combinatorial type problems for triangulation graphs

THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES COMBINATORIAL TYPE PROBLEMS FOR TRIANGULATION GRAPHS By WILLIAM E. WOOD A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2006 UMI Number: 3232464 3232464 2006 Copyright 2006 by Wood, William E. UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 All rights reserved. by ProQuest Information and Learning Company. The members of the Committee approve the dissertation of William E. Wood defended on July 6, 2006. Philip Bowers Professor Directing Dissertation Lois Hawkes Outside Committee Member Steve Bellenot Committee Member Eric Klassen Committee Member Craig Nolder Committee Member Jack Quine Committee Member The Office of Graduate Studies has verified and approved the above named committee members. ii Dedicated to my parents, Bob and Sue Wood, and to my godmother, Dee Gelbach, for offering their love and support through everything I’ve done. Even the very silly things. iii ACKNOWLEDGEMENTS I owe a lot to a great many people for getting me this far. Thanks first go to my committee, Steve Bellenot, Phil Bowers, Lois Hawkes, Eric Klassen, Craig Nolder, and Jack Quine for enduring reading this document. I have dedicated this work to my parents, Bob and Sue Wood, and my Aunt Dee. I would be nowhere without them. Ken Stephenson has been like a second advisor to me. Monica Hurdal taught me lots in my early graduate career, and I thank her for allowing me to work as her research assistant. Ara Basmajian, Soo Bong Chae, Peter Doyle, Zheng-Xu He, Karsten Henckell, David Mullins, Eirini Poimenidiou, and Jim Tanton were some early influences without whom I would never have gotten here. I also had a terrific group of faculty and fellow graduate students at Florida State University whose counsel and enthusiasm I have found invaluable. Thanks also to David Dickerson for his contribution to this work. I also thank the many baristas at assorted Tallahassee coffee shops for transforming portions of my meager income into the caffeinated beverages that fueled this thesis. Finally, I owe a great debt to my advisor, Phil Bowers. Phil has been a mentor to me since long before I ever became a graduate student. I am greatly honored to have him as an advisor and as a friend. iv TYPESETTING NOTES This thesis was written in L A T E X 2 ε using the excellent WinEdt 5 editor. The circle packing pictures were generated by Ken Stephenson’s CirclePack program. Most of the remaining diagrams were drawn in XFig 3.2, with some drawn in Maple 9.5, OpenOffice 1.1.4, and Mayura 4.3. v TABLE OF CONTENTS List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. DISCRETE CONFORMAL TYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Classical conformal type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 From surfaces to graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Vertex and edge extremal length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Random walks and electric networks . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Circle packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Equilateral type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.8 The type problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. BOUNDED REFINEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Combinatorial extremal length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Extremal metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Shadow paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Extremal length of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.5 Electrical versus VEL type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.6 Bounded refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.7 Refinement and edge extremal length . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.8 Outdegree of planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.9 Refinement and vertex extremal length . . . . . . . . . . . . . . . . . . . . . . . . 44 3.10 Refinement and type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.11 Beyond triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4. TYPE INVARIANTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1 Parabolic vertex growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Hyperbolic trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Explicit parabolic extremal metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Growth versus asymmetry: displacing growth . . . . . . . . . . . . . . . . . . . 62 4.5 Growth versus asymmetry: trapping growth . . . . . . . . . . . . . . . . . . . . 64 4.6 Slow growth in hyperbolic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7 k-fuzz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.8 Dual graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 vi 4.9 Outer spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5. EQUILATERAL SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.1 Quasiconformal mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Equilateral surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3 Bounded degree convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.4 Unbounded degree convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 APPENDIX: A CONFORMAL EXTREMAL METRIC ON THE PLANE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 vii LIST OF FIGURES 2.1 Conducting surfaces and the Riemann map . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The constant degree five, six, and seven packings . . . . . . . . . . . . . . . . . 14 2.3 Circle packings approximate the Riemann map . . . . . . . . . . . . . . . . . . . 18 2.4 Conductance for the tailored random walk . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Packing with alternating generations of degree 6 and 7. . . . . . . . . . . . . 22 3.1 The diamond configuration D n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Resistors in series and parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Construction of a VEL-parabolic, EEL-hyperbolic electric network . . . . 36 3.4 Incident-adjacent edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Barycentric and hexagonal refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Shadow paths in hex refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 Adding a binary tree with an unbounded refinement . . . . . . . . . . . . . . . 