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Shearing deformations of Hitchin representations and the Atiyah–Bott–Goldman symplectic form

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Abstract

The Hitchin component \(\mathrm {Hit}_n(S)\) of a closed surface S is a preferred component of the character variety \(\mathcal {X}_{\mathrm {PSL}_n(\mathbb {R})}(S)\) consisting of homomorphisms from the fundamental group \(\pi _1(S)\) to the Lie group \(\mathrm {PSL}_n(\mathbb {R})\) , whose elements enjoy remarkable geometric and dynamical properties. We consider a certain type of deformations of the elements of \(\mathrm {Hit}_n(S)\), called shearing deformations, and compute their pairing for the Atiyah–Bott–Goldman symplectic form of the character variety.

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Acknowledgements

This work was carried out while the author was visiting the University of Southern California, with funding provided by the Scientific and Technological Research Council of Turkey (TÜBİTAK). She is grateful to Professor Francis Bonahon for his endless support, and to her supervisor Professor Yaşar Sözen for giving her this opportunity. She is also very thankful to the referee for their careful reading and comments.

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  1. Department of Mathematics, Hacettepe University, 06800, Ankara, Turkey

    Hatice Zeybek

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  1. Hatice Zeybek
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Correspondence to Hatice Zeybek.

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This research was supported by the Scientific and Technological Research Council of Turkey, TÜBİTAK (BİDEB-2214/A).

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Zeybek, H. Shearing deformations of Hitchin representations and the Atiyah–Bott–Goldman symplectic form. Geom Dedicata 213, 401–421 (2021). https://doi.org/10.1007/s10711-020-00588-6

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