The aim of this article is to explain the deep relationships between circle-packings and combinatorial moduli of curves, and to compare the approaches to Cannon’s conjecture to which they lead.
Le but de cette note est de tenter d’expliquer les liens étroits qui unissent la théorie des empilements de cercles et des modules combinatoires et de comparer les approches à la conjecture de J.W. Cannon qui en découlent.
Revised:
Accepted:
DOI: 10.5802/aif.2488
Classification: 52C26, 30C62, 30F10, 30F40
Keywords: Circle packings, quasiconformal, modulus of curves
@article{AIF_2009__59_6_2175_0, author = {Ha\"Issinsky, Peter}, title = {Empilements de cercles et modules combinatoires}, journal = {Annales de l'Institut Fourier}, pages = {2175--2222}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {6}, year = {2009}, doi = {10.5802/aif.2488}, zbl = {1189.30080}, mrnumber = {2640918}, language = {fr}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2488/} }
TY - JOUR TI - Empilements de cercles et modules combinatoires JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 2175 EP - 2222 VL - 59 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2488/ UR - https://zbmath.org/?q=an%3A1189.30080 UR - https://www.ams.org/mathscinet-getitem?mr=2640918 UR - https://doi.org/10.5802/aif.2488 DO - 10.5802/aif.2488 LA - fr ID - AIF_2009__59_6_2175_0 ER -
HaÏssinsky, Peter. Empilements de cercles et modules combinatoires. Annales de l'Institut Fourier, Volume 59 (2009) no. 6, pp. 2175-2222. doi : 10.5802/aif.2488. https://aif.centre-mersenne.org/articles/10.5802/aif.2488/
[1] Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966 | MR: 2241787 | Zbl: 0138.06002
[2] Conformal invariants : topics in geometric function theory, McGraw-Hill Book Co., New York, 1973 (McGraw-Hill Series in Higher Mathematics) | MR: 357743 | Zbl: 0272.30012
[3] Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987 (Lecture Notes in Math.) Tome 1351, Springer, Berlin, 1988, pp. 52-68 | MR: 982072 | Zbl: 0662.46037
[4] Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math., Tome 150 (2002) no. 1, pp. 127-183 | Article | MR: 1930885 | Zbl: 1037.53023
[5] Rigidity for quasi-Möbius group actions, J. Differential Geom., Tome 61 (2002) no. 1, pp. 81-106 | MR: 1949785 | Zbl: 1044.37015
[6] Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol., Tome 9 (2005), p. 219-246 (electronic) | Article | MR: 2116315 | Zbl: 1087.20033
[7] The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989) (Oxford Sci. Publ.), Oxford Univ. Press, New York, 1991, pp. 315-369 | MR: 1130181 | Zbl: 0764.57002
[8] The combinatorial Riemann mapping theorem, Acta Math., Tome 173 (1994) no. 2, pp. 155-234 | Article | MR: 1301392 | Zbl: 0832.30012
[9] Squaring rectangles : the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus : groups, geometry and special functions (Brooklyn, NY, 1992) (Contemp. Math.) Tome 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133-212 | MR: 1292901 | Zbl: 0818.20043
[10] Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. Math., Tome 24 (1999) no. 2, pp. 265-304 | MR: 1724092 | Zbl: 0939.20048
[11] Recognizing constant curvature discrete groups in dimension , Trans. Amer. Math. Soc., Tome 350 (1998) no. 2, pp. 809-849 | Article | MR: 1458317 | Zbl: 0910.20024
[12] Un principe variationnel pour les empilements de cercles, Invent. Math., Tome 104 (1991) no. 3, pp. 655-669 | Article | MR: 1106755 | Zbl: 0745.52010
[13] Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., Tome 53 (1981), pp. 53-73 | Article | Numdam | MR: 623534 | Zbl: 0474.20018
[14] Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001 | MR: 1800917 | Zbl: 0985.46008
[15] Quasiconformal maps in metric spaces with controlled geometry, Acta Math., Tome 181 (1998) no. 1, pp. 1-61 | Article | MR: 1654771 | Zbl: 0915.30018
[16] Modulus and the Poincaré inequality on metric measure spaces, Math. Z., Tome 245 (2003) no. 2, pp. 255-292 | Article | MR: 2013501 | Zbl: 1037.31009
[17] Conformal Assouad dimension and modulus, Geom. Funct. Anal., Tome 14 (2004) no. 6, pp. 1278-1321 | Article | MR: 2135168 | Zbl: 1108.28008
[18] Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys, Tome 88 (1936), pp. 141-164
[19] Modern dimension theory, Bibliotheca Mathematica, Vol. VI. Edited with the cooperation of the “Mathematisch Centrum” and the “Wiskundig Genootschap” at Amsterdam, Interscience Publishers John Wiley & Sons, Inc., New York, 1965 | MR: 208571
[20] Dimension conforme et sphère à l’infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math., Tome 14 (1989) no. 2, pp. 177-212 | MR: 1024425 | Zbl: 0722.53028
[21] The convergence of circle packings to the Riemann mapping, J. Differential Geom., Tome 26 (1987) no. 2, pp. 349-360 | MR: 906396 | Zbl: 0694.30006
[22] Square tilings with prescribed combinatorics, Israel J. Math., Tome 84 (1993) no. 1-2, pp. 97-118 | Article | MR: 1244661 | Zbl: 0788.05019
[23] Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), Tome 6 (1982) no. 3, pp. 357-381 | Article | MR: 648524 | Zbl: 0496.57005
[24] Metric and geometric quasiconformality in Ahlfors regular Loewner spaces, Conform. Geom. Dyn., Tome 5 (2001), p. 21-73 (electronic) | Article | MR: 1872156 | Zbl: 0981.30015
[25] Quasi-Möbius maps, J. Analyse Math., Tome 44 (1984/85), pp. 218-234 | Article | MR: 801295 | Zbl: 0593.30022
Cited by Sources: