Algebraic & Geometric Topology

Length functions of Hitchin representations

Guillaume Dreyer

Full-text: Open access


Given a Hitchin representation ρ:π1(S) PSLn(), we construct n continuous functions iρ:C Höl(S) 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) PSLn() is of the form ± expiρ(γ): such functions generalize to higher rank Thurston’s length function of Fuchsian representations. Identities and differentiability properties of these lengths iρ, as well as applications to eigenvalue estimates, are also considered.

Article information

Algebr. Geom. Topol., Volume 13, Number 6 (2013), 3153-3173.

Received: 11 September 2012
Revised: 4 February 2013
Accepted: 19 March 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Hitchin representation Anosov representation length function Hölder geodesic current


Dreyer, Guillaume. Length functions of Hitchin representations. Algebr. Geom. Topol. 13 (2013), no. 6, 3153--3173. doi:10.2140/agt.2013.13.3153.

Export citation


  • F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71–158
  • F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139–162
  • F Bonahon, Geodesic laminations with transverse Hölder distributions, Ann. Sci. École Norm. Sup. 30 (1997) 205–240
  • F Bonahon, Transverse Hölder distributions for geodesic laminations, Topology 36 (1997) 103–122
  • F Bonahon, G Dreyer, Hitchin representations and geodesic laminations, in preparation
  • F Bonahon, G Dreyer, Parametrizing Hitchin components
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657–661
  • G Dreyer, Thurston's cataclysms for Anosov representations
  • A Fathi, F Laudenbach, V Poénaru, Thurston's work on surfaces, Mathematical Notes 48, Princeton Univ. Press (2012)
  • V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006) 1–211
  • É Ghys, P de la Harpe, Editors, Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser, Boston, MA (1990)
  • W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557–607
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, (S M Gernsten, editor), Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
  • M Gromov, Three remarks on geodesic dynamics and fundamental group, Enseign. Math. 46 (2000) 391–402
  • O Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, J. Differential Geom. 80 (2008) 391–431
  • O Guichard, A Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012) 357–438
  • N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449–473
  • F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51–114
  • G A Margulis, Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. 17, Springer, Berlin (1991)
  • D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 3rd edition, Ergeb. Math. Grenzgeb. 34, Springer, Berlin (1994)
  • R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton Univ. Press (1992)
  • M Pollicott, R Sharp, Length asymptotics in higher Teichmuller theory, in preparation
  • W P Thurston, Minimal stretch maps between hyperbolic surfaces
  • W P Thurston, The geometry and topology of $3$–manifolds, Princeton University (1982)
  • A Weil, On discrete subgroups of Lie groups, Ann. of Math. 72 (1960) 369–384