Subdivision Rules, 3-Manifolds, and Circle Packings
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Simplicial Complexes
46 III Complexes III.1 Simplicial Complexes There are many ways to represent a topological space, one being a collection of simplices that are glued to each other in a structured manner. Such a collection can easily grow large but all its elements are simple. This is not so convenient for hand-calculations but close to ideal for computer implementations. In this book, we use simplicial complexes as the primary representation of topology. Rd k Simplices. Let u0; u1; : : : ; uk be points in . A point x = i=0 λiui is an affine combination of the ui if the λi sum to 1. The affine hull is the set of affine combinations. It is a k-plane if the k + 1 points are affinely Pindependent by which we mean that any two affine combinations, x = λiui and y = µiui, are the same iff λi = µi for all i. The k + 1 points are affinely independent iff P d P the k vectors ui − u0, for 1 ≤ i ≤ k, are linearly independent. In R we can have at most d linearly independent vectors and therefore at most d+1 affinely independent points. An affine combination x = λiui is a convex combination if all λi are non- negative. The convex hull is the set of convex combinations. A k-simplex is the P convex hull of k + 1 affinely independent points, σ = conv fu0; u1; : : : ; ukg. We sometimes say the ui span σ. Its dimension is dim σ = k. We use special names of the first few dimensions, vertex for 0-simplex, edge for 1-simplex, triangle for 2-simplex, and tetrahedron for 3-simplex; see Figure III.1. -
Arxiv:2008.00328V3 [Math.DS] 28 Apr 2021 Uniformly Hyperbolic and Topologically Mixing
ERGODICITY AND EQUIDISTRIBUTION IN STRICTLY CONVEX HILBERT GEOMETRY FENG ZHU Abstract. We show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank- one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures equipped with their Hilbert metric. More specifically, such structures admit a finite Sullivan measure; with respect to this measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects. In [Mar69], Margulis established counting results for uniform lattices in constant negative curvature, or equivalently for closed hyperbolic manifolds, by means of ergodicity and equidistribution results for the geodesic flows on these manifolds with respect to suitable measures. Thomas Roblin, in [Rob03], obtained analogous ergodicity and equidistribution results in the more general setting of CAT(−1) spaces. These results include ergodicity of the horospherical foliations, mixing of the geodesic flow, orbital equidistribution of the group, equidistribution of primitive closed geodesics, and, in the geometrically finite case, asymptotic counting estimates. Gabriele Link later adapted similar techniques to prove similar results in the even more general setting of rank-one isometry groups of Hadamard spaces in [Lin20]. These results do not directly apply to manifolds endowed with strictly convex projective structures, considered with their Hilbert metrics and associated Bowen- Margulis measures, since strictly convex Hilbert geometries are in general not CAT(−1) or even CAT(0) (see e.g. [Egl97, App. B].) Nevertheless, these Hilbert geometries exhibit substantial similarities to Riemannian geometries of pinched negative curvature. In particular, there is a good theory of Busemann functions and of Patterson-Sullivan measures on these geometries. -
[Math.GT] 9 Oct 2020 Symmetries of Hyperbolic 4-Manifolds
Symmetries of hyperbolic 4-manifolds Alexander Kolpakov & Leone Slavich R´esum´e Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4- vari´et´es non-compactes compl`etes arithm´etiques hyperboliques M, `avolume fini, telles M G + M G que Isom ∼= , ou Isom ∼= . Pour y parvenir, nous utilisons essentiellement la g´eom´etrie de poly`edres de Coxeter dans l’espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C¸a nous permet d’obtenir une borne sup´erieure universelle pour le volume minimal d’une 4-vari´et´ehyperbolique ayant le groupe G comme son groupe d’isom´etries, par rap- port de l’ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-vari´et´es hyperboliques ayant G comme le groupe d’isom´etries. Abstract In this paper, for each finite group G, we construct the first explicit examples of non- M M compact complete finite-volume arithmetic hyperbolic 4-manifolds such that Isom ∼= G + M G , or Isom ∼= . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group. -
Apollonian Circle Packings: Dynamics and Number Theory
APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND NUMBER THEORY HEE OH Abstract. We give an overview of various counting problems for Apol- lonian circle packings, which turn out to be related to problems in dy- namics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lec- tures given at RIMS, Kyoto in the fall of 2013. Contents 1. Counting problems for Apollonian circle packings 1 2. Hidden symmetries and Orbital counting problem 7 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 15 5. Expanders and Sieve 19 References 25 1. Counting problems for Apollonian circle packings An Apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of Apollonius of Perga: Theorem 1.1 (Apollonius of Perga, 262-190 BC). Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Figure 1. Pictorial proof of the Apollonius theorem 1 2 HEE OH Figure 2. Possible configurations of four mutually tangent circles Proof. We give a modern proof, using the linear fractional transformations ^ of PSL2(C) on the extended complex plane C = C [ f1g, known as M¨obius transformations: a b az + b (z) = ; c d cz + d where a; b; c; d 2 C with ad − bc = 1 and z 2 C [ f1g. As is well known, a M¨obiustransformation maps circles in C^ to circles in C^, preserving angles between them. -
F-VECTORS of BARYCENTRIC SUBDIVISIONS 1. Introduction This Work Is Concerned with the Effect of Barycentric Subdivision on the E
f-VECTORS OF BARYCENTRIC SUBDIVISIONS FRANCESCO BRENTI AND VOLKMAR WELKER Abstract. For a simplicial complex or more generally Boolean cell complex ∆ we study the behavior of the f- and h-vector under barycentric subdivision. We show that if ∆ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney-Davis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary complex of a simple polytope. For a general (d − 1)-dimensional simplicial complex ∆ the h- polynomial of its n-th iterated subdivision shows convergent be- havior. More precisely, we show that among the zeros of this h- polynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d. 1. Introduction This work is concerned with the effect of barycentric subdivision on the enumerative structure of a simplicial complex. More precisely, we study the behavior of the f- and h-vector of a simplicial complex, or more generally, Boolean cell complex, under barycentric subdivisions. In our main result we show that if the h-vector of a simplicial complex or Boolean cell complex is non-negative then the h-polynomial (i.e., the generating polynomial of the h-vector) of its barycentric subdivi- sion has only real simple zeros. Moreover, if one applies barycentric subdivision iteratively then there is a limiting behavior of the zeros of the h-polynomial, with one zero going to infinity and the other zeros converging. -
On Eigenvalues of Geometrically Finite Hyperbolic Manifolds With
On Eigenvalues of Geometrically Finite Hyperbolic Manifolds with Infinite Volume Xiaolong Hans Han April 6, 2020 Abstract Let M be an oriented geometrically finite hyperbolic manifold of dimension n ≥ 3 with infinite volume. Then for all k ≥ 0, we provide a lower bound on the k-th eigenvalue of the Laplacian operator acting on functions on M by a constant and the k-th eigenvalue of some neighborhood Mf of the thick part of the convex core. 1. Introduction As analytic data of a manifold, the spectrum of the Laplacian operator acting on func- tions contains rich information about the geometry and topology of a manifold. For example, by results of Cheeger and Buser, with a lower bound on Ricci curvature, the first eigenvalue is equivalent to the Cheeger constant of a manifold, which depends on the geometry of the manifold. The rigidity and richness of hyperbolic geometry make the connection even more intriguing: by Canary [4], for infinite volume, topologically tame hyperbolic 3-manifolds, the 2 bottom of the L -spectrum of −∆, denoted by λ0, is zero if and only if the manifold is not geometrically finite. If λ0 of a geometrically finite, infinite volume hyperbolic manifold is 1, by Canary and Burger [2], the manifold is either the topological interior of a handlebody or an R-bundle over a closed surface, which is a very strong topological restriction. Conversely, if we have data about the geometry of a manifold, we can convert it into data about its spectrum, like lower bounds on eigenvalues, dimensions of eigenspaces, existence of embedded eigenvalues, etc. -
ISOMETRY TRANSFORMATIONS of HYPERBOLIC 3-MANIFOLDS Sadayoshi KOJIMA 0. Introduction by a Hyperbolic Manifold, We Mean a Riemanni
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 29 (1988) 297-307 297 North-Holland ISOMETRY TRANSFORMATIONS OF HYPERBOLIC 3-MANIFOLDS Sadayoshi KOJIMA Department of Information Science, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo, Japan Received 1 December 1986 Revised 10 June 1987 The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold. AMS (MOS) Subj. Class.: 57S17, 53C25, 57M99 3-manifold hyperbolic manifold isometry 0. Introduction By a hyperbolic manifold, we mean a Riemannian manifold with constant sectional curvature -1. It is known by several ways that the isometry group of a hyperbolic manifold of finite volume is finite. On the other hand, Greenberg has shown in [3] that every finite group is realized as the full conformal automorphism group of some Riemann surface of hyperbolic type. Since such a Riemann surface, which is a 2-manifold with a conformal structure, inherits a unique conformally equivalent hyperbolic structure through the uniformization theorem, there are no limitations on the group structure of the full isometry group of a hyperbolic manifold in dimension 2. In this paper, we first give a quick proof of the theorem of Greenberg quoted above, and then along the same line we show, raising the dimension by one. Theorem. Every jinite group is realized as the full isometry group of some closed hyperbolic 3 -manifold. An immediate corollary to this is the following. -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. -
Traversing Three-Manifold Triangulations and Spines 11
