Short loop decompositions of surfaces and the geometry of Jacobians

TL;DR

This paper explores the geometry of Riemannian surfaces through embedded graphs, providing bounds on homologically independent curves, pants decompositions, and systolic areas, with implications for Jacobians and group theory.

Contribution

It introduces new bounds on homologically independent loops, pants decompositions, and systolic areas, extending previous results and demonstrating sharpness through hyperbolic surface constructions.

Findings

Bound on lengths of homologically independent curves proportional to log(g)

Existence of pants decompositions with total length bounded by n log(n)

Lower bounds on systolic area related to first Betti number

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 $\lambda \in (0,1)$ 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 %systole of Jacobians 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.