Lengthening deformations of singular hyperbolic tori
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, p. 1239-1260

Let S be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of S lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when S becomes Euclidean, i.e. very small.

Soit S un tore muni d’une métrique hyperbolique admettant un trou ou une singularité conique. Nous décrivons quelles déformations infinitésimales de S allongent (ou raccourcissent) toutes les géodésiques fermées. Nous étudions aussi comment la réponse à cette question dégénère lorsque S devient euclidienne, c’est-à-dire très petite.

@article{AFST_2015_6_24_5_1239_0,
     author = {Gu\'eritaud, Fran\c cois},
     title = {Lengthening deformations of singular hyperbolic tori},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {5},
     year = {2015},
     pages = {1239-1260},
     doi = {10.5802/afst.1483},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2015_6_24_5_1239_0}
}
Guéritaud, François. Lengthening deformations of singular hyperbolic tori. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 5, pp. 1239-1260. doi : 10.5802/afst.1483. http://www.numdam.org/item/AFST_2015_6_24_5_1239_0/

[1] Bonahon (F.).— Low-dimensional Geometry: from Euclidean Surfaces to Hyperbolic Knots, Student Math. Library (Vol. 49), AMS 2009, 384pp. | MR 2522946 | Zbl 1176.57001

[2] Bowditch (B. H.).— Markoff triples and quasifuchsian groups, Proc. London Math. Soc. 77, p. 697-736 (1998). | MR 1643429 | Zbl 0928.11030

[3] Conway (J.H.), Guy (R. K.).— The Book of Numbers, Springer Verlag, New York (1994). | Zbl 0866.00001

[4] Charette (V.).— Non-proper affine actions of the holonomy group of a punctured torus, Forum Math. 18, no. 1, p. 121-135 (2006). | MR 2206247 | Zbl 1096.53041

[5] Charette (V.), Drumm (T. A.), Goldman (W. M.).— Affine deformations of a three-holed sphere, Geometry & Topology 14, p. 1355-1382 (2010). | MR 2653729 | Zbl 1202.57001

[6] Charette (V.), Drumm (T. A.), Goldman (W. M.).— Finite-sided deformation spaces of complete affine 3-manifolds, J. of Topology 7 (1), p. 225-246 (2014). | MR 3180618 | Zbl 1297.30070

[7] Charette (V.), Drumm (T. A.), Goldman (W. M.).— Proper affine deformations of two-generator Fuchsian groups, arXiv:1501.04535. | MR 3180618

[8] Danciger (J.), Guéritaud (F.), Kassel (F.).— Geometry and topology of complete Lorentz spacetimes of constant curvature, Annales de l’ÉNS, 4e série, tome 49, fascicule 1, p. 1-57 (2016).

[9] Danciger (J.), Guéritaud (F.), Kassel (F.).— Margulis spacetimes via the arc complex, Inventions Mathematicae.

[10] Drumm (T. A.).— Linear holonomy of Margulis space-times, J. Diff. Geom. 38, no. 3, p. 679-690 (1993). | MR 1243791 | Zbl 0784.53040

[11] Ford (L. R.).— The fundamental region for a Fuchsian group, Bull. AMS 31, p. 531-539 (1935). | MR 1561111

[12] Goldman (W. M.), Labourie (F.), Margulis (G.).— Proper Affine Actions and Geodesic Flows of hyperbolic surfaces, Annals of Math. 170 no. 3, p. 1051-1083 (2009). | MR 2600870 | Zbl 1193.57001

[13] Goldman (W. M.), Labourie (F.), Margulis (G. A.), Minsky (Y.).— Complete flat Lorentz 3-manifolds and laminations on hyperbolic surfaces, in preparation.

[14] Goldman (W. M.).— The modular group action on real SL(2)-characters of a one-holed torus, Geometry & Topology 7, p. 443-486 (2003). | MR 2026539 | Zbl 1037.57001

[15] Hardy (G. H.), Wright (E. M.).— An Introduction to the Theory of Numbers, 5th ed. Clarendon Press, Oxford (1979). | MR 568909 | Zbl 0086.25803

[16] Margulis (G.).— Free properly discontinuous groups of affine transformations, Dokl. Akad. Nauk. SSSR 272, p. 937-940 (1983). | MR 722330

[17] Thurston (W. P.).— Minimal stretch maps between hyperbolic surfaces, 1986 preprint, arXiv:math/9801039v1.