Hitchin characters and geodesic laminations

Guillaume Dreyer (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.; and Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)

Abstract

For a closed surface $S$, the Hitchin component $\mathrm{Hit}_n (S)$ is a preferred component of the character variety consisting of group homomorphisms from the fundamental group $\pi_1(S)$ to the Lie group $\mathrm{PSL}_n (\mathbb{R})$. We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination $\lambda$ on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space $\mathcal{T}(S)$ by shearing coordinates associated with $\lambda$, corresponding to the case $n=2$. However, significantly new ideas are needed in this higher-dimensional case. The article concludes with a few applications.

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This research was partially supported by the grants DMS-0604866, DMS-1105402 and DMS-1406559 from the U.S. National Science Foundation, and by a Fellowship from the Simons Foundation (grant 301050). In addition, the authors gratefully acknowledge support from the NSF grants DMS-1107452, 1107263 and 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).

Received 26 May 2015

Received revised 19 March 2017

Accepted 8 April 2017

Published 27 November 2017