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Peter HaÏssinsky
Empilements de cercles et modules combinatoires
(Circle packings and combinatorial moduli)
Annales de l'institut Fourier, 59 no. 6 (2009), p. 2175-2222, doi: 10.5802/aif.2488
Article: subscription required (your ip address: 50.16.17.90) | Reviews MR 2640918 | Zbl 1189.30080
Class. Math.: 52C26, 30C62, 30F10, 30F40
Keywords: Circle packings, quasiconformal, modulus of curves
[1] Lars V. Ahlfors, 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] Lars V. Ahlfors, Conformal invariants : topics in geometric function theory, McGraw-Hill Book Co., 1973, McGraw-Hill Series in Higher Mathematics MR 357743 | Zbl 0272.30012
[3] B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Springer, 1988, p. 52–68 MR 982072 | Zbl 0662.46037
[4] Mario Bonk & Bruce Kleiner, “Quasisymmetric parametrizations of two-dimensional metric spheres”, Invent. Math. 150 (2002) no. 1, p. 127-183
Article | MR 1930885 | Zbl 1037.53023
[5] Mario Bonk & Bruce Kleiner, “Rigidity for quasi-Möbius group actions”, J. Differential Geom. 61 (2002) no. 1, p. 81-106
Article | MR 1949785 | Zbl 1044.37015
[6] Mario Bonk & Bruce Kleiner, “Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary”, Geom. Topol. 9 (2005), p. 219-246 (electronic)
Article | MR 2116315 | Zbl 1087.20033
[7] James W. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Univ. Press, 1991, p. 315–369 MR 1130181 | Zbl 0764.57002
[8] James W. Cannon, “The combinatorial Riemann mapping theorem”, Acta Math. 173 (1994) no. 2, p. 155-234
Article | MR 1301392 | Zbl 0832.30012
[9] James W. Cannon, William J. Floyd & Walter R. Parry, Squaring rectangles : the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus : groups, geometry and special functions (Brooklyn, NY, 1992), Amer. Math. Soc., 1994, p. 133–212 MR 1292901 | Zbl 0818.20043
[10] James W. Cannon, William J. Floyd & Walter R. Parry, “Sufficiently rich families of planar rings”, Ann. Acad. Sci. Fenn. Math. 24 (1999) no. 2, p. 265-304 MR 1724092 | Zbl 0939.20048
[11] James W. Cannon & Eric L. Swenson, “Recognizing constant curvature discrete groups in dimension $3$”, Trans. Amer. Math. Soc. 350 (1998) no. 2, p. 809-849
Article | MR 1458317 | Zbl 0910.20024
[12] Yves Colin de Verdière, “Un principe variationnel pour les empilements de cercles”, Invent. Math. 104 (1991) no. 3, p. 655-669
Article | MR 1106755 | Zbl 0745.52010
[13] Mikhael Gromov, “Groups of polynomial growth and expanding maps”, Inst. Hautes Études Sci. Publ. Math. 53 (1981), p. 53-73
Numdam | MR 623534 | Zbl 0474.20018
[14] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, 2001 MR 1800917 | Zbl 0985.46008
[15] Juha Heinonen & Pekka Koskela, “Quasiconformal maps in metric spaces with controlled geometry”, Acta Math. 181 (1998) no. 1, p. 1-61
Article | MR 1654771 | Zbl 0915.30018
[16] Stephen Keith, “Modulus and the Poincaré inequality on metric measure spaces”, Math. Z. 245 (2003) no. 2, p. 255-292
Article | MR 2013501 | Zbl 1037.31009
[17] Stephen Keith & Tomi Laakso, “Conformal Assouad dimension and modulus”, Geom. Funct. Anal. 14 (2004) no. 6, p. 1278-1321
Article | MR 2135168 | Zbl 1108.28008
[18] Paul Koebe, “Kontaktprobleme der konformen Abbildung”, Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys 88 (1936), p. 141-164 JFM 62.1217.04
[19] Jun-iti Nagata, 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] Pierre Pansu, “Dimension conforme et sphère à l’infini des variétés à courbure négative”, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989) no. 2, p. 177-212 MR 1024425 | Zbl 0722.53028
[21] Burt Rodin & Dennis Sullivan, “The convergence of circle packings to the Riemann mapping”, J. Differential Geom. 26 (1987) no. 2, p. 349-360
Article | MR 906396 | Zbl 0694.30006
[22] Oded Schramm, “Square tilings with prescribed combinatorics”, Israel J. Math. 84 (1993) no. 1-2, p. 97-118
Article | MR 1244661 | Zbl 0788.05019
[23] William P. Thurston, “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”, Bull. Amer. Math. Soc. (N.S.) 6 (1982) no. 3, p. 357-381
Article | MR 648524 | Zbl 0496.57005
[24] Jeremy Tyson, “Metric and geometric quasiconformality in Ahlfors regular Loewner spaces”, Conform. Geom. Dyn. 5 (2001), p. 21-73 (electronic)
Article | MR 1872156 | Zbl 0981.30015
[25] Jussi Väisälä, “Quasi-Möbius maps”, J. Analyse Math. 44 (1984/85), p. 218-234
Article | MR 801295 | Zbl 0593.30022
Peter HaÏssinsky
Empilements de cercles et modules combinatoires
(Circle packings and combinatorial moduli)
Annales de l'institut Fourier, 59 no. 6 (2009), p. 2175-2222, doi: 10.5802/aif.2488
Article: subscription required (your ip address: 50.16.17.90) | Reviews MR 2640918 | Zbl 1189.30080
Class. Math.: 52C26, 30C62, 30F10, 30F40
Keywords: Circle packings, quasiconformal, modulus of curves
Résumé - Abstract
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.
