From curves to currents

MARTINEZ GRANADO, Didac; Thurston, Dylan P.

doi:10.1017/fms.2021.68

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From curves to currents

MARTINEZ GRANADO, Didac; Thurston, Dylan P.

2021 • In Forum of Mathematics, Sigma, 9

Peer Reviewed verified by ORBi

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https://hdl.handle.net/10993/67008

DOI
10.1017/fms.2021.68

arXiV
2004.01550v3

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Keywords :

Curve counting; Extremal length; Geodesic currents; Analysis; Geometry and Topology; Mathematics - Geometric Topology; Mathematics - Dynamical Systems

Abstract :

[en] Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

Disciplines :

Mathematics

Author, co-author :

MARTINEZ GRANADO, Didac ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Thurston, Dylan P. ; Department of Mathematics, Indiana University, Bloomington, United States

External co-authors :

yes

Language :

English

Title :

From curves to currents

Publication date :

29 November 2021

Journal title :

Forum of Mathematics, Sigma

eISSN :

2050-5094

Publisher :

Cambridge University Press

Volume :

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S2050509421000682

Funders :

NSF - National Science Foundation

Funding text :

We thank Francisco Arana, Martin Bridgeman, Maxime Fortier Bourque, Maria Beatrice Pozzetti, Kasra Rafi, and Tengren Zhang for helpful conversations and the anonymous referee for careful reading and suggestions. The first author was supported by the Mathematics Department Indiana University Bloomington, via the Hazel King Thompson fellowship and the Indiana University Graduate School under the Dissertation Research Fellowship. The second author was supported by the National Science Foundation under Grant Numbers DMS-1507244 and DMS-2110143.

Commentary :

60 pages; v2: clarify counting results and role of weighted curves; v3: Incorporate erratum on definition of "essential crossing", published separately by Forum of Math, Sigma. The numbers have been adjusted to match in the two versions

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since 30 December 2025

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