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Rigidity of Quasiconformal Maps on Carnot Groups RIGIDITY OF QUASICONFORMAL MAPS ON CARNOT GROUPS Mark Medwid A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2017 Committee: Xiangdong Xie, Advisor Alexander Tarnovsky, Graduate Faculty Representative Mihai Staic Juan Bes´ Copyright c August 2017 Mark Medwid All rights reserved iii ABSTRACT Xiangdong Xie, Advisor Quasiconformal mappings were first utilized by Grotzsch¨ in the 1920’s and then later named by Ahlfors in the 1930’s. The conformal mappings one studies in complex analysis are locally angle-preserving: they map infinitesimal balls to infinitesimal balls. Quasiconformal mappings, on the other hand, map infinitesimal balls to infinitesimal ellipsoids of a uniformly bounded ec- centricity. The theory of quasiconformal mappings is well-developed and studied. For example, quasiconformal mappings on Euclidean space are almost-everywhere differentiable. A result due to Pansu in 1989 illustrated that quasiconformal mappings on Carnot groups are almost-everywhere (Pansu) differentiable, as well. It is easy to show that a biLipschitz map is quasiconformal but the converse does not hold, in general. There are many instances, however, where globally defined quasiconformal mappings on Carnot groups are biLipschitz. In this paper we show that, under cer- tain conditions, a quasiconformal mapping defined on an open subset of a Carnot group is locally biLipschitz. This result is motivated by rigidity results in geometry (for example, the theorem by Mostow in 1968). Along the way we develop background material on geometric group theory and show its connection to quasiconformal mappings. iv ACKNOWLEDGMENTS I would like to acknowledge, first and foremost, my wife, Heather. She’s put up with the long nights and anxiety that accompany graduate school; she constantly cheered me on, told me to be more confident, and agreed to put her life on hold for a number of years while I finished my degree. I’m not sure my finishing graduate school would be possible without her support. Next, I would like to thank my advisor, Xiangdong Xie, who exposed me to a new field of mathematics. He very patiently put up with my na¨ıve questions, ignorance and incorrect proofs; he slowly helped me to become a better mathematician. Beyond academics, he gave me lots of helpful career and life advice that I’ll be sure to apply in the future. He is also one of the first people to encourage me to graduate early – this lead to me accepting a new tenure-track job. I would also like to acknowledge and thank the other members of my dissertation commit- tee (Mihai Staic, Juan Bes´ and Alexander Tarnovsky) for their thoughtful commentary on my manuscript and insightful questions during my defense. Furthermore, I would like to acknowledge my fellow colleagues (both former and present) for their many helpful discussions on life, graduate school and mathematics. I formed many valuable friendships during graduate school. There are too many to be included here, but I will attempt to do so: Jake Laubacher, David Walmsley, Todd Romutis, John Haman and Sam Carolus shared an office with me and kept my day-to-day life more interesting. This was a room filled with exceedingly promising young mathematicians and I was honored to be a part of it. I’d also like to thank Robert Kelvey, Jeff Norton and Leo Pinheiro (who are all now on to greener pastures) for their many helpful conversations with me. My family, in particular, my parents, deserve special mention for all their support over the ten consecutive years I’ve been in school since graduating from high school in 2007. They’ve always had my back, no matter what. Finally, I feel that the Math & Stats Department at BGSU deserves special mention. The department supported my education expenses and my livelihood, and there are so many stellar faculty and staff members that helped form me both as a mathematician and as a teacher. v TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION . 1 1.1 The Word Metric . 2 1.2 Quasi-Isometries . 3 1.3 Quasi-Isometric Rigidity . 9 CHAPTER 2 PRELIMINARIES . 11 2.1 BiLipschitz Mappings . 11 2.2 Quasiconformality . 12 2.3 Quasisymmetric Mappings . 17 CHAPTER 3 QUASI-ISOMETRIES ON HYPERBOLIC SPACE . 24 3.1 The Morse Lemma . 24 3.2 The Gromov Boundary . 35 CHAPTER 4 CARNOT GROUPS . 42 4.1 Lie Algebras and Lie Groups . 42 4.2 The Exponential Map & Carnot-Caratheodory´ Distance . 46 4.3 Pansu’s Differentiation Theorem . 51 CHAPTER 5 MAIN RESULTS . 