Algebraic & Geometric Topology

Horowitz–Randol pairs of curves in $q$–differential metrics

Anja Bankovic

Full-text: Open access


The Euclidean cone metrics coming from q–differentials on a closed surface of genus g2 define an equivalence relation on homotopy classes of closed curves, where two classes are equivalent if they have the equal length in every such metric. We prove an analogue of the result of Randol for hyperbolic metrics (building on the work of Horowitz): for every integer q1, the corresponding equivalence relation has arbitrarily large equivalence classes. In addition, we describe how these equivalence relations are related to each other.

Article information

Algebr. Geom. Topol., Volume 14, Number 5 (2014), 3107-3139.

Received: 19 January 2014
Accepted: 31 January 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds

compact surfaces flat metrics hyperbolic metrics


Bankovic, Anja. Horowitz–Randol pairs of curves in $q$–differential metrics. Algebr. Geom. Topol. 14 (2014), no. 5, 3107--3139. doi:10.2140/agt.2014.14.3107.

Export citation


  • R Abraham, Bumpy metrics, from: “Global analysis”, (S-S Chern, S Smale, editors), Amer. Math. Soc. (1970) 1–3
  • J W Anderson, Variations on a theme of Horowitz, from: “Kleinian groups and hyperbolic $3$–manifolds”, (Y Komori, V Markovic, C Series, editors), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press, Cambridge (2003) 307–341
  • D V Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982) 675–709, 896 In Russian; translated in Math. USSR-Izv. 21 (1983) 1–29
  • F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139–162
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • M Duchin, C J Leininger, K Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010) 231–277
  • H M Farkas, I Kra, Riemann surfaces, 2nd edition, Graduate Texts in Mathematics 71, Springer, New York (1992)
  • A Fathi, F Laudenbach, V Poénaru (editors), Travaux de Thurston sur les surfaces, revised 2nd edition, Astérisque 66–67, Soc. Math. France, Paris (1979)
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • E Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939) 261–304
  • R D Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972) 635–649
  • C M Judge, Conformally converting cusps to cones, Conform. Geom. Dyn. 2 (1998) 107–113
  • I Kapovich, G Levitt, P Schupp, V Shpilrain, Translation equivalence in free groups, Trans. Amer. Math. Soc. 359 (2007) 1527–1546
  • C J Leininger, Equivalent curves in surfaces, Geom. Dedicata 102 (2003) 151–177
  • J D Masters, Length multiplicities of hyperbolic $3$–manifolds, Israel J. Math. 119 (2000) 9–28
  • H Masur, J Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. 134 (1991) 455–543
  • Y N Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992) 151–217
  • W D Neumann, Notes on geometry and $3$–manifolds, from: “Low dimensional topology”, (K B ör öczky, Jr, W D Neumann, A Stipsicz, editors), Bolyai Soc. Math. Stud. 8, János Bolyai Math. Soc., Budapest (1999) 191–267
  • R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton Univ. Press (1992)
  • K Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol. 9 (2005) 179–202
  • B Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity, Proc. Amer. Math. Soc. 78 (1980) 455–456
  • K Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. 5, Springer, Berlin (1984)