Geometry & Topology

Collar lemma for Hitchin representations

Gye-Seon Lee and Tengren Zhang

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

There is a classical result known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface intersect each other, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for every hyperbolic structure on the surface. In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.

Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2243-2280.

Dates
Received: 22 December 2015
Revised: 10 July 2016
Accepted: 8 August 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1508437641

Digital Object Identifier
doi:10.2140/gt.2017.21.2243

Mathematical Reviews number (MathSciNet)
MR3654108

Zentralblatt MATH identifier
1367.57010

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 30F60: Teichmüller theory [See also 32G15] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Keywords
hyperbolic surfaces convex real projective surfaces collar lemma Hitchin representations

Citation

Lee, Gye-Seon; Zhang, Tengren. Collar lemma for Hitchin representations. Geom. Topol. 21 (2017), no. 4, 2243--2280. doi:10.2140/gt.2017.21.2243. https://projecteuclid.org/euclid.gt/1508437641


Export citation

References

  • L Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960) 94–97
    Mathematical Reviews (MathSciNet): MR0111834
    Zentralblatt MATH: 0090.05101
    Digital Object Identifier: doi:10.1090/S0002-9904-1960-10413-2
    Project Euclid: euclid.bams/1183523461
  • M Bridgeman, R Canary, F Labourie, A Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal. 25 (2015) 1089–1179
    Mathematical Reviews (MathSciNet): MR3385630
    Zentralblatt MATH: 1360.37078
    Digital Object Identifier: doi:10.1007/s00039-015-0333-8
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, New York (1999)
    Mathematical Reviews (MathSciNet): MR1744486
    Zentralblatt MATH: 0988.53001
  • M Burger, M B Pozzetti, Maximal representations, non Archimedean Siegel spaces, and buildings, preprint (2015)
    arXiv: 1509.01184
    Zentralblatt MATH: 06779922
  • P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser, Boston (1992)
    Mathematical Reviews (MathSciNet): MR1183224
    Zentralblatt MATH: 1239.32001
  • S Choi, The Margulis lemma and the thick and thin decomposition for convex real projective surfaces, Adv. Math. 122 (1996) 150–191
    Mathematical Reviews (MathSciNet): MR1405450
    Digital Object Identifier: doi:10.1006/aima.1996.0058
  • S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657–661
    Mathematical Reviews (MathSciNet): MR1145415
    Zentralblatt MATH: 0810.57005
    Digital Object Identifier: doi:10.1090/S0002-9939-1993-1145415-8
  • D Cooper, D D Long, S Tillmann, On convex projective manifolds and cusps, Adv. Math. 277 (2015) 181–251
    Mathematical Reviews (MathSciNet): MR3336086
    Zentralblatt MATH: 06431971
    Digital Object Identifier: doi:10.1016/j.aim.2015.02.009
  • K Corlette, Flat $G$–bundles with canonical metrics, J. Differential Geom. 28 (1988) 361–382
    Mathematical Reviews (MathSciNet): MR965220
    Zentralblatt MATH: 0676.58007
    Digital Object Identifier: doi:10.4310/jdg/1214442469
    Project Euclid: euclid.jdg/1214442469
  • D B A Epstein, A Marden, V Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. 159 (2004) 305–336
    Mathematical Reviews (MathSciNet): MR2052356
    Zentralblatt MATH: 1064.30044
    Digital Object Identifier: doi:10.4007/annals.2004.159.305
  • W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791–845
    Mathematical Reviews (MathSciNet): MR1053346
    Zentralblatt MATH: 0711.53033
    Digital Object Identifier: doi:10.4310/jdg/1214444635
    Project Euclid: euclid.jdg/1214444635
  • O Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, J. Differential Geom. 80 (2008) 391–431
    Mathematical Reviews (MathSciNet): MR2472478
    Zentralblatt MATH: 1223.57015
    Digital Object Identifier: doi:10.4310/jdg/1226090482
    Project Euclid: euclid.jdg/1226090482
  • O Guichard, A Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012) 357–438
    Mathematical Reviews (MathSciNet): MR2981818
    Zentralblatt MATH: 1270.20049
    Digital Object Identifier: doi:10.1007/s00222-012-0382-7
  • L Keen, Collars on Riemann surfaces, from “Discontinuous groups and Riemann surfaces” (L Greenberg, editor), Ann. of Math. Studies 79, Princeton Univ. Press (1974) 263–268
    Mathematical Reviews (MathSciNet): MR0379833
  • F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51–114
    Mathematical Reviews (MathSciNet): MR2221137
    Zentralblatt MATH: 1103.32007
    Digital Object Identifier: doi:10.1007/s00222-005-0487-3
  • F Labourie, G McShane, Cross ratios and identities for higher Teichmüller–Thurston theory, Duke Math. J. 149 (2009) 279–345
    Mathematical Reviews (MathSciNet): MR2541705
    Zentralblatt MATH: 1182.30075
    Digital Object Identifier: doi:10.1215/00127094-2009-040
    Project Euclid: euclid.dmj/1248182808
  • J P Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43 (1976) 829–840
    Mathematical Reviews (MathSciNet): MR0432921
    Zentralblatt MATH: 0341.30020
    Digital Object Identifier: doi:10.1215/S0012-7094-76-04364-7
    Project Euclid: euclid.dmj/1077311954
  • N Tholozan, Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$, preprint (2015)
    arXiv: 1506.04640
  • T Zhang, Degeneration of Hitchin representations along internal sequences, Geom. Funct. Anal. 25 (2015) 1588–1645
    Mathematical Reviews (MathSciNet): MR3426063
    Digital Object Identifier: doi:10.1007/s00039-015-0342-7

MSP

Geometry & Topology

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more