This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into - known as the n-Hitchin component - to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains all -Hitchin components as well as the set of negatively curved metrics on the surface.
@article{PMIHES_2007__106__139_0,
author = {Labourie, Fran\c{c}ois},
title = {Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {139--213},
publisher = {Springer},
volume = {106},
year = {2007},
doi = {10.1007/s10240-007-0009-5},
language = {en},
url = {http://www.numdam.org/articles/10.1007/s10240-007-0009-5/}
}
TY - JOUR
AU - Labourie, François
TI - Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle
JO - Publications Mathématiques de l'IHÉS
PY - 2007
SP - 139
EP - 213
VL - 106
PB - Springer
UR - http://www.numdam.org/articles/10.1007/s10240-007-0009-5/
DO - 10.1007/s10240-007-0009-5
LA - en
ID - PMIHES_2007__106__139_0
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%0 Journal Article
%A Labourie, François
%T Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle
%J Publications Mathématiques de l'IHÉS
%D 2007
%P 139-213
%V 106
%I Springer
%U http://www.numdam.org/articles/10.1007/s10240-007-0009-5/
%R 10.1007/s10240-007-0009-5
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%F PMIHES_2007__106__139_0
Labourie, François. Cross ratios, surface groups, $PSL(n,\mathbf {R})$ and diffeomorphisms of the circle. Publications Mathématiques de l'IHÉS, Tome 106 (2007), pp. 139-213. doi : 10.1007/s10240-007-0009-5. http://www.numdam.org/articles/10.1007/s10240-007-0009-5/
1. , Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Tr. Mat. Inst. Steklova, vol. 90, 1967. | MR | Zbl
2. , , The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond., Ser. A, 308 (1983), 523-615 | MR | Zbl
3. , The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), 139-162 | MR | Zbl
4. , Sur le birapport au bord des CAT(-1)-espaces, Publ. Math., Inst. Hautes Études Sci., 83 (1996), 95-104 | Numdam | MR | Zbl
5. , , , Maximal surface group representations in isometry groups of classical hermitian symmetric spaces, Geom. Dedicata, 122 (2006), 185-213 | MR | Zbl
6. , , , Surface group representations and U(p,q)-Higgs bundles, J. Differ. Geom., 64 (2003), 111-170 | MR | Zbl
7. , , , Moduli spaces of holomorphic triples over compact Riemann surfaces, Math. Ann., 328 (2004), 299-351 | MR | Zbl
8. , , , , Maximal representations of surface groups: symplectic Anosov structures, Pure Appl. Math. Q., 1 (2005), 543-590 | MR
9. , , , Surface group representations with maximal Toledo invariant, C. R., Math., Acad. Sci. Paris, 336 (2003), 387-390 | MR | Zbl
10. , Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, 1951, pp. 29-55. | MR | Zbl
11. , , Moduli spaces of local systems and higher Teichmüller theory, Publ. Math., Inst. Hautes Études Sci., 103 (2006), 1-211 | Numdam | MR | Zbl
12. , , Representations of the fundamental group of a closed oriented surface in Sp(4,ℝ), Topology, 43 (2004), 831-855 | MR | Zbl
13. , Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Sci. Éc. Norm. Supér., IV. Sér., 20 (1987), 251-270 | Numdam | Zbl
14. , The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), 200-225 | MR | Zbl
15. , Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., 85 (1986), 263-302 | MR | Zbl
16. , Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991), Proc. Symp. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43-96. | MR | Zbl
17. , Components of spaces of representations and stable triples, Topology, 40 (2001), 823-850 | MR | Zbl
18. O. Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, to appear in J. Differ. Geom, (2005).
19. , Cocycles, Hausdorff measures and cross ratios, Ergodic Theory Dyn. Syst., 17 (1997), 1061-1081 | MR | Zbl
20. , The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., III. Ser., 55 (1987), 59-126 | MR | Zbl
21. , Lie groups and Teichmüller space, Topology, 31 (1992), 449-473 | MR | Zbl
22. F. Labourie, Cross ratios, Anosov representations and the energy functional on Teichmüller space, to appear in Ann. Sci. Éc. Norm. Supér., IV. Sér.
23. , Anosov flows, surface groups and curves in projective space, Invent. Math., 165 (2006), 51-114 | MR | Zbl
24. F. Labourie and G. McShane, Cross ratios and identities for higher Teichmüller-Thurston theory, math.DG/0611245, 2006.
25. , Structure au bord des variétés à courbure négative, Séminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994-1995, Sémin. Théor. Spectr. Géom., vol. 13, Univ. Grenoble I, Saint, 1995, pp. 97-122. | Numdam | MR | Zbl
26. , Simple geodesics and a series constant over Teichmüller space, Invent. Math., 132 (1998), 607-632 | MR | Zbl
27. , Le spectre marqué des longueurs des surfaces à courbure négative, Ann. Math. (2), 131 (1990), 151-162 | MR | Zbl
28. , Sur la géometrie symplectique de l'espace des géodésiques d'une variété à courbure négative, Rev. Mat. Iberoam., 8 (1992), 441-456 | Zbl
29. , The symplectic structure on moduli space, The Floer Memorial Volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 627-635. | MR | Zbl
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