×

Horowitz-Randol pairs of curves in \(q\)-differential metrics. (English) Zbl 1305.30019

Summary: The Euclidean cone metrics coming from \(q\)-differentials on a closed surface of genus \(g \geq 2\) 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 \(q \geq 1\), the corresponding equivalence relation has arbitrarily large equivalence classes. In addition, we describe how these equivalence relations are related to each other.

MSC:

30F10 Compact Riemann surfaces and uniformization
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] R Abraham, Bumpy metrics (editors S S Chern, S Smale), Amer. Math. Soc. (1970) 1 · Zbl 0215.23301
[2] J W Anderson, Variations on a theme of Horowitz (editors Y Komori, V Markovic, C Series), London Math. Soc. Lecture Note Ser. 299, Cambridge Univ. Press, Cambridge (2003) 307 · Zbl 1046.57015 · doi:10.1017/CBO9780511542817.015
[3] D V Anosov, Generic properties of closed geodesics, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982) 675, 896 · Zbl 0512.58014
[4] F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139 · Zbl 0653.32022 · doi:10.1007/BF01393996
[5] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999) · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9
[6] M Duchin, C J Leininger, K Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010) 231 · Zbl 1207.53052 · doi:10.1007/s00222-010-0262-y
[7] H M Farkas, I Kra, Riemann surfaces, Graduate Texts in Mathematics 71, Springer (1992) · Zbl 0764.30001 · doi:10.1007/978-1-4612-2034-3
[8] A Fathi, F Laudenbach, V Poénaru, editors, Travaux de Thurston sur les surfaces, Astérisque 66-67, Soc. Math. France (1979) 284
[9] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002) · Zbl 1044.55001
[10] E Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939) 261 · Zbl 0024.08003
[11] R D Horowitz, Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math. 25 (1972) 635 · Zbl 1184.20009 · doi:10.1002/cpa.3160250602
[12] C M Judge, Conformally converting cusps to cones, Conform. Geom. Dyn. 2 (1998) 107 · Zbl 1053.53506 · doi:10.1090/S1088-4173-98-00024-1
[13] I Kapovich, G Levitt, P Schupp, V Shpilrain, Translation equivalence in free groups, Trans. Amer. Math. Soc. 359 (2007) 1527 · Zbl 1119.20037 · doi:10.1090/S0002-9947-06-03929-8
[14] C J Leininger, Equivalent curves in surfaces, Geom. Dedicata 102 (2003) 151 · Zbl 1037.57013 · doi:10.1023/B:GEOM.0000006579.44245.92
[15] J D Masters, Length multiplicities of hyperbolic \(3\)-manifolds, Israel J. Math. 119 (2000) 9 · Zbl 0963.57006 · doi:10.1007/BF02810661
[16] H Masur, J Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. of Math. 134 (1991) 455 · Zbl 0774.58024 · doi:10.2307/2944356
[17] Y N Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992) 151 · Zbl 0763.53042
[18] W D Neumann, Notes on geometry and \(3\)-manifolds (editors K Böröczky Jr., W D Neumann, A Stipsicz), Bolyai Soc. Math. Stud. 8, János Bolyai Math. Soc., Budapest (1999) 191 · Zbl 0944.57012
[19] R C Penner, J L Harer, Combinatorics of train tracks, Annals of Mathematics Studies 125, Princeton Univ. Press (1992) · Zbl 0765.57001
[20] K Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol. 9 (2005) 179 · Zbl 1082.30037 · doi:10.2140/gt.2005.9.179
[21] B Randol, The length spectrum of a Riemann surface is always of unbounded multiplicity, Proc. Amer. Math. Soc. 78 (1980) 455 · Zbl 0447.30034 · doi:10.2307/2042345
[22] K Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. 5, Springer, Berlin (1984) · Zbl 0547.30001 · doi:10.1007/978-3-662-02414-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.