Constructing equivariant maps for representations
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 393-428

We show that if Γ is a discrete subgroup of the group of the isometries of k , and if ρ is a representation of Γ into the group of the isometries of n , then any ρ-equivariant map F: k n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable ρ-equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if Γ is of divergence type and ρ is non-elementary, then there exists a measurable map D: k n conjugating the actions of Γ and ρ(Γ). Related applications are discussed.

On montre que pour chaque groupe discrète d’isométries G de l’espace hyperbolique de dimension k, chaque représentation R de G dans le groupe Isom( n ) et pour chaque application R-équivariante F de k en n , il existe une extension de F dans le sens faible des mesures. On obtient donc, comme conséquence de ce fait, une extension d’un résultat de Besson, Courtois et Gallot sur l’existence d’une application équivariante qui n’augmente pas le volume. En plus, avec une hypothèse supplémentaire, on montre que notre extension faible est effectivement une vraie application mesurable du bord à l’infini de k . On utilise alors ce résultat pour obtenir une version mesurable du résultat de Cannon et Thurston sur l’existence de courbes de Peano équivariantes. Enfin, on discute quelques applications.

DOI : https://doi.org/10.5802/aif.2434
Classification:  57M50,  37A99
Keywords: Hyperbolic spaces, discrete groups, isometries, representation, equivariant, barycenter, natural map, volume
@article{AIF_2009__59_1_393_0,
     author = {Francaviglia, Stefano},
     title = {Constructing equivariant maps for representations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     pages = {393-428},
     doi = {10.5802/aif.2434},
     mrnumber = {2514869},
     zbl = {1171.57016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_1_393_0}
}
Francaviglia, Stefano. Constructing equivariant maps for representations. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 393-428. doi : 10.5802/aif.2434. http://www.numdam.org/item/AIF_2009__59_1_393_0/

[1] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego Functions of bounded variation and free discontinuity problems, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (2000) | MR 1857292 | Zbl 0957.49001

[2] Ambrosio, Luigi; Lisini, Stefano; Savaré, Giuseppe Stability of flows associated to gradient vector fields and convergence of iterated transport maps (2005) (Preprint SNS, available version on http://cvgmt.sns.it/papers/amblissav051) | MR 2258529 | Zbl 1099.49027

[3] Benedetti, Riccardo; Petronio, Carlo Lectures on hyperbolic geometry, Springer-Verlag, Berlin, Universitext (1992) | MR 1219310 | Zbl 0768.51018

[4] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal., Tome 5 (1995), pp. 731-799 | Article | MR 1354289 | Zbl 0851.53032

[5] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Minimal entropy and Mostow’s rigidity theorems, Ergodic Theory Dynam. Systems, Tome 16 (1996) no. 4, pp. 623-649 | Article | MR 1406425 | Zbl 0887.58030

[6] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Lemme de Schwarz réel et applications géométriques, Acta Mathematica, Tome 183 (1999) no. 2, pp. 145-169 | Article | MR 1738042 | Zbl 1035.53038

[7] Bishop, Christopher J.; Jones, Peter W. Hausdorff dimension and Kleinian groups, Acta Mathematica, Tome 179 (1997) no. 1, pp. 1-39 | Article | MR 1484767 | Zbl 0921.30032

[8] Canary, Richard D. Ends of hyperbolic 3-manifolds, J. Amer. Math. Soc., Tome 6 (1993) no. 1, pp. 1-35 | MR 1166330 | Zbl 0810.57006

[9] Cannon, James W.; Thurston, William P. Group invariant Peano curves, Preprint (1989)

[10] Dellacherie, Claude; Meyer, Paul-André Probabilities and potential, North-Holland Publishing Co., Amsterdam, North-Holland Mathematics Studies, Tome 29 (1978) | MR 521810 | Zbl 0494.60001

[11] Douady, Adrien; Earle, Clifford J. Conformally natural extension of homeomorphisms of the circle, Acta Math., Tome 157 (1986) no. 1-2, pp. 23-48 | Article | MR 857678 | Zbl 0615.30005

[12] Dunfield, Nathan M. Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math., Tome 136 (1999) no. 3, pp. 623-657 | Article | MR 1695208 | Zbl 0928.57012

[13] Francaviglia, Stefano Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, Int. Math. Res. Not. (2004) no. 9, pp. 425-459 | Article | MR 2040346 | Zbl 1088.57015

[14] Francaviglia, Stefano; Klaff, Ben Maximal volume representations are Fuchsian, Geom. Dedicata, Tome 117 (2006), pp. 111-124 | Article | MR 2231161 | Zbl 1096.51004

[15] Kapovich, Michael Hyperbolic manifolds and discrete groups, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 183 (2001) | MR 1792613 | Zbl 0958.57001

[16] Klaff, B. Boundary slopes of knots in closed 3 -manifolds with cyclic fundamental group, University Illinois-Chicago (2003) (Ph. D. Thesis)

[17] Mahan, Mj Cannon-Thurston Maps and Bounded Geometry (arXiv:math.GT/0701725)

[18] Mahan, Mj Ending Laminations and Cannon-Thurston Maps (arXiv:math.GT/07017 25)

[19] Mcmullen, Curtis T. Local connectivity, Kleinian groups and geodesics on the blowup of the torus., Invent. Math., Tome 146 (2001) no. 1, pp. 35-91 | Article | MR 1859018 | Zbl 1061.37025

[20] Minsky, Yair N. On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds, J. Amer. Math. Soc., Tome 7 (1994) no. 3, pp. 539-588 | MR 1257060 | Zbl 0808.30027

[21] Miyachi, Hideki Moduli of continuity of Cannon-Thurston maps, Spaces of Kleinian groups, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 329 (2006), pp. 121-149 | MR 2258747 | Zbl 1098.30032

[22] Nicholls, Peter J. The ergodic theory of discrete groups, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 143 (1989) | MR 1041575 | Zbl 0674.58001

[23] Roblin, Thomas Sur l’ergodicité rationnelle et les propriétés ergodiques du flot géodésique dans les variétés hyperboliques, Ergodic Theory Dynam. Systems, Tome 20 (2000) no. 6, pp. 1785-1819 | Article | MR 1804958 | Zbl 0968.37012

[24] Scorza, Irène Fractal curves in the limit sets of simply degenerate once punctured torus groups (Preprint)

[25] Soma, Teruhiko Equivariant, almost homeomorphic maps between S 1 and S 2 , Proc. Amer. Math. Soc., Tome 123 (1995) no. 9, pp. 2915-2920 | MR 1277134 | Zbl 0855.57012

[26] Stein, Elias M. Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N.J., Princeton Mathematical Series, No. 30 (1970) | MR 290095

[27] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 171-202 | Article | Numdam | MR 556586 | Zbl 0439.30034

[28] Sullivan, Dennis Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., Tome 153 (1984) no. 3-4, pp. 259-277 | Article | MR 766265 | Zbl 0566.58022

[29] Thurston, W. P. The geometry and topology of 3 -manifolds, Princeton University Mathematics Department, Mimeographed notes (1979)

[30] Yue, Chengbo Dimension and rigidity of quasi-Fuchsian representations, Ann. of Math. (2), Tome 143 (1996) no. 2, pp. 331-355 | Article | MR 1381989 | Zbl 0843.22019

[31] Yue, Chengbo The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., Tome 348 (1996) no. 12, pp. 4965-5005 | Article | MR 1348871 | Zbl 0864.58047