47 3.8 Zig-zag refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Circuit reductions for a binary tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Sector of the constant degree six triangulation . . . . . . . . . . . . . . . . . . . . 61 4.3 A parabolic tree with exponential growth . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Trapping growth inside diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5 Quadratic growth in a hyperbolic triangulation . . . . . . . . . . . . . . . . . . . 68 4.6 Parabolic graphs with hyperbolic k-fuzz . . . . . . . . . . . . . . . . . . . . . . . . 72 4.7 The graph G # is a refinement of both G and G ∗ . . . . . . . . . . . . . . . . . . 73 4.8 Geodesic vertex paths in S(n) may not extend. . . . . . . . . . . . . . . . . . . . 75 4.9 A vertex in S O (n) has two neighbors in S O (n) . . . . . . . . . . . . . . . . . . . . 76 4.10 A vertex in S O (n) cannot have three neighbors in S O (n) . . . . . . . . . . . . 77 4.11 S O (n) is connected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.12 The outer sphere skeleton G O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.1 log log z as a conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 viii ABSTRACT The main result in this thesis bounds the combinatorial modulus of a ring in a triangulation graph in terms of the modulus of a related ring. The bounds depend only on how the rings are related and not on the rings themselves. This may be used to solve the combinatorial type problem in a variety of situations, most significantly in graphs with unbounded degree. Other results regarding the type problem are presented along with several examples illustrating the limits of the results. ix [...]... 19 1 The VEL type of G is the same as its CP type 2 The RW type, electrical type, and EEL type of G are the same 3 If G has bounded degree, then all described forms of combinatorial type – electrical type, RW type, CP type, VEL type, EEL type, and EQ type – are the same 4 There is a graph that is EEL-hyperbolic and VEL-parabolic In other words, there is really only one notion of type for a graph of... construction will be discussed in Section 5.2 This surface has a classical conformal type as described in Section 2.1, and we define the equilateral type of G to be the classical conformal type of |G|eq Equilateral type is defined as the classical type of a surface that is described purely combinatorially As such, equilateral type offers a very direct link between the classical and discrete theories Indeed,... then survey how to use this and other techniques to solve the type problem for certain classes of graphs We also examine another form of combinatorial type introduced by Bowers and Stephenson that offers a direct connection between the combinatorics and the geometry Chapter 2 surveys the field and introduces the various forms of conformal type and how they may be applied discretely We also fix a lot of... a feel for the objects of concern and to motivate the forthcoming results Some of the details are bypassed for material that will be developed later, will not play a role in our proofs, or for which there are established references (which we provide) We also get a few technical definitions and notations out of the way 2.1 Classical conformal type We begin with a review of classical conformal type, presenting... degree and two for unbounded degree It is not known where EQ -type fits for unbounded degree graphs Parts (1) and (4) are proved by He and Schramm in [HS95] We detail the construction of a VEL-parabolic and EEL-hyperbolic graph in Chapter 3 The equivalence between electrical and RW -type as outlined in Section 2.4 is detailed in [DS84], whereas the equivalence of EEL type and electrical type may be found... the nature of the type dichotomy For example, if one has a graph of known type, to what extent could one modify it without changing the type? Our main theorem gives a fairly precise answer to this question We also will present several examples illustrating exactly how far is too far Many of the results of discrete conformal geometry use geometric methods, specifically those of quasiconformal mappings... non-compact simply connected Riemann surfaces S Let f be a conformal homeomorphism of S into a region in C The modulus, extremal length, and conformal type are defined on S as the corresponding notions in f (S), and the theorems discussed still hold 2.2 From surfaces to graphs In the combinatorial setting, our fundamental objects are no longer surfaces, but graphs We make this connection in the next section,... radius See [RS87] or [Ste05] for a proof, or [Aha90],[Aha94] for an alternate approach and generalizations Our interest is in the discrete notion of conformal type the theory of circle packings provides A disk triangulation graph is circle pack hyperbolic (CP-hyperbolic) if it is the contacts graph of a circle packing of the unit disk Otherwise it is the contacts graph for a packing of the plane and... of this chapter is to develop combinatorial extremal length for graphs and prove our featured theorem, which says that if edges and vertices are added to a graph by a sufficiently reasonable process, then the change in the extremal length is bounded in a way that does not depend on the graph An immediate consequence is that these processes preserve combinatorial type 3.1 Combinatorial extremal length... circle packing is always a triangulation We use the term “complex” to describe more general cell structures, but never for a circle packing.) For a circle c in a packing, its flower is c along with the circles tangent to c As we add layers of circles about c, we define the nth generation of circles about c as the circles whose combinatorial distance to c is n (technically we mean the combinatorial distance . SCIENCES COMBINATORIAL TYPE PROBLEMS FOR TRIANGULATION GRAPHS By WILLIAM E. WOOD A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree. theorems. 3 CHAPTER 2 DISCRETE CONFORMAL TYPE We overview the various forms of type in this chapter. Our goal is to get a feel for the objects of concern and to motivate the forthcoming results. Some. CONFORMAL TYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Classical conformal type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 From surfaces to graphs