55 5.1 Rigidity of Quasisymmetric Mappings on Fibered Metric Spaces . 55 5.2 Rigidity of Quasiconformal Maps on Two-Step Carnot Groups . 66 BIBLIOGRAPHY . 76 vi LIST OF FIGURES Figure Page 2.1 An artist’s rendering of what a quasiconformal map might look like. 13 3.1 The Morse Lemma. 25 3.2 Two cases for triangles in trees. 31 3.3 Asymptotic rays in the upper-half plane model of H2. 36 1 CHAPTER 1 INTRODUCTION The purpose of this dissertation is to develop the theory accompanying the main result of the author’s research efforts while studying at Bowling Green State University, as well as the proof of the result itself. The author assumes knowledge of basic group theory, analysis, and point-set topology; anything beyond this will be developed in the pages contained hereafter. The following is the author’s main result: Theorem. Let G = V1 V2 be a two-step Carnot group and let W1 V1 be a subspace and set ⊕ ✓ W := [W ,W ]. Let W = W W ; W is a Lie sub-algebra of G and so corresponds to a Lie 2 1 1 1 ⊕ 2 subgroup. Suppose that E G is an open set. If f : E G is a quasiconformal mapping that ✓ ! permutes the left cosets of W , then f is locally biLipschitz. Many of the terms contained in the above theorem are ostensibly foreign to the reader. The basic idea is the following: Carnot groups are spaces that are “nice” enough to carry some notion of differentiability. The definition is fairly similar in flavor to the familiar one from calculus. One then becomes interested in a certain type of (almost everywhere) differentiable function f that we name “quasiconformal.” As in calculus, f needs only to be defined on an open subset for the purposes of the theorem. If f moves a particular subset in a nice enough way, then it turns out that f is locally biLipschitz – to wit, inside a smaller open set, f distorts distances by no more than some bounded factor. This is perhaps enough to remind one of the famed Mean Value Theorem of first year calculus: Theorem. Let f :[a, b] R be continuous and differentiable on (a, b). Then there exists some ! number c in (a, b) so that f(b) f(a) = f 0(c) (b a). | − | · − When f has a bounded derivative, then this implies f distorts the lengths of intervals by a bounded factor. The main result of this dissertation is, of course, different in many ways from the 2 Mean Value Theorem, requiring a different setting and assumptions, but the punchline to this quick corollary carries a similar spirit. With the above context, the main theorem is an analysis theorem. However, the study of such theorems also has some deep ties to the world of geometric group theory that will be fleshed out in the body of the dissertation. While quasiconformal analysis is used in several branches of math- ematics, it is through geometric group theory that the author was first exposed to the study of quasiconformal mappings. Details for theorems and definitions will be provided (when useful) so as to make this dissertation a stand-alone document for the readers; however, the introductory material is, for the most part, well studied. Many excellent papers and books exist for provid- ing copious details on these subjects – recommendations will be provided when relevant for the interested reader. 1.1 The Word Metric Recall that a group is finitely generated if there exists a finite set S so that G = S . In what h i follows, unless specified, G denotes a finitely generated group. One can endow G with a metric space structure as follows. Definition 1.1.1. (Kapovich, 2013) Suppose that G is a finitely generated group with generating set S = s1,s2,...,sd . The Cayley graph of G with respect to S is a graph Γ(G, S) with vertex { } set G and edges are constructed according to the rule there is an edge connecting g and h g = h s✏ for some 1 i d, ✏ 1, 1 . () · i 2{− } Definition 1.1.2. (Kapovich, 2013) Let G be a finitely generated group with generating set S, and suppose Γ(G, S) is the Cayley graph of G with respect to S. Endow Γ with the shortest path metric d – the restriction of d to G, dS, is called the word metric on G with respect to S and is denoted (G, dS). One should note that, if e is the identity element of G, then for any g G, d (e, g) is 2 S the reduced word length of g with letters in S. Suppose h, g G; write both h and g as re- 2 3 ✏ ✏ ✏ ✏ duced words with letters in S, i.e. h = s 1h s kh ,g = s 1g s kg . Then the sequence i(1h) ··· i(kh) i(1g) ··· i(kg) ✏1g ✏1g ✏kg 1 (h, hs ,...,hs s − ,hg) h hg i(1g) i(1g) i(kg 1) gives an edge-path joining and , so this implies that ··· − d(e, g) d(h, hg).
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