SHORT LOOP DECOMPOSITIONS OF SURFACES AND THE GEOMETRY OF JACOBIANS

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 Cλ such that every closed Riemannian surface of genus g whose area is normalized at 4pi(g − 1) has at least [λg] homologically independent loops of length at most Cλ 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 4pi(g+ n 2 −1) admits a pants decomposition whose total length (sum of the lengths) does not exceed Cg n log(n+ 1) for some constant Cg 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. 1