Volume 13, issue 6 (2013)

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Length functions of Hitchin representations

Guillaume Dreyer

Algebraic & Geometric Topology 13 (2013) 3153–3173
Abstract

Given a Hitchin representation $\rho :\phantom{\rule{0.3em}{0ex}}{\pi }_{1}\left(S\right)\to {PSL}_{n}\left(ℝ\right)$, we construct $n$ continuous functions defined on the space of Hölder geodesic currents such that, for a closed, oriented curve $\gamma$ in $S$, the eigenvalue of the matrix $\rho \left(\gamma \right)\in {PSL}_{n}\left(ℝ\right)$ is of the form $±exp\phantom{\rule{0.3em}{0ex}}{\ell }_{i}^{\rho }\left(\gamma \right)$: such functions generalize to higher rank Thurston’s length function of Fuchsian representations. Identities and differentiability properties of these lengths ${\ell }_{i}^{\rho }$, as well as applications to eigenvalue estimates, are also considered.

Keywords
Hitchin representation, Anosov representation, length function, Hölder geodesic current
Mathematical Subject Classification
Primary: 57M50
Secondary: 57M05, 37F30, 22E40
Publication
Revised: 4 February 2013
Accepted: 19 March 2013
Published: 11 September 2013
Authors
 Guillaume Dreyer Department of Mathematics University of Notre Dame 255 Hurley Hall Notre Dame, IN 46556 USA