Expansion complexes for finite subdivision rules. I

Cannon, J.; Floyd, W.; Parry, W.

doi:10.1090/S1088-4173-06-00126-3

Expansion complexes for finite subdivision rules. I
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by J. W. Cannon, W. J. Floyd and W. R. Parry

Conform. Geom. Dyn. 10 (2006), 63-99

DOI: https://doi.org/10.1090/S1088-4173-06-00126-3

Published electronically: March 22, 2006

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Abstract:

This paper develops the basic theory of conformal structures on finite subdivision rules. The work depends heavily on the use of expansion complexes, which are defined and discussed in detail. It is proved that a finite subdivision rule with bounded valence and mesh approaching $0$ is conformal (in the combinatorial sense) if there is a partial conformal structure on the model subdivision complex with respect to which the subdivision map is conformal. This gives a new approach to the difficult combinatorial problem of determining when a finite subdivision rule is conformal.

References

Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469–481. MR 1096169, DOI 10.1090/S0894-0347-1991-1096169-1
Philip L. Bowers and Kenneth Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68. MR 1479069, DOI 10.1090/S1088-4173-97-00014-3
James W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155–234. MR 1301392, DOI 10.1007/BF02398434
J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 265–304. MR 1724092
J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153–196. MR 1875951, DOI 10.1090/S1088-4173-01-00055-8
J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, Constructing rational maps from subdivision rules, Conform. Geom. Dyn. 7 (2003), 76–102. MR 1992038, DOI 10.1090/S1088-4173-03-00082-1

Expansion complexes for finite subdivision rules II

J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), no. 2, 809–849. MR 1458317, DOI 10.1090/S0002-9947-98-02107-2

Theory of Functions of a Complex Variable

Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745, DOI 10.1007/978-1-4684-9930-8
David Gabai, Homotopy hyperbolic $3$-manifolds are virtually hyperbolic, J. Amer. Math. Soc. 7 (1994), no. 1, 193–198. MR 1205445, DOI 10.1090/S0894-0347-1994-1205445-3
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Lee Mosher, Geometry of cubulated $3$-manifolds, Topology 34 (1995), no. 4, 789–814. MR 1362788, DOI 10.1016/0040-9383(94)00050-6
Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823

Entropy formula for the Ricci flow and its geometric applications

Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

Ricci flow with surgery on 3-manifolds

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