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An introduction to pressure metrics for higher Teichmüller spaces

Published online by Cambridge University Press:  17 March 2017

MARTIN BRIDGEMAN ,
RICHARD CANARY  and
ANDRÉS SAMBARINO
MARTIN BRIDGEMAN
Affiliation:
Boston College, Chestnut Hill, MA 02467, USA
RICHARD CANARY
Affiliation:
University of Michigan, Ann Arbor, MI 41809, USA
ANDRÉS SAMBARINO
Affiliation:
Université Pierre et Marie Curie (Paris VI), 75005 Paris, France
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Abstract

We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2001 - 2035
Copyright
© Cambridge University Press, 2017 

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References

Ahlfors, L. and Beurling, A.. The boundary correspondence under quasiconformal mappings. Acta Math. 96 (1956), 125–142.Google Scholar
Anderson, J. W. and Rocha, A.. Analyticity of Hausdorff dimension of limit sets of Kleinian groups. Ann. Acad. Sci. Fenn. Math. 22 (1997), 349–364.Google Scholar
Benoist, Y.. Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7 (1997), 1–47.Google Scholar
Benoist, Y.. Propriétés asymptotiques des groupes linéaires II. Adv. Stud. Pure Math. 26 (2000), 33–48.Google Scholar
Benoist, Y.. Convexes divisibles I. Algebraic Groups and Arithmetic (Tata Institute of Fundamental Research Studies in Mathematics, 17) . TIFR Publications, Mumbai, 2004, pp. 339–374.Google Scholar
Benoist, Y.. Convexes divisibles III. Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 793–832.Google Scholar
Bers, L.. Spaces of Kleinian groups. Maryland Conference in Several Complex Variables I (Lecture Notes in Mathematics, 155) . Springer, Berlin, 1970, pp. 9–34.Google Scholar
Bonahon, F.. The geometry of Teichmüller space via geodesic currents. Invent. Math. 92 (1988), 139–162.Google Scholar
Bowen, R.. Periodic orbits of hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, Berlin, 1975.Google Scholar
Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.Google Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181–202.Google Scholar
Bradlow, S., Garcia-Prada, O. and Gothen, P.. Deformations of maximal representations in Sp(4, ℝ). Q. J. Math. 63 (2012), 795–843.Google Scholar
Bridgeman, M.. Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space. Geom. Topol. 14 (2010), 799–831.Google Scholar
Bridgeman, M. and Canary, R.. Simple length rigidity for Kleinian surface groups and applications. Preprint, 2015, arXiv:1509.02510.Google Scholar
Bridgeman, M., Canary, R. and Labourie, F.. Simple length rigidity for Hitchin representations. In preparation.Google Scholar
Bridgeman, M., Canary, R., Labourie, F. and Sambarino, A.. The pressure metric for Anosov representations. Geom. Funct. Anal. 25 (2015), 1089–1179.Google Scholar
Bridgeman, M. and Taylor, E.. An extension of the Weil–Petersson metric to quasi-fuchsian space. Math. Ann. 341 (2008), 927–943.Google Scholar
Burger, M.. Intersection, the Manhattan curve and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not. IMRN 7 (1993), 217–225.Google Scholar
Burger, M., Iozzi, A., Labourie, F. and Wienhard, A.. Maximal representations of surface groups: symplectic Anosov structures. Pure Appl. Math. Q. 1 (2005), 555–601.Google Scholar
Burger, M., Iozzi, A. and Wienhard, A.. Surface group representations with maximal Toledo invariant. Ann. of Math. (2) 172 (2010), 517–566.Google Scholar
Burger, M., Iozzi, A. and Wienhard, A.. Maximal representations and Anosov structures. In preparation.Google Scholar
Canary, R. D., Minsky, Y. N. and Taylor, E. C.. Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds. J. Geom. Anal. 9 (1999), 17–40.Google Scholar
Champetier, C.. Petite simplification dans les groupes hyperboliques. Ann. Fac. Sci. Toulouse Math. (6) 3 (1994), 161–221.Google Scholar
Chu, T.. The Weil–Petersson metric in moduli space. Chinese J. Math. 4 (1976), 29–51.Google Scholar
Corlette, K. and Iozzi, A.. Limit sets of discrete groups of isometries of exotic hyperbolic spaces. Trans. Amer. Math. Soc. 351 (1999), 1507–1530.Google Scholar
Crampon, M.. Entropies of compact strictly convex projective manifolds. J. Mod. Dynam. 3 (2009), 511–547.Google Scholar
Dal’Bo, F. and Kim, I.. A criterion of conjugacy for Zariski dense subgroups. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 647–650.Google Scholar
Darvishzadeh, M. and Goldman, W.. Deformation spaces of convex real projective structures and hyperbolic structures. J. Korean Math. Soc. 33 (1996), 625–639.Google Scholar
Daskalopoulos, G. and Wentworth, R.. Classification of Weil–Petersson isometries. Amer. J. Math. 125 (2003), 941–975.Google Scholar
Fock, V. and Goncharov, A.. Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211.Google Scholar
Ghosh, S.. Anosov structure on Margulis space time. Preprint, available at http://front.math.ucdavis.edu/1412.8211.Google Scholar
Ghosh, S.. The pressure metric on the Margulis multiverse. Geom. Dedicata, accepted. Preprint, available at http://front.math.ucdavis.edu/1505.00534.Google Scholar
Goldman, W. and Labourie, F.. Geodesics in Margulis spacetimes. Ergod. Th. & Dynam. Sys. 32 (2012), 643–651.Google Scholar
Goldman, W., Labourie, F. and Margulis, G.. Proper affine actions and geodesic flows of hyperbolic surfaces. Ann. of Math. (2) 170 (2009), 1051–1083.Google Scholar
Gromov, M.. Hyperbolic groups. Essays in Group Theory (MSRI Publications, 8) . Springer, New York, 1987, pp. 75–263.Google Scholar
Gueritaud, F., Guichard, O., Kassel, F. and Wienhard, A.. Anosov representations and proper actions. Geom. Topology, accepted. Preprint, available at http://front.math.ucdavis.edu/1502.03811.Google Scholar
Guichard, O.. In preparation.Google Scholar
Guichard, O. and Wienhard, A.. Anosov representations: domains of discontinuity and applications. Invent. Math. 190 (2012), 357–438.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Hitchin, N.. Lie groups and Teichmüller space. Topology 31 (1992), 449–473.Google Scholar
Huang, Z.. Asymptotic flatness of the Weil–Petersson metric on Teichmüller space. Geom. Dedicata 110 (2005), 81–102.Google Scholar
Humphreys, J.. Introduction to Lie Algebras and Representation Theory. Springer, Berlin, 1972.Google Scholar
Johannson, K.. Homotopy Equivalences of 3-manifolds with Boundary (Lecture Notes in Mathematics, 761) . Springer, Berlin, 1979.Google Scholar
Johnson, D. and Millson, J.. Deformation spaces associated to compact hyperbolic manifolds. Discrete Groups and Geometric Analysis (Progress in Mathematics, 67) . Birkhäuser, Boston, 1987, pp. 48–106.Google Scholar
Kapovich, M., Leeb, B. and Porti, J.. Morse actions of discrete groups on symmetric spaces. Preprint, available at http://front.math.ucdavis.edu/1403.7671.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.Google Scholar
Katok, A., Knieper, G., Pollicott, M. and Weiss, H.. Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98 (1989), 581–597.Google Scholar
Labourie, F.. Anosov flows, surface groups and curves in projective space. Invent. Math. 165 (2006), 51–114.Google Scholar
Labourie, F.. Cross ratios, Anosov representations and the energy functional on Teichmüller space. Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), 437–469.Google Scholar
Labourie, F. and Wentworth, R.. Variations along the fuchsian locus. Preprint, available at http://arxiv.org/1506.01686.Google Scholar
Li, Q.. Teichmüller space is totally geodesic in Goldman space. Asian J. Math. 20 (2016), 21–46.Google Scholar
Livšic, A. N.. Cohomology of dynamical systems. Math. USSR Izv. 6 (1972), 1296–1320.Google Scholar
Loftin, J.. The compactification of the moduli space of convex RP 2 surfaces. I. J. Differential Geom. 68 (2004), 223–276.Google Scholar
Lubotzky, A. and Magid, A.. Varieties of representations of finitely generated groups. Mem. Amer. Math. Soc. 58 (1985), no. 336.Google Scholar
Manning, A.. Topological entropy for geodesic flows. Ann. of Math. (2) 110 (1979), 567–573.Google Scholar
Margulis, G.. Free completely discontinuous groups of affine transformations. Sov. Math. Dokl. 28 (1983), 435–439.Google Scholar
Masur, H.. Extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43 (1976), 623–635.Google Scholar
