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The University of Chicago Some Theorems in Kleinian Groups and Complex Dynamics a Dissertation Submitted to the Faculty of the D THE UNIVERSITY OF CHICAGO SOME THEOREMS IN KLEINIAN GROUPS AND COMPLEX DYNAMICS A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY YAN MARY HE CHICAGO, ILLINOIS JUNE 2018 Copyright c 2018 by Yan Mary He All Rights Reserved To my parents and in memory of Yichen Yang TABLE OF CONTENTS ACKNOWLEDGMENTS . vi ABSTRACT . vii 1 INTRODUCTION . 1 1.1 Basmajian-type identities . .1 1.1.1 Statement of results . .4 1.2 Displacement problem for free Fuchsian groups . .6 2 PRELIMINARIES . 10 2.1 Hyperbolic geometry . 10 2.2 Kleinian groups . 11 2.2.1 Actions on the boundary sphere and limit sets . 11 2.2.2 Marked Schottky space and Schottky groups . 12 2.3 Complex dynamics of quadratic polynomials . 12 2.4 Thermodynamic formalism . 14 2.4.1 Symbolic coding . 14 2.4.2 Transfer operator, pressure and Hausdorff dimension . 16 2.4.3 Application I: Limit set of Schottky groups . 19 2.4.4 Application II: Julia set of quadratic polynomials . 20 3 BASMAJIAN-TYPE IDENTITIES AND HAUSDORFF DIMENSION OF LIMIT SETS . 22 3.1 Some linear examples . 22 3.1.1 Middle-third Cantor set . 23 3.1.2 Semigroups of similarities . 24 3.2 Schottky Groups: the Complexified Basmajian Identity . 25 3.2.1 Complex length . 26 3.2.2 Double cosets, finite state automata and orthogeodesics . 26 3.2.3 Cross ratios . 28 3.2.4 Complexified Basmajian’s identity . 29 3.2.5 Monodromy . 31 3.3 Quadratic Polynomials . 31 3.3.1 Basmajian-type identity . 32 3.3.2 Monodromy . 34 3.3.3 Hausdorff dimension one locus . 34 3.4 Proof of the Convergence Theorems . 35 3.4.1 Quadratic polynomials: proof of Theorem 3.3.2 . 35 3.4.2 Schottky groups: proof of Theorem 3.2.6 . 37 3.5 Basmajian identity on the modular surface . 40 iv 4 DISPLACEMENT OF GENERATORS OF FREE FUCHSIAN GROUPS . 44 4.1 Conformal densities and paradoxical decompositions . 44 4.1.1 Conformal densities . 44 4.1.2 Culler-Shalen’s paradoxical decompositions . 45 4.2 The finite co-area case . 47 4.3 The general case . 51 4.4 Quantitative geometry of hyperbolic surfaces . 53 4.4.1 The two-dimensional Margulis constant . 53 4.4.2 Lengths of a pair of closed loops . 54 REFERENCES . 56 v ACKNOWLEDGMENTS I consider myself extremely lucky to have both Danny Calegari and Peter Shalen as my advisors. They not only have taught me a great deal of mathematics over the past several years, but more importantly have influenced me to become a good person with integrity and tenacity. I thank Danny Calegari for being a great source of mathematics, for passing down the value of having broad mathematical interests to the next generation and for teaching me not to be afraid of difficult problems, both in life and in mathematics. I thank Peter Shalen for being an amazing role model. I admire his talent and his sense of humor, and I’m thankful for his patience and encouragement which kept me going during many difficult times and sustained my interests in mathematics. I thank Juliette Bavard for her wonderful friendship and inspiring collaboration. I thank everyone in the Les Fous family - Juliette Bavard, Lucas Bakes, Maxime Bergeron, Moreno Bonutti, Marco Guaraco, Sebastian Hurtado, Jacek Jendrej, Rohit Nagpal, Roseanne Raphael and Yuchin Sun, as well as Ying Bao, Minh Pham and Nhung Hoang for providing a warm family and delicious food in Hyde Park. On both professional and personal levels, I am grateful to many of my math friends, in- cluding Spyros Alexakis, Ara Basmajian, John Friedlander, Andrey Gogolev, Quoc Ho, Yong Hou, Lisa Jeffrey, Sarah Koch, Paul Selick, Fang Wang and Yun Yang, for their mentorship and friendship at various stages of my mathematical development. For the past decade, from Toronto to Chicago, I am extremely grateful to my parents for their unconditional love, sacrifice, support and money. I also thank them for accepting me becoming a wrong type of Doctor. vi ABSTRACT This thesis is an amalgam of the two manuscripts [16] and [17] that the author completed during her graduate studies at the University of Chicago. Article [16] was published in Ergodic Theory and Dynamical Systems and is reprinted with permission here. The first part of the thesis contains results obtained in [16], where we study Basmajian- type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit nontrivial monodromy. The second part of