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Conformal Geometry and Dynamics

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On the character variety of the three-holed projective plane


Authors: Sara Maloni and Frédéric Palesi
Journal: Conform. Geom. Dyn. 24 (2020), 68-108
MSC (2010): Primary 57M50, 20E05, 37A15
DOI: https://doi.org/10.1090/ecgd/349
Published electronically: March 3, 2020
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Abstract: We study the (relative) $ \mathrm {SL}(2,\mathbb{C})$ character variety of the three-holed projective plane and the action of the mapping class group on it. We describe a domain of discontinuity for this action, which strictly contains the set of primitive stable representations defined by Minsky, and also the set of convex-cocompact characters.


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Additional Information

Sara Maloni
Affiliation: Department of Mathematics, University of Virginia, Kerchof Hall, Charlottesville, Virginia 22904-4137
Email: sm4cw@virginia.edu

Frédéric Palesi
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France
Email: frederic.palesi@univ-amu.fr

DOI: https://doi.org/10.1090/ecgd/349
Keywords: Three-holed projective plane, character varieties, mapping class group, Bowditch set, domain of discontinuity
Received by editor(s): December 21, 2017
Received by editor(s) in revised form: July 18, 2019, November 11, 2019, December 16, 2019, and December 19, 2019
Published electronically: March 3, 2020
Additional Notes: The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 RNMS: “Geometric Structures and Representation Varieties” (the GEAR Network). This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2015 semester. The authors are grateful to the organizers of the program for the invitations to participate, and to the MSRI and its staff for their hospitality and generous support.
The first author was partially supported by NSF grants DMS-1506920, DMS-1650811, DMS-1839968 and DMS-1848346.
The second author was partially supported by the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement n\(^{∘}\) FP7-246918, and by ANR VALET (ANR-13-JS01-0010) and the work has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02).
Article copyright: © Copyright 2020 American Mathematical Society