Masur, H. and Wolf, M.. The Weil–Petersson isometry group. Geom. Dedicata 93 (2002), 177–190.Google Scholar
McMullen, C.. Thermodynamics, dimension and the Weil–Petersson metric. Invent. Math. 173 (2008), 365–425.Google Scholar
Mineyev, I.. Flows and joins of metric spaces. Geom. Topol. 9 (2005), 403–482.Google Scholar
Nie, X.. On the Hilbert geometry of simplicial Tits sets. Ann. Inst. Fourier (Grenoble) 65 (2015), 1005–1030.Google Scholar
Parreau, A.. Compactification d’espaces de représentations de groupes de type fini. Math. Z. 272 (2012), 51–86.Google Scholar
Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
Patterson, S.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.Google Scholar
Pollicott, M.. Symbolic dynamics for Smale flows. Amer. J. Math. 109 (1987), 183–200.Google Scholar
Pollicott, M. and Sharp, R.. Length asymptotics in higher Teichmüller theory. Proc. Amer. Math. Soc. 142 (2014), 101–112.Google Scholar
Pollicott, M. and Sharp, R.. A Weil–Petersson metric on spaces of metric graphs. Geom. Dedicata 172 (2014), 229–244.Google Scholar
Potrie, R. and Sambarino, A.. Eigenvalues and entropy of a Hitchin representation. Preprint, available athttp://front.math.ucdavis.edu/1411.5405.Google Scholar
Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, London, 1978.Google Scholar
Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.Google Scholar
Sambarino, A.. Quantitative properties of convex representations. Comment. Math. Helv. 89 (2014), 443–488.Google Scholar
Sambarino, A.. The orbital counting problem for hyperconvex representations. Ann. Inst. Fourier (Grenoble) 65 (2015), 1755–1797.Google Scholar
Schmutz, P.. Die parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen. Comment. Math. Helv. 68 (1993), 278–288.Google Scholar
Shub, M.. Global Stability of Dynamical Systems. Springer, New York, 1987.Google Scholar
Slodkowski, Z.. Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 347–355.Google Scholar
Storm, P.. Hyperbolic convex cores and simplicial volume. Duke Math. J. 140 (2007), 281–319.Google Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.Google Scholar
Tapie, S.. A variation formula for the topological entropy of convex-cocompact manifolds. Ergod. Th. & Dynam. Sys. 31 (2011), 1849–1864.Google Scholar
Thurston, W.. Hyperbolic geometry and 3-manifolds. Low-Dimensional Topology (Bangor, 1979). Cambridge University Press, Cambridge, 1982, pp. 9–25.Google Scholar
Thurston, W. P.. Hyperbolic structures on 3-manifolds, I: deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), 203–246.Google Scholar
Tromba, A.. On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil–Petersson metric. Manuscipta Math. 56 (1986), 475–497.Google Scholar
Wolf, M.. The Teichmüller theory of harmonic maps. J. Differential Geom. 29 (1989), 449–479.Google Scholar
Wolf, M.. The Weil–Petersson Hessian of length on Teichmüller space. J. Differential Geom. 91 (2012), 129–169.Google Scholar
Wolpert, S.. Noncompleteness of the Weil–Petersson metric for Teichmüller space. Pacific J. Math. 61 (1975), 573–577.Google Scholar
Wolpert, S.. Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85 (1986), 119–145.Google Scholar
Wolpert, S.. Thurston’s Riemannian metric for Teichmüller space. J. Differential Geom. 23 (1986), 143–174.Google Scholar
Wolpert, S.. Geometry of the Weil–Petersson completion of Teichmüller space. Surveys in Differential Geometry, Vol. VIII (Boston, MA, 2002). International Press of Boston, Boston, 2003, pp. 357–393.Google Scholar
Xu, B.. Incompleteness of pressure metric on Teichmüller space of a bordered surface. Preprint, available at http://front.math.ucdavis.edu/1608.03937.Google Scholar
Yue, C.. The ergodic theory of discrete isometry groups on manifolds of variable negative curvature. Trans. Amer. Math. Soc. 348 (1996), 4965–5005.Google Scholar
Zhang, T.. The degeneration of convex RP 2 structures on surfaces. Proc. Lond. Math. Soc. (3) 111 (2015), 967–1012.Google Scholar
Zhang, T.. Degeneration of Hitchin sequences along internal sequences. Geom. Funct. Anal. 25 (2015), 1588–1645.Google Scholar