TRAVERSING THREE-MANIFOLD TRIANGULATIONS AND SPINES J. HYAM RUBINSTEIN, HENRY SEGERMAN AND STEPHAN TILLMANN Abstract. A celebrated result concerning triangulations of a given closed 3–manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3–manifolds. These results build on classical work that goes back to Alexander, Newman, Moise, and Pachner. The key special case of 1–vertex triangulations of closed 3–manifolds was independently proven by Matveev and Piergallini. The general result for closed 3–manifolds can be found in work of Benedetti and Petronio, and Amendola gives a proof for topologically finite non-compact 3–manifolds. These results (and their proofs) are phrased in the dual language of spines. The purpose of this note is threefold. We wish to popularise Amendola’s result; we give a combined proof for both closed and non-compact manifolds that emphasises the dual viewpoints of triangulations and spines; and we give a proof replacing a key general position argument due to Matveev with a more combinatorial argument inspired by the theory of subdivisions. 1. Introduction Suppose you have a finite number of triangles. If you identify edges in pairs such that no edge remains unglued, then the resulting identification space looks locally like a plane and one obtains a closed surface, a two-dimensional manifold without boundary. The classification theorem for surfaces, which has its roots in work of Camille Jordan and August Möbius in the 1860s, states that each closed surface is topologically equivalent to a sphere with some number of handles or crosscaps. -
Thurston Obstructions and Ahlfors Regular Conformal Dimension Peter Haïssinsky, Kevin Pilgrim
Thurston obstructions and Ahlfors regular conformal dimension Peter Haïssinsky, Kevin Pilgrim To cite this version: Peter Haïssinsky, Kevin Pilgrim. Thurston obstructions and Ahlfors regular conformal dimension. 2007. hal-00152983v2 HAL Id: hal-00152983 https://hal.archives-ouvertes.fr/hal-00152983v2 Preprint submitted on 22 Jun 2007 (v2), last revised 13 Jun 2008 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thurston obstructions and Ahlfors regular conformal dimension Peter Ha¨ıssinsky, Univ. Provence and Kevin M. Pilgrim, Indiana University Bloomington June 22, 2007 Abstract Let f : S2 → S2 be an expanding branched covering map of the sphere to itself with finite postcritical set Pf . Associated to f is a canonical quasisymmetry class G(f) of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show inf H.dim(X) ≥ max 2, sup{Q|λ(fΓ,Q)=1} . X∈G(f) Γ 2 Γ Γ The supremum is taken over all multicurves Γ ⊂ S − Pf . The map fΓ,Q : R → R is defined by 1−Q ′ fΓ,Q(γ)= deg(f : δ → γ) [γ ], ′ ∼ ′ [γX]∈Γ δXγ ′ 2 where the second sum is over all preimages δ of γ freely homotopic to γ in S −Pf , and λ(fΓ,Q) is its Perron-Frobenius leading eigenvalue. -
Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions
Finite Subdivision Rules from Matings of Quadratic Functions: Existence and Constructions Mary E. Wilkerson Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics William J. Floyd, Chair Peter E. Haskell Leslie D. Kay John F. Rossi April 24, 2012 Blacksburg, Virginia Keywords: Finite Subdivision Rules, Matings, Hubbard Trees, Combinatorial Dynamics Copyright 2012, Mary E. Wilkerson Finite Subdivision Rules from Matings of Quadratic Functions Mary E. Wilkerson (ABSTRACT) Combinatorial methods are utilized to examine preimage iterations of topologically glued polynomials. In particular, this paper addresses using finite subdivision rules and Hubbard trees as tools to model the dynamic behavior of mated quadratic functions. Several methods of construction of invariant structures on modified degenerate matings are detailed, and examples of parameter-based families of matings for which these methods succeed (and fail) are given. Acknowledgments There are several wonderful, caring, and helpful people who deserve the greatest of my appreciation for helping me through this document and for my last several years at Virginia Tech: • My family and friends, I owe you thanks for your love and support. • Robert and the Tolls of Madness, thank you for keeping me sane by providing an outlet to help me get away from grad school every now and then. • All of my previous mentors and professors in the VT Math Department, I can't express how far I feel that I've come thanks to the opportunities you've given me to learn and grow. Thank you all for giving me a chance and guiding me to where I am today. -
Constructing Subdivision Rules from Rational Maps
CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 11, Pages 128–136 (August 14, 2007) S 1088-4173(07)00167-1 CONSTRUCTING SUBDIVISION RULES FROM RATIONAL MAPS J. W. CANNON, W. J. FLOYD, AND W. R. PARRY Abstract. This paper deepens the connections between critically finite ra- tional 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 ◦n is the subdivision map of a finite subdivision rule. This enables one to give essentially combinatorial models for the dynamics of such iterates. We are interested here in connections between finite subdivision rules and ra- tional 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 agroupG on its space at infinity, and finite subdivision rules were developed to give models for the action of such 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 com- plex SR is a 2-sphere, then the subdivision map σR is a critically finite branched map of this 2-sphere.