This is the last Arxiv version. It contains the same content as the published version in JoT, for which DMATH and FNR unlocked Open Access rights. This article was accepted and published in Journal of Topology under Open Access conditions.
[en] Every geodesic current on a hyperbolic surface has an associated dual space. If the current is a lamination, this dual embeds isometrically into a real tree. We show that, in general, the dual space is a Gromov hyperbolic metric tree-graded space, and express its Gromov hyperbolicity constant in terms of the geodesic current. In the case of geodesic currents with no atoms and full support, such as those coming from certain higher rank representations, we show the duals are homeomorphic to the surface. We also analyze the completeness of the dual and the properties of the action of the fundamental group of the surface on the dual. Furthermore, we compare two natural topologies in the space of duals.
Disciplines :
Mathematics
Author, co-author :
De Rosa, Luca; Department of Mathematics, ETH Zurich, Zurich, Switzerland
MARTINEZ GRANADO, Didac ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
FNR - Luxembourg National Research Fund National Science Foundation National Science Foundation
Funding number :
17145118
Funding text :
We thank Marc Burger, Indira Chatterji, Michael Kapovich, Giuseppe Martone, Eduardo Oreg\u00F3n\u2010Reyes, Anne Parreau, Beatrice Pozzetti, Jenya Sapir and Dylan Thurston for useful conversations. We are grateful to the referee for their invaluable feedback. Their careful reading of the previous version revealed several errors and offered insightful suggestions that enhanced the clarity of the manuscript. The first author would like to thank Raphael Appenzeller, Benjamin Br\u00FCck, Matthew Cordes, Xenia Flamm and Francesco Fournier Facio for insightful conversations. The second author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 \u2018RNMS: Geometric Structures and Representation Varieties\u2019 (the GEAR Network), and from the Luxembourg National research Fund AFR/Bilateral\u2010ReSurface 22/17145118, and thanks Marc Burger for his invitation to ETH Z\u00FCrich, during which a portion of this work was done.
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