la Recherche ScientifiqueShort loop decompositions of surfaces and the geometry of JacobiansHAL CCSD2012[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG][MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT]Sabourau, Stéphane2016-11-16 22:05:342022-09-29 14:21:152016-11-18 09:44:33enJournal articleshttps://hal.archives-ouvertes.fr/hal-00795807/document10.1007/s00039-012-0147-xapplication/binary1Given 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 $\lambda \in (0,1)$ there exists a constant $C_\lambda$ such that every closed Riemannian surface of genus $g$ whose area is normalized at $4\pi(g-1)$ has at least $[\lambda g]$ homologically independent loops of length at most $C_\lambda \log(g)$. 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 $4\pi (g+\frac{n}{2}-1)$ 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~$\Z$ in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal.