Constructing equivariant maps for representations
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 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: 10.5802/aif.2434
Classification: 57M50, 37A99
Keywords: Hyperbolic spaces, discrete groups, isometries, representation, equivariant, barycenter, natural map, volume
Mot clés : espace hyperbolique, discrète group, isométries, représentation, équivariant, barycentre, application naturelle, volume
Francaviglia, Stefano 1

1 Dipartimento di Matematica Applicata “U.Dini” via Buonarroti 1/c 56127 Pisa (Italy)
@article{AIF_2009__59_1_393_0,
     author = {Francaviglia, Stefano},
     title = {Constructing equivariant maps for representations},
     journal = {Annales de l'Institut Fourier},
     pages = {393--428},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2434},
     zbl = {1171.57016},
     mrnumber = {2514869},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2434/}
}
TY  - JOUR
AU  - Francaviglia, Stefano
TI  - Constructing equivariant maps for representations
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 393
EP  - 428
VL  - 59
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2434/
DO  - 10.5802/aif.2434
LA  - en
ID  - AIF_2009__59_1_393_0
ER  - 
%0 Journal Article
%A Francaviglia, Stefano
%T Constructing equivariant maps for representations
%J Annales de l'Institut Fourier
%D 2009
%P 393-428
%V 59
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2434/
%R 10.5802/aif.2434
%G en
%F 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/articles/10.5802/aif.2434/

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[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., Volume 146 (2001) no. 1, pp. 35-91 | DOI | MR | Zbl

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

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

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

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

[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., Volume 123 (1995) no. 9, pp. 2915-2920 | MR | Zbl

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

[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 | DOI | Numdam | MR | Zbl

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

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

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

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

Cited by Sources: