On the conformal gauge of a compact metric space

Carrasco Piaggio, Matias

Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, pp. 495-548.

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension ${dim}_{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute ${dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.

Dans cet article, on étudie la jauge conforme Ahlfors régulière d’un espace métrique compact et sa dimension conforme ${dim}_{AR}(X,d)$. À l’aide d’une suite de recouvrements finis de $(X,d)$, on construit des distances dans sa jauge Ahlfors régulière de dimension de Hausdorff contrôlée. On obtient ainsi une description combinatoire, à homéomorphismes bi-Lipschitz près, de toutes les métriques dans la jauge. On montre comment calculer ${dim}_{AR}X$ à partir de modules combinatoires en considérant un exposant critique $Q_N$.

Classification: 30L10, 51F99, 20F67, 30C65, 28A78
Keywords: Ahlfors regular, conformal gauge, conformal dimension, combinatorial modulus, Gromov-hyperbolic
Keywords: Ahlfors régulier, jauge conforme, dimension conforme, module combinatoire, Gromov-hyperbolique

[1] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., 1973. | MR

[2] M. Bonk, Quasiconformal geometry of fractals, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, 1349-1373. | MR

[3] M. Bonk, J. Heinonen & P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001). | MR

[4] M. Bonk & B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), 127-183. | MR

[5] M. Bonk & B. Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol. 9 (2005), 219-246. | MR

[6] M. Bourdon & B. Kleiner, Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn. 7 (2013), 39-107. | MR

[7] M. Bourdon & B. Kleiner, Some applications of ${\ell }_p$-cohomology to boundaries of Gromov hyperbolic spaces, preprint arXiv:1203.1233.

[8] M. Bourdon & H. Pajot, Cohomologie ${\ell }_p$ et espaces de Besov, J. reine angew. Math. 558 (2003), 85-108. | MR

[9] J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155-234. | MR

[10] M. Carrasco Piaggio, Jauge conforme des espaces métriques compacts, Thèse, Université Aix-Marseille, 2011.

[11] M. Carrasco Piaggio, Conformal dimension and canonical splittings of hyperbolic groups, preprint arXiv:1301.6492.

[12] M. Christ, A $T\left(b\right)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601-628. | MR

[13] M. Coornaert, T. Delzant & A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Math. 1441, Springer, 1990. | MR

[14] G. David & S. Semmes, Fractured fractals and broken dreams, Oxford Lecture Series in Mathematics and its Applications 7, The Clarendon Press Oxford Univ. Press, 1997. | MR

[15] G. Elek, The $l_p$-cohomology and the conformal dimension of hyperbolic cones, Geom. Dedicata 68 (1997), 263-279. | MR

[16] É. Ghys & P. De La Harpe (éds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser, 1990. | MR

[17] P. Haïssinsky, Empilements de cercles et modules combinatoires, Ann. Inst. Fourier (Grenoble) 59 (2009), 2175-2222. | Numdam | MR

[18] P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, d'après Mostow, Pansu, Bourdon, Pajot, Bonk, Kleiner, Séminaire Bourbaki, vol. 2007/08, exp. no 993, Astérisque 326 (2009), 321-362. | Numdam | MR

[19] P. Haïssinsky & K. M. Pilgrim, Thurston obstructions and Ahlfors regular conformal dimension, J. Math. Pures Appl. 90 (2008), 229-241. | MR

[20] P. Haïssinsky & K. M. Pilgrim, Coarse expanding conformal dynamics, Astérisque 325 (2009). | MR

[21] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, 2001. | MR

[22] S. Keith & T. Laakso, Conformal Assouad dimension and modulus, Geom. Funct. Anal. 14 (2004), 1278-1321. | MR

[23] B. Kleiner, The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity, in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, 743-768. | MR

[24] L. V. Kovalev, Conformal dimension does not assume values between zero and one, Duke Math. J. 134 (2006), 1-13. | MR

[25] G. Lupo-Krebs & H. Pajot, Dimensions conformes, espaces Gromov-hyperboliques et ensembles autosimilaires, in Séminaire de Théorie Spectrale et Géométrie. Vol. 22. Année 2003-2004, Sémin. Théor. Spectr. Géom. 22, Univ. Grenoble I, 2004, 153-182. | Numdam | MR

[26] J. M. Mackay & J. T. Tyson, Conformal dimension; theory and application, University Lecture Series 54, Amer. Math. Soc., 2010. | MR

[27] P. Pansu, Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212. | MR

[28] S. Semmes, Metric spaces and mappings seen at many scales, in Metric structures for Riemannian and Non-Riemmannian spaces (M. Gromov, éd.), Birkhäuser, 2001.

[29] P. Tukia & J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114. | MR

[30] J. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 23 (1998), 525-548. | MR

[31] A. L. VolʼBerg & S. V. Konyagin, On measures with the doubling condition, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 666-675; English translation: Math. USSR-Izv. 30 (1988), 629-638. | MR

[32] J.-M. Wu, Hausdorff dimension and doubling measures on metric spaces, Proc. Amer. Math. Soc. 126 (1998), 1453-1459. | MR