Walter R. Parry

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.

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.

Homology of classical Lie groups made discrete. II. H2, H3, and relations with scissors congruences☆

The present work extends a number of earlier results; see [S, 7, 15-181. For example, the following fundamental exact sequence (essentially due to Bloch and Wigner in somewhat different form, but not published by them) can be found in DuPont and Sah [7]: o~o/z-tH3(SL(2,@))~~(@)~n:(a=~)~~,(a=)~o. (2.12) The group Y(C) is defined in (1.4))( 1.8) and is isomorphic to the group PC considered in DuPont and Sah [7]. Also, the study of the scissors congruence problem had led to two exact sequences in DuPont [S, Theorems 1.3 and 1.43 that roughly correspond to the f -eigenspaces of (2.12) under the action of complex conjugation: O~A~H,(SU(2))~~S3/L~[WO([W/Z)-*H,(SU(2))-tO; (0.1)

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.

Constructing subdivision rules from rational maps

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f is the subdivision map of a finite subdivision rule. We are interested here in connections between finite subdivision rules and rational maps. Finite subdivision rules arose out of our attempt to resolve Cannon’s Conjecture: If G is a Gromov-hyperbolic group whose space at infinity is a 2-sphere, then G has a cocompact, properly discontinuous, isometric action on hyperbolic 3-space. Cannon’s Conjecture can be reduced (see, for example, the Cannon-Swenson paper [5]) to a conjecture about (combinatorial) conformality for the action of such a group G on its space at infinity, and finite subdivision rules were developed to give models for the action of a Gromov-hyperbolic group on the 2-sphere at infinity. There is also a connection between finite subdivision rules and rational maps. If R is an orientation-preserving finite subdivision rule such that the subdivision complex SR is a 2-sphere, then the subdivision map σR is a critically finite branched map of this 2-sphere. In joint work [3] with Kenyon we consider these subdivision maps under the additional hypotheses that R has bounded valence (this is equivalent to its not having periodic critical points) and mesh approaching 0. In [3, Theorem 3.1] we show that if R is conformal (in the combinatorial sense) then the subdivision map σR is equivalent to a rational map. The converse follows from [4, Theorem 4.7]. In this paper we consider the problem of when a rational map f can be equivalent to the subdivision map of a finite subdivision rule. Since a subdivision complex has only finitely many vertices, such a rational map must be critically finite. We specialize here to the case that f has no periodic critical points. Our main theorem, which has also been proved by Bonk and Meyer [1] when f has a hyperbolic orbifold (which includes all but some well-understood examples), is the following: Date: March 15, 2007. 2000 Mathematics Subject Classification. Primary 37F10, 52C20; Secondary 57M12.

Nearly Euclidean Thurston maps

We introduce and study a class of Thurston maps from the 2-sphere to itself which we call nearly Euclidean Thurston (NET) maps. These are simple generalizations of Euclidean Thurston maps.

Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds

The twisted face-pairing construction of our earlier papers gives an ecient way of generating, mechanically and with little eort, myriads of relatively simple face-pairing descriptions of interesting closed 3-manifolds. The corresponding description in terms of surgery, or Dehn-lling, reveals the twist construction as a carefully organized surgery on a link. In this paper, we work out the relationship between the twisted face-pairing description of closed 3-manifolds and the more common descriptions by surgery and Heegaard diagrams. We show that all Heegaard diagrams have a natural decomposition into subdiagrams called Heegaard cylinders, each of which has a natural shape given by the ratio of two positive integers. We characterize the Heegaard diagrams arising naturally from a twisted face-pairing description as those whose Heegaard cylinders all have inte- gral shape. This characterization allows us to use the Kirby calculus and standard tools of Heegaard theory to attack the problem of nding which closed, orientable 3-manifolds have a twisted face-pairing description.

Twisted face-pairing 3-manifolds

This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing e of a faceted 3-ball P yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of twisted face-pairing shows how to correct this defect of the quotient pseudomanifold P/∈ systematically. The method describes how to modify P by edge subdivision and how to modify any orientation-reversing face-pairing ∈ of P by twisting, so as to yield an infinite parametrized family of face-pairings (Q, δ) whose quotient complexes Q/δ are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings ∈, it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex Q/δ of a twisted face-pairing is a closed 3-manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to Q. (2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct? We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings. We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds Q/δ and of the original pseudomanifold P/∈ (with vertices removed). We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurstons eight 3-dimensional geometries.

Axioms for Translation Length Functions

This paper arose from the paper [3] of Culler and Morgan concerning group actions on ℝ-trees. An ℝ-tree is a nonempty metric space T such that every two points in T are joined by a unique arc (image of an injective continuous function from a closed interval in ℝ to T) and that arc is the image of an isometry from a closed interval in ℝ to T. This generalizes the usual notion of simplicial tree. Roughly speaking, the difference is that in a simplicial tree, the branching occurs at a discrete set of points, namely, the vertices, whereas in an ℝ-tree, there is no restriction on where branching may occur. The notion of ℝ-tree arose indirectly in the work of Lyndon [5]and Chiswell [2]concerning Lyndon length functions. The first definition was given by Tits in [7].

Squaring rectangles for dumbbells

The theorem on squaring a rectangle (see Schramm [Israel J. Math. 84 (1993)] and Cannon-Floyd-Parry [Contemp. Math. 169, AMS, 1994]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemann mapping.