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The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 ICM Proceedings
, 2006
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Constructing rational maps from subdivision rules
, 2003
"... Suppose R is an orientationpreserving 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 2sphere. If R has mesh approaching 0 and SR is a 2sphere, it is proved in Theorem 3.1 th ..."
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Cited by 14 (3 self)
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Suppose R is an orientationpreserving 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 2sphere. If R has mesh approaching 0 and SR is a 2sphere, 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.
Expansion complexes for finite subdivision rules
 I, Conform. Geom. Dyn
"... Abstract. 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 onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an ..."
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Cited by 12 (5 self)
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Abstract. 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 onetile 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 onetile 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. 1.
Twisted facepairing 3manifolds
 Trans. Amer. Math. Soc
"... Abstract. This paper is an enriched version of our introductory paper on twisted facepairing 3manifolds. Just as every edgepairing of a 2dimensional disk yields a closed 2manifold, so also every facepairing ɛ of a faceted 3ball P yields a closed 3dimensional pseudomanifold. In dimension 3, t ..."
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Cited by 9 (5 self)
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Abstract. This paper is an enriched version of our introductory paper on twisted facepairing 3manifolds. Just as every edgepairing of a 2dimensional disk yields a closed 2manifold, so also every facepairing ɛ of a faceted 3ball P yields a closed 3dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3manifold at some of its vertices. The method of twisted facepairing shows how to correct this defect of the quotient pseudomanifold P/ɛ systematically. The method describes how to modify P by edge subdivision and how to modify any orientationreversing facepairing ɛ of P by twisting, so as to yield an infinite parametrized family of facepairings (Q, δ) whose quotient complexes Q/δ are all closed orientable 3manifolds. The method is so efficient that, starting even with almost trivial facepairings ɛ, it yields a rich family of highly nontrivial, yet relatively simple, 3manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by
SUBDIVISION RULES FOR SPECIAL CUBULATED GROUPS
"... Abstract. We find explicit subdivision rules for all special cubulated groups. A subdivision rule for a group produces a sequence of tilings on a sphere which encode all quasiisometric information for a group. We show how these tilings detect properties such as growth, ends, divergence, etc. We inc ..."
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Abstract. We find explicit subdivision rules for all special cubulated groups. A subdivision rule for a group produces a sequence of tilings on a sphere which encode all quasiisometric information for a group. We show how these tilings detect properties such as growth, ends, divergence, etc. We include figures of several worked out examples. 1.