TY - JOUR

T1 - Short Loop Decompositions of Surfaces and the Geometry of Jacobians

AU - Balacheff, Florent

AU - Parlier, Hugo

AU - Sabourau, Stéphane

N1 - Funding Information:
tole, Jacobian, period lattice, Schottky problem Bers’ constant, pants decomposition. Teichmüller and moduli spaces, systolic area of groups 2010 Mathematics Subject Classification: Primary: 30F10, secondary: 32G20, 53C22 H.P.’s research supported by Swiss National Science Foundation grant number PP00P2 128557

PY - 2012/2

Y1 - 2012/2

N2 - 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.

AB - 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.

KW - Jacobian

KW - pants decomposition

KW - period lattice

KW - Riemann surfaces

KW - Schottky problem Bers' constant

KW - short homology basis

KW - simple closed geodesics

KW - systole

KW - systolic area of groups

KW - Teichmüller and moduli spaces

UR - http://www.scopus.com/inward/record.url?scp=84858864717&partnerID=8YFLogxK

U2 - 10.1007/s00039-012-0147-x

DO - 10.1007/s00039-012-0147-x

M3 - Article

AN - SCOPUS:84858864717

SN - 1016-443X

VL - 22

SP - 37

EP - 73

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 1

ER -