Conformal dimension and combinatorial modulus of compact metric spaces

Carrasco Piaggio, Matias

Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 141-145.

Lʼobjet principal de cette Note est lʼétude de la dimension conforme Ahlfors régulière (${\dim }_{\mathit{AR}}X$) dʼun espace métrique compact X. Cʼest un invariant numérique de quasisymétrie, introduit par P. Pansu. Elle joue actuellement un rôle important en théorie géométrique des groupes et en dynamique conforme. On montre comment calculer ${\dim }_{\mathit{AR}}X$ à partir de modules combinatoires en considérant un exposant critique Q. Comme conséquence de lʼégalité ${\dim }_{\mathit{AR}}X=Q$, on obtient un critère général de dimension un. Les conditions sont données en termes de points de coupure locale de X. On donne par ailleurs des applications de ces résultats aux bords des groupes hyperboliques et aux ensembles de Julia des fractions rationnelles semi-hyperboliques.

In this Note we study the Ahlfors regular conformal dimension (${\dim }_{\mathit{AR}}X$) of a compact metric space X. This is a quasisymmetric numerical invariant, introduced by P. Pansu. It plays nowadays an important role in geometric group theory and in conformal dynamics. We show how to compute ${\dim }_{\mathit{AR}}X$ using a critical exponent Q associated to the combinatorial modulus. As a consequence of the equality ${\dim }_{\mathit{AR}}X=Q$, we obtain a general criterion ensuring that the AR conformal dimension is 1. The conditions are stated in terms of local cut points of X. Finally, we give applications of these results to the boundaries of Gromov hyperbolic groups and to the Julia sets of semi-hyperbolic rational maps.

[1] Bonk, M.; Kleiner, B. Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math., Volume 150 (2002), pp. 127-183

[2] Bonk, M.; Kleiner, B. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol., Volume 9 (2005), pp. 219-246

[3] M. Bourdon, B. Kleiner, Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups, 2010.

[4] Bourdon, M.; Pajot, H. Cohomologie ${\ell }_p$ et espaces de Besov, J. Reine Angew. Math., Volume 558 (2003), pp. 85-108

[5] Bowditch, B. Cut points and canonical splittings of hyperbolic groups, Acta Math., Volume 180 (1998) no. 2, pp. 145-186

[6] Canary, R.; Minsky, Y.; Taylor, E. Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, J. Geom. Anal., Volume 9 (1999), pp. 18-40

[7] Cannon, J. The combinatorial Riemann mapping theorem, Acta Math., Volume 173 (1994) no. 2, pp. 155-234

[8] M. Carrasco Piaggio, Jauge conforme des espaces métriques compacts, Ph.D. thesis, Aix-Marseille Université, 2011.

[9] Dunwoody, M.J. The accessibility of finitely presented groups, Invent. Math., Volume 81 (1985) no. 3, pp. 449-457

[10] Elek, G. The ${\ell }_p$-cohomology and the conformal dimension of hyperbolic cones, Geom. Dedicata, Volume 68 (1997), pp. 263-279

[11] Haïssinsky, P. Empilements de cercles et modules combinatoires, Ann. Inst. Fourier, Volume 59 (2009) no. 6, pp. 2175-2222

[12] Haïssinsky, P.; Pilgrim, K. Coarse expanding conformal dynamics, Astérisque, Volume 325 (2009)

[13] Heinonen, J. Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001

[14] Mackay, J. Spaces with conformal dimension greater than one, Duke Math. J., Volume 153 (2010) no. 2, pp. 211-227

[15] Pansu, P. Dimension conforme et sphère à lʼinfini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 14 (1989), pp. 177-212

[16] Stallings, J.R. On torsion-free groups with infinitely many ends, Ann. of Math. (2), Volume 88 (1968), pp. 312-334