James W. Cannon

The combinatorial Riemann mapping theorem

The combinatorial Riemann mapping theorem is designed to supply a surface with local quasiconformal coordinates compatible with local combinatorial data. This theorem was discovered in an at tempt to show that certain negatively curved groups have constant curvature. A potential application is that of finding local coordinates on which a given group acts uniformly quasiconformally. The classical Riemann mapping theorem may also be viewed as supplying local coordinates (take a ring and map it conformally, by the classical theorem, onto a right circular cylinder; pull the resulting flat coordinates back to the ring as canonical local coordinates). This coordinatization role is disguised in the classical case by the fact that a Riemann surface comes preequipped with local coordinates in the desired conformal class. In the combinatorial case we begin with a topological surface having no presupplied quasiconformal structure and our task is that of discovering the local coordinates (again by pulling coordinates back from an appropriate right circular cylinder). The combinatorial data are supplied by coverings of the surface called shinglings. A shingle is a compact connected set. A shingling is a locally finite cover of the surface by shingles. (A shingling is like a tiling except that shingles are allowed to overlap while tiles usually do not overlap.) A shingling may be viewed as a combinatorial approximation to the surface. A given shingling, being locally finite, gives only a first approximation to a local quasiconformal structure on the surface. The total structure can only be determined by a sequence of finer and finer shinglings. The problem becomes that of

The combinatorial structure of the Hawaiian earring group

Abstract In this paper we study the combinatorial structure of the Hawaiian earring group, by showing that it can be represented as a group of transfinite words on a countably infinite alphabet exactly analogously to the representation of a finite rank free group as finite words on a finite alphabet. We define a big free group similarly as the group of transfinite words on given set, and study their group theoretic structure.

Finite subdivision rules

We introduce and study finite subdivision rules. A finite subdivision rule R consists of a finite 2-dimensional CW complex SR, a subdivision R(SR) of SR, and a continuous cellular map φR : R(SR) → SR whose restriction to each open cell is a homeomorphism. If R is a finite subdivision rule, X is a 2-dimensional CW complex, and f : X → SR is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide X to obtain an infinite sequence of tilings. We wish to determine when this sequence of tilings is conformal in the sense of Cannon’s combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be replaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed.

Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space IHI3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the 2-sphere.

One-dimensional sets and planar sets are aspherical

Abstract We give a relatively short proof of the theorem that planar sets are aspherical. The first proof of this theorem, by third author Andreas Zastrow, was considerably longer.

The big fundamental group, big Hawaiian earrings, and the big free groups ✩

Abstract In this second paper in a series of three we generalize the notions of fundamental group and Hawaiian earring. In the first paper we generalized the notion of free group to that of a big free group . In the current article we generalize the notion of fundamental group by defining the big fundamental group of a topological space. We also describe big Hawaiian earrings , which are generalizations of the classical Hawaiian earring. We then prove that the big fundamental group of a big Hawaiian earring is a big free group.

Constructing rational maps from subdivision rules

Suppose R is a finite subdivision rule with an edge pairing. Then the subdivision map σR is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If R has mesh approaching 0 and SR is a 2-sphere, it is proved in Theorem 3.1 that if R is conformal then σR is realizable by a rational map. Furthermore, a general construction is given which, starting with a one tile rotationally invariant finite subdivision rule, produces a finite subdivision rule Q with an edge pairing such that σQ is realizable by a rational map. In this paper we illustrate a technique for constructing critically finite rational maps. The starting point for the construction is an orientation-preserving finite subdivision rule R with an edge pairing. For such a finite subdivision rule the CW-complex SR is a surface, and the map σR : SR → SR is a branched covering. If SR is orientable, then unless σR is a homeomorphism or a covering of the torus, SR is a 2-sphere and σR is critically finite. In the latter case, SR has an orbifold structure OR and σR induces a map τR : T (OR) → T (OR) on the Teichmüller space of the orbifold. By work of Thurston, σR can be realized by a rational map exactly if τR has a fixed point. Alternatively, we prove that σR can be realized by a rational map if R has mesh approaching 0 and is conformal. We next give a general construction which, starting with a one tile rotationally invariant finite subdivision rule R, produces an orientation-preserving finite subdivision rule Q with an edge pairing such that Q is conformal if and only if R is conformal; we then show in Theorem 3.2 that σQ is realizable by a rational map. We next give several examples of orientation-preserving finite subdivision rules with edge pairings. For each example R for which the associated map σR can be realized by a rational map, we explicitly construct a rational map realizing it. We conclude with some questions. A motivation for this work is the Bowers-Stephenson paper [1]. In that paper they construct an expansion complex for the pentagonal subdivision rule (see Figure 4) and numerically approximate the expansion constant. In Example 4.4 we consider an associated finite subdivision ruleQ with an edge pairing and construct a rational map fQ(z) = 2z(z+9/16)5 27(z−3/128)3(z−1)2 which realizes σQ. The expansion constant for the pentagonal subdivision rule is (f ′ Q(0)) 1/5 = (−324)1/5. We thank Curt McMullen, Kevin Pilgrim, and the referee for helpful comments. Date: September 30, 2002. 1991 Mathematics Subject Classification. Primary 37F10, 52C20; Secondary 57M12.

Solvgroups are not almost convex

We show that no cocompact discrete group based on solvgeometry, Sol, is almost convex. This reflects the geometry of Sol, and implies that the Cayley graph of a cocompact discrete group based on Sol cannot be efficiently constructed by finitely many local replacement rules.

Characterization of taming sets on 2-spheres

1. Suppose that a 2-sphere S in E3 is tame modulo a closed subset F, and suppose that F is tame (i.e., F lies on a tame 2-sphere in E3). Is S tame? And if S is not tame, at which points of F can S be wild? These questions are answered by Theorem 1.1. The author is deeply indebted to C. E. Burgess, who in private conversation pointed out how Lemma 2.3 and linking arguments can be used to establish special cases of Theorem 1.1 (see [6]). If E is a positive number, let Fe denote the closed subset of F consisting of those points of F which lie in components of F of diameter equal to or greater than E. Let F# denote the subset of F which consists of those points of F which are degenerate components of F. The set F# is not necessarily closed.

Expansion complexes for finite subdivision rules. II

This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a one-tile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant partial conformal structure, and hence is conformal. The paper next considers one-tile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex.