Large-scale rank and geometry of the Lipschitz metric on geodesic currents
Abstract
For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by 8g−9. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to 8g−9 and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to 3g−3. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.
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BibTeX
@misc{martinezgranado2026largescale,
title = {{Large-scale rank and geometry of the Lipschitz metric on geodesic currents}},
author = {Dídac Martínez-Granado and Jenya Sapir},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03359},
eprint = {2610.03359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03359},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Large-scale rank and geometry of the Lipschitz metric on geodesic currents
AU - Dídac Martínez-Granado
AU - Jenya Sapir
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.03359
UR - https://arxiv.org/abs/2610.03359
AB - For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by 8g−9. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to 8g−9 and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to 3g−3. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.
ER -