Length functions of Hitchin representations

Dreyer, Guillaume

doi:10.2140/agt.2013.13.3153

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Length functions of Hitchin representations

Guillaume Dreyer

Algebraic & Geometric Topology 13 (2013) 3153–3173

DOI: 10.2140/agt.2013.13.3153

Abstract

Given a Hitchin representation $ρ : π_{1} (S) \to {PSL}_{n} (ℝ)$ , we construct $n$ continuous functions $ℓ_{i}^{ρ} : C^{Höl} (S) \to ℝ$ defined on the space of Hölder geodesic currents $C^{Höl} (S)$ such that, for a closed, oriented curve $γ$ in $S$ , the $i^{th}$ eigenvalue of the matrix $ρ (γ) \in {PSL}_{n} (ℝ)$ is of the form $\pm exp ℓ_{i}^{ρ} (γ)$ : such functions generalize to higher rank Thurston’s length function of Fuchsian representations. Identities and differentiability properties of these lengths $ℓ_{i}^{ρ}$ , 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

References

Publication

Received: 11 September 2012

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

Volume 13, issue 6 (2013)