the thesis follows mainly from [17]. In particular, we prove an inequal- ity that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the log(2k − 1) Theorem for Kleinian groups due to Anderson- Canary-Culler-Shalen [1]. As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of a pair of based loops, which improves a result of Buser’s. vii CHAPTER 1 INTRODUCTION The main body of this thesis, Chapter 3 and 4, is respectively an expanded version of the manuscripts [16] and [17] that the author completed during her graduate studies at the University of Chicago. After giving some preliminaries in Chapter 2, we study Basmajian- type identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems in Chapter 3 and we discuss the displacement problem for free Fuchsian groups in Chapter 4. 1.1 Basmajian-type identities If C ⊂ [0; 1] is a Cantor set of zero measure, the Hausdorff dimension of C is the limit of −1 −(log an= log n) , where an is the length of the nth biggest component of [0; 1] − C. There is no obvious analog of this theorem for an arbitrary Cantor set C ⊂ C, but for a family of Cantor sets Cz ⊂ C depending holomorphically on a complex parameter z, we can sometimes obtain such a relation. Note that if the measure of C ⊂ [0; 1] is zero, then an satisfy an identity 1 X 1 = an: n=1 For Cz ⊂ C depending on z, we obtain by analytic continuation a formal holomorphic family of identities 1 X S(z) = an(z): n=1 Thus it is natural to investigate the conditions under which the right hand side is absolutely summable. In Chapter 3, we study holomorphic families of Cantor sets Cz associated to familiar 1 1-dimensional complex dynamical systems, and for such families we introduce identities of this form (which have a natural geometric interpretation when Cz ⊂ R) and show that the right hand sides are absolutely summable if and only if the Hausdorff dimension of Cz is strictly less than 1. The identities themselves are of independent interest — even in the case of C ⊂ R, where a special case is Basmajian’s orthospectrum identity for a hyperbolic surface. If Σ is a compact hyperbolic surface with geodesic boundary, an orthogeodesic γ ⊂ Σ is a properly immersed geodesic arc perpendicular to @Σ. Basmajian [2] proved the following identity: X length(γ) length(@Σ) = 2 log coth (1.1) 2 γ where the sum is taken over all orthogeodesics γ in Σ. The geometric meaning of this identity is that there is a canonical decomposition of @Σ into a Cantor set (of zero measure), plus a countable collection of complementary intervals, one for each orthogeodesic, whose length depends only on the length of the corresponding orthogeodesic. For simplicity, one can look at surfaces with a single boundary component. A hyperbolic structure on Σ is the same as a discrete faithful representation ρ : π1(Σ) ! PSL(2; R) acting on the upper half-plane model in the usual way. After conjugation, we may assume that ρ(@Σ) stabilizes the positive imaginary axis `. Orthogeodesics correspond to double cosets of π1(@Σ) in π1(Σ). For each nontrivial double coset π1(@Σ)απ1(@Σ), the hyperbolic geodesic α(`) corresponds to another boundary component of Σe, and the contribution to Basmajian’s identity from this term is log(ρ(α)(0)/ρ(α)(1)); hence X length(@Σ) = log(ρ(α)(0)/ρ(α)(1)): α If we deform ρ to some nearby representation ρz : π1(Σ) ! PSL(2; C), and replace each ρ by ρz above, we obtain a formula for the complex length of ρz(@Σ). This is the desired 2 complexification of Basmajian’s identity. The identity makes sense, and is absolutely convergent, exactly throughout the subset S<1 of Schottky space S where the Hausdorff dimension of the limit set of ρz(π1(Σ)) is less than one. Since log is multivalued, it is important to choose the correct branch at a real representation, and then follow the branch by analytic continuation. The space S<1 is not simply-connected and some terms exhibit monodromy (in the form of integer multiples of 2πi) when they are analytically continued around homotopically nontrivial loops. Based on Sullivan’s dictionary between Kleinian groups and rational maps, we develop a 2 parallel theory for quadratic polynomials fc(z) = z + c. If the complex parameter c is real and is in the interval (−∞; −2), the Julia set Jc of the quadratic polynomial fc is a Cantor set of zero measure contained in the real line. Thus there is a natural Basmajian-type identity associated to Jc.
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