Bibliography
[2] Lars V. Ahlfors, Conformal invariants : topics in geometric function theory, McGraw-Hill Book Co., 1973, McGraw-Hill Series in Higher Mathematics MR 357743 | Zbl 0272.30012
[3] B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Springer, 1988, p. 52–68 MR 982072 | Zbl 0662.46037
[4] Mario Bonk & Bruce Kleiner, “Quasisymmetric parametrizations of two-dimensional metric spheres”, Invent. Math. 150 (2002) no. 1, p. 127-183
Article | MR 1930885 | Zbl 1037.53023
[5] Mario Bonk & Bruce Kleiner, “Rigidity for quasi-Möbius group actions”, J. Differential Geom. 61 (2002) no. 1, p. 81-106
Article | MR 1949785 | Zbl 1044.37015
[6] Mario Bonk & Bruce Kleiner, “Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary”, Geom. Topol. 9 (2005), p. 219-246 (electronic)
Article | MR 2116315 | Zbl 1087.20033
[7] James W. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Univ. Press, 1991, p. 315–369 MR 1130181 | Zbl 0764.57002
[8] James W. Cannon, “The combinatorial Riemann mapping theorem”, Acta Math. 173 (1994) no. 2, p. 155-234
Article | MR 1301392 | Zbl 0832.30012
[9] James W. Cannon, William J. Floyd & Walter R. Parry, Squaring rectangles : the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus : groups, geometry and special functions (Brooklyn, NY, 1992), Amer. Math. Soc., 1994, p. 133–212 MR 1292901 | Zbl 0818.20043
[10] James W. Cannon, William J. Floyd & Walter R. Parry, “Sufficiently rich families of planar rings”, Ann. Acad. Sci. Fenn. Math. 24 (1999) no. 2, p. 265-304 MR 1724092 | Zbl 0939.20048
[11] James W. Cannon & Eric L. Swenson, “Recognizing constant curvature discrete groups in dimension $3$”, Trans. Amer. Math. Soc. 350 (1998) no. 2, p. 809-849
Article | MR 1458317 | Zbl 0910.20024
[12] Yves Colin de Verdière, “Un principe variationnel pour les empilements de cercles”, Invent. Math. 104 (1991) no. 3, p. 655-669
Article | MR 1106755 | Zbl 0745.52010
[13] Mikhael Gromov, “Groups of polynomial growth and expanding maps”, Inst. Hautes Études Sci. Publ. Math. 53 (1981), p. 53-73
Numdam | MR 623534 | Zbl 0474.20018
[14] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, 2001 MR 1800917 | Zbl 0985.46008
[15] Juha Heinonen & Pekka Koskela, “Quasiconformal maps in metric spaces with controlled geometry”, Acta Math. 181 (1998) no. 1, p. 1-61
Article | MR 1654771 | Zbl 0915.30018
[16] Stephen Keith, “Modulus and the Poincaré inequality on metric measure spaces”, Math. Z. 245 (2003) no. 2, p. 255-292
Article | MR 2013501 | Zbl 1037.31009
[17] Stephen Keith & Tomi Laakso, “Conformal Assouad dimension and modulus”, Geom. Funct. Anal. 14 (2004) no. 6, p. 1278-1321
Article | MR 2135168 | Zbl 1108.28008
[18] Paul Koebe, “Kontaktprobleme der konformen Abbildung”, Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys 88 (1936), p. 141-164 JFM 62.1217.04
[19] Jun-iti Nagata, 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] Pierre Pansu, “Dimension conforme et sphère à l’infini des variétés à courbure négative”, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989) no. 2, p. 177-212 MR 1024425 | Zbl 0722.53028
[21] Burt Rodin & Dennis Sullivan, “The convergence of circle packings to the Riemann mapping”, J. Differential Geom. 26 (1987) no. 2, p. 349-360
Article | MR 906396 | Zbl 0694.30006
[22] Oded Schramm, “Square tilings with prescribed combinatorics”, Israel J. Math. 84 (1993) no. 1-2, p. 97-118
Article | MR 1244661 | Zbl 0788.05019
[23] William P. Thurston, “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”, Bull. Amer. Math. Soc. (N.S.) 6 (1982) no. 3, p. 357-381
Article | MR 648524 | Zbl 0496.57005
[24] Jeremy Tyson, “Metric and geometric quasiconformality in Ahlfors regular Loewner spaces”, Conform. Geom. Dyn. 5 (2001), p. 21-73 (electronic)
Article | MR 1872156 | Zbl 0981.30015
[25] Jussi Väisälä, “Quasi-Möbius maps”, J. Analyse Math. 44 (1984/85), p. 218-234
Article | MR 801295 | Zbl 0593.30022
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