Horospheres in Teichmüller space and mapping class group
Annales de l'Institut Fourier, Volume 73 (2023) no. 4, pp. 1677-1707.

We study the geometry of horospheres in Teichmüller space of Riemann surfaces of genus g with n punctures, where 3g-3+n2. We show that every C 1 -diffeomorphism of Teichmüller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and metric balls, we obtain a new proof of Royden’s Theorem that the isometry group of the Teichmüller metric is the extended mapping class group.

Nous étudions la géométrie des horosphéres dans l’espace de Teichmüller des surfaces de Riemann de genre g avec n perforations, oú 3g-3+n2. Nous montrons que chaque C 1 -difféomorphisme de l’espace de Teichmüller dans lui-même qui préserve les horosphéres est un élément du groupe mapping class étendu. En utilisant la relation entre les horosphéres et les boules métriques, nous obtenons une nouvelle preuve du théoréme de Royden qui affirme que le groupe d’isométrie de la métrique de Teichmüller est le groupe mapping class étendu.

Published online:
DOI: 10.5802/aif.3556
Classification: 32G15, 30F30, 30F60
Keywords: Extremal length, horosphere, mapping class group, Teichmüller space.
Mot clés : longueur extrémale, horosphère, groupe mapping class, espace de Teichmüller
Su, Weixu 1; Tan, Dong 2

1 School of Mathematics Sun Yat-sen University 210275, Guangzhou (P. R. China)
2 Guangxi Center for Mathematical Research Guangxi University 530000, Nanning (P. R. China)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Su, Weixu and Tan, Dong},
     title = {Horospheres in {Teichm\"uller} space and mapping class group},
     journal = {Annales de l'Institut Fourier},
     pages = {1677--1707},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {73},
     number = {4},
     year = {2023},
     doi = {10.5802/aif.3556},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3556/}
AU  - Su, Weixu
AU  - Tan, Dong
TI  - Horospheres in Teichmüller space and mapping class group
JO  - Annales de l'Institut Fourier
PY  - 2023
SP  - 1677
EP  - 1707
VL  - 73
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3556/
DO  - 10.5802/aif.3556
LA  - en
ID  - AIF_2023__73_4_1677_0
ER  - 
%0 Journal Article
%A Su, Weixu
%A Tan, Dong
%T Horospheres in Teichmüller space and mapping class group
%J Annales de l'Institut Fourier
%D 2023
%P 1677-1707
%V 73
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3556/
%R 10.5802/aif.3556
%G en
%F AIF_2023__73_4_1677_0
Su, Weixu; Tan, Dong. Horospheres in Teichmüller space and mapping class group. Annales de l'Institut Fourier, Volume 73 (2023) no. 4, pp. 1677-1707. doi : 10.5802/aif.3556. https://aif.centre-mersenne.org/articles/10.5802/aif.3556/

[1] Bourque, Maxime Fortier; Rafi, Kasra Non-convex balls in the Teichmüller metric, J. Differ. Geom., Volume 110 (2018) no. 3, pp. 379-412 | Zbl

[2] Busemann, Herbert The geometry of geodesics, Courier Corporation, 2012

[3] Earle, Clifford J.; Kra, Irwin On isometries between Teichmüller spaces, Duke Math. J., Volume 41 (1974) no. 3, pp. 583-591 | Zbl

[4] Fathi, Albert; Laudenbach, François; Poénaru, Valentin Thurston’s Work on Surfaces, Mathematical Notes, 48, Princeton University Press, 2021 | DOI

[5] Gardiner, Frederick P. Measured foliations and the minimal norm property for quadratic differentials, Acta Math., Volume 152 (1984) no. 1, pp. 57-76 | DOI | MR | Zbl

[6] Gardiner, Frederick P. Teichmüller theory and quadratic differentials, Pure Appl. Math., John Wiley & Sons, 1987

[7] Gardiner, Frederick P.; Masur, Howard Extremal length geometry of Teichmüller space, Complex Var. Elliptic Equ., Volume 16 (1991) no. 2-3, pp. 209-237 | Zbl

[8] Harvey, Willam J. Boundary structure of the modular group, Riemann Surfaces Related Topics (Annals of Mathematics Studies), Volume 97, Princeton University Press, 2016, pp. 245-252

[9] Hubbard, John; Masur, Howard Quadratic differentials and foliations, Acta Math., Volume 142 (1979) no. 1, pp. 221-274 | DOI | MR | Zbl

[10] Ivanov, Nikolai V. Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115, American Mathematical Society, 1992 | DOI | MR

[11] Ivanov, Nikolai V. Automorphisms of complexes of curves and of Teichmüller spaces, Int. Math. Res. Not., Volume 1997 (1997) no. 14, pp. 651-666 | DOI | MR

[12] Ivanov, Nikolai V. Isometries of Teichmüller spaces from the point of view of Mostow rigidity, Topology, ergodic theory, real algebraic geometry (Advances in the Mathematical Sciences), Volume 202, American Mathematical Society, 2001, pp. 131-149 | MR | Zbl

[13] Jeffers, Jason Lost theorems of geometry, Am. Math. Mon., Volume 107 (2000) no. 9, pp. 800-812 | DOI | MR | Zbl

[14] Kerckhoff, Steven P. The asymptotic geometry of Teichmüller space, Topology, Volume 19 (1980) no. 1, pp. 23-41 | DOI | MR | Zbl

[15] Korkmaz, Mustafa Complexes of curves and mapping class groups, Ph. D. Thesis, Michigan State University (1996) | DOI | MR

[16] Lenzhen, Anna; Masur, Howard Criteria for the divergence of pairs of Teichmüller geodesics, Geom. Dedicata, Volume 144 (2010) no. 1, pp. 191-210 | DOI | Zbl

[17] Luo, Feng Automorphisms of the complex of curves, Topology, Volume 39 (2000) no. 2, pp. 283-298 | MR | Zbl

[18] Masur, Howard Transitivity properties of the horocyclic and geodesic flows on moduli space, J. Anal. Math., Volume 39 (1981) no. 1, pp. 1-10 | DOI | MR | Zbl

[19] Masur, Howard The Teichmüller flow is Hamiltonian, Proc. Am. Math. Soc., Volume 123 (1995) no. 12, pp. 3739-3747 | Zbl

[20] Masur, Howard Dense geodesics in moduli space, Riemann Surfaces Related Topics (Annals of Mathematics Studies), Volume 97, Princeton University Press, 2016, pp. 417-438

[21] Minsky, Yair N. Teichmüller geodesics and ends of hyperbolic 3-manifolds, Elsevier, 1993

[22] Royden, Halsey L. Automorphisms and Isometries of Teichmüller Space, Advances in the Theory of Riemann Surfaces (Annals of Mathematics Studies), Volume 66, Princeton University Press, 1971, pp. 369-384 | DOI | Zbl

[23] Tan, Dong Totally geodesic homeomorphisms between Teichmüller spaces, Ann. Acad. Sci. Fenn., Math., Volume 45 (2020) no. 2, pp. 673-685 | MR | Zbl

[24] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Am. Math. Soc., Volume 19 (1988) no. 2, pp. 417-431 | DOI | MR | Zbl

[25] Walsh, Cormac The horoboundary and isometry group of Thurston’s Lipschitz metric, Handbook of Teichmüller theory. Volume IV (IRMA Lectures in Mathematics and Theoretical Physics), Volume 19, European Mathematical Society, 2014, pp. 327-353 | DOI | MR | Zbl

[26] Walsh, Cormac The asymptotic geometry of the Teichmüller metric, Geom. Dedicata, Volume 200 (2019) no. 1, pp. 115-152 | DOI | MR | Zbl

Cited by Sources: