English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Given a Riemannian surface, we consider a naturally embedded graph whichcaptures part of the topology and geometry of the surface. By studying thisgraph, we obtain results in three different directions.  First, we find bounds on the lengths of homologically independent curves onclosed Riemannian surfaces. As a consequence, we show that for any $\lambda \in(0,1)$ there exists a constant $C_\lambda$ such that every closed Riemanniansurface 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 thehomological systole of genus $g$ surfaces. We construct hyperbolic surfacesshowing that our general result is sharp. We also extend the upper boundobtained by P. Buser and P. Sarnak on the minimal norm of nonzero periodlattice vectors of Riemann surfaces %systole of Jacobians of Riemann surfacesin their geometric approach of the Schottky problem to almost $g$ homologicallyindependent vectors.  Then, we consider the lengths of pants decompositions on complete Riemanniansurfaces in connexion with Bers' constant and its generalizations. Inparticular, 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 pantsdecomposition 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 presentablenontrivial groups with no free factor isomorphic to $\Z$ in terms of its firstBetti number. The asymptotic behavior of this lower bound is optimal.

Short loop decompositions of surfaces and the geometry of Jacobians