Expansion complexes for finite subdivision rules. I

Authors:
J. W. Cannon, W. J. Floyd and W. R. Parry

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

MSC (2000):
Primary 30F45, 52C20; Secondary 20F67, 52C26

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

Published electronically:
March 22, 2006

MathSciNet review:
2218641

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Mladen Bestvina and Geoffrey Mess,
*The boundary of negatively curved groups*, J. Amer. Math. Soc.**4**(1991), no. 3, 469–481. MR**1096169**, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1090/S1088-4173-03-00082-1
EXPii J. W. Cannon, W. J. Floyd, and W. R. Parry, - 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 https://doi.org/10.1090/S0002-9947-98-02107-2
CC C. Caratheodory, - 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 https://doi.org/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** - David Gabai,
*Homotopy hyperbolic $3$-manifolds are virtually hyperbolic*, J. Amer. Math. Soc.**7**(1994), no. 1, 193–198. MR**1205445**, DOI https://doi.org/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** - 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 https://doi.org/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 https://doi.org/10.1016/0040-9383%2894%2900050-6 - Zeev Nehari,
*Conformal mapping*, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952. MR**0045823**
P1 G. Perelman,

*Expansion complexes for finite subdivision rules II*, preprint, available from http://www.math.vt.edu/people/floyd.

*Theory of Functions of a Complex Variable*, Vol. II, Chelsea, New York, 1960.

*Entropy formula for the Ricci flow and its geometric applications*, preprint, available from http://www.arXiv.org/abs/math.DG/0211159. P2 G. Perelman,

*Finite extinction time for the solutions to the Ricci flow on certain three-manifolds*, preprint, available from http://www.arXiv.org/abs/math.DG/0307245. P3 G. Perelman,

*Ricci flow with surgery on 3-manifolds*, preprint, available from http: //www.arXiv.org/abs/math.DG/0303109 . CP K. Stephenson,

*CirclePack*, software, available from http://www.math.utk.edu/˜kens. T W. P. Thurston, Lecture notes, CBMS Conference, University of Minnesota at Duluth, 1983.

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Additional Information

**J. W. Cannon**

Affiliation:
Department of Mathematics Brigham Young University, Provo, Utah 84602

Email:
cannon@math.byu.edu

**W. J. Floyd**

Affiliation:
Department of Mathematics Virginia Tech, Blacksburg, Virginia 24061

MR Author ID:
67750

Email:
floyd@math.vt.edu

**W. R. Parry**

Affiliation:
Department of Mathematics Eastern Michigan University, Ypsilanti, Michigan 48197

MR Author ID:
136390

Email:
walter.parry@emich.edu

Keywords:
Conformality,
expansion complex,
finite subdivision rule

Received by editor(s):
November 22, 2004

Published electronically:
March 22, 2006

Additional Notes:
This research was supported in part by NSF grants DMS-9803868, DMS-9971783, DMS-10104030, and DMS-0203902

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.