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Relatively Dominated Representations Relatively Dominated Representations by Feng Zhu A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2020 Doctoral Committee: Professor Richard D. Canary, Chair Associate Professor Steven Abney Professor Lizhen Ji Associate Professor Sarah C. Koch Professor Ralf J. Spatzier Feng Zhu [email protected] ORCID iD: 0000-0003-1450-8473 © Feng Zhu 2020 DEDICATION To '舅l, for having planted the first seeds of the discipline in my young mind. ii ACKNOWLEDGMENTS Thank you, Dick Canary, for suggesting this question, for many helpful discussions, and for being a wonderful Ph.D. advisor. Thank you also to my dissertation committee for your help. I would like to thank the mathematical community and the department for taking me in and for their support, without which this journey would not have been possible. I thank in particular Jean-Philippe Burelle, Jeff Danciger, Matt Durham, Ilya Gekhtman, Fanny Kassel, Wouter van Limbeek, Sara Maloni, Jason Manning, Max Riestenberg, Andres´ Sambarino and Kostas Tsouvalas for their insights, advice, and encouragement, and Jairo Bochi for pointing out (via Andres´ Sambarino) the work of [QTZ19]. The author was partially supported by U.S. National Science Foundation (NSF) grant DMS 1564362 “FRG: Geometric Structures on Higher Teichmuller¨ Spaces”, and acknowledges support from NSF grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Represen- tation Varieties” (the GEAR Network). This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC starting grant DiGGeS, grant agreement No 715982). iii PREFACE This thesis is about certain relatively hyperbolic discrete subgroups of semisimple Lie groups. If those are not words that inspire familiarity, Chapter 0 contains some background and context which will hopefully be helpful. That chapter is written with the curious reader who may or may not be a mathematician, but is at least not too daunted by things like matrices and vector spaces, in mind. For the mathematical reader who has some familiarity with at least some of those words, Chapter 1 provides a more traditional introduction, including, towards its end, an outline of the contents of the rest of the thesis and how they are organized. To our world today and its pressing challenges, this thesis, in the best and worst tradition of Hardy’s apology, contributes just about nothing, beyond a fleeting glimpse of a beautiful edifice of abstract thought, and the real but ephemeral possibility of a more thorough understanding of the structure of space, broadly construed. Those are not entirely nothing, nevertheless, and I hope Chapter 0 will help bring that message to an at least marginally larger audience. iv TABLE OF CONTENTS Dedication ........................................... ii Acknowledgments ....................................... iii Preface ............................................. iv List of Figures ......................................... vii List of Appendices ....................................... viii Abstract ............................................. ix Chapter 1 Background for a Generalish Reader ........................... 1 1.1 Fundamental groups of manifolds.........................1 1.2 Discrete subgroups of Lie groups..........................1 1.3 Geometric structures and holonomy representations................2 1.4 The joys of negative curvature...........................4 1.5 Teichmuller¨ theory, classical and higher......................5 1.6 Geometric group theory...............................7 1.7 Aside: applications.................................8 1.7.1 Negatively-curved space-time and limiting cases.............8 1.7.2 Poincare´ and mixed-curvature embeddings................9 1.7.3 Quantiative comparisons between geometric objects...........9 2 Introduction ......................................... 10 3 Preliminaries ........................................ 13 3.0 Hyperbolic spaces and groups........................... 13 3.1 Relatively hyperbolic groups............................ 16 3.1.1 A Bowditch–Yaman criterion for relative hyperbolicity.......... 22 3.1.2 Geodesics in the cusped space....................... 23 3.1.3 Reparametrizing projected geodesics................... 27 3.2 Singular value decompositions........................... 30 4 Dominated Representations, d’apres` Bochi–Potrie–Sambarino ............. 32 5 Relatively Dominated Representations .......................... 34 v 5.1 Dual representations................................. 36 5.2 Discreteness, faithfulness, proximal elements................... 36 5.3 Relative quasi-isometric embedding........................ 37 5.4 Upper domination and almost-unipotence..................... 39 6 Relative Domination Implies Relative Hyperbolicity ................... 42 6.1 Existence and transversality of limits........................ 42 6.2 Relative domination implies relative hyperbolicity................. 50 6.2.1 The limit set................................ 51 6.2.2 Dynamics on the limit set......................... 52 6.2.3 Perfectness................................. 55 6.2.4 Geometrically-finite convergence group action.............. 55 6.2.5 Peripherals are maximal parabolics.................... 57 6.2.6 Summary of argument........................... 58 7 Limit Maps ......................................... 59 8 Examples ........................................... 62 8.1 In rank one..................................... 62 8.2 A higher rank example............................... 63 9 Relation to Kapovich–Leeb ................................. 66 9.1 Relatively dominated implies relatively RCA................... 66 9.2 Uniform regularity and distortion, and equivalence of notions........... 67 10 Extending the Definition .................................. 70 Appendices ........................................... 72 Bibliography .......................................... 84 vi LIST OF FIGURES FIGURE p 1.1 Failure of the Morse lemma in the Euclidean plane: the broken path acb is a ( 2; 0)- 1 quasigeodesic independent of the length L of the geodesic ab, but c is 2 L far from ab..............................................4 1.2 Classification of surfaces: surfaces of genus 0, 1, 2 and 3, or in the notation above Σ0, Σ1, Σ2, and Σ3. Of these, the last two admit hyperbolic metrics. TikZ finesse on last 3 diagrams courtesy of Salman Siddiqi...........................6 1.3 The grey curves form a pants decomposition, here assigned lengths 1, 2 and 3; the red curves are drawn to indicate twist parameters growing across the pants curves. This data specifies a hyperbolic metric on the Σ2 shown here.................6 1.4 A cusped hyperbolic surface...............................8 3.1 A triangle is contained in an ideal triangle; an ideal triangle with illustrative vertical and horizontal paths (green and red, resp.)........................ 13 3.2 Schematic illustration of geometrically-finite quasi-Fuchsian with a accidental parabolic 21 3.3 Solid lines here indicated geodesics in X, dotted lines indicate projected geodesics.. 26 3.4 The red path γ^ in the cusped space may be considered as a relative path (Γ^; H^ ) as described above. The peripheral excursions η^1 and η^2 are the parts of γ^ lying inside the horoballs (colored grey); their projections η1 and η2 are the corresponding fuchsia paths with the same endpoints. Depths of some of the vertices along η1 are labelled as an illustrative example................................... 28 vii LIST OF APPENDICES A Linear Algebraic Lemmas ................................. 72 B A Local Version of Quas–Thieullen–Zarrabi ....................... 74 viii ABSTRACT Convex cocompact subgroups of rank-one semisimple Lie groups such as PSL(2,R) form a structurally stable class of quasi-isometrically embedded discrete subgroups which are naturally associated to negatively-curved geometric structures. Anosov representations give a higher-rank analogue of convex cocompactness which shares many of its good geometric and dynamical properties, and have become important objects of study in higher Teichmueller theory. In this thesis we introduce relatively dominated representations as a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness—a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We prove that groups admitting relatively dominated representations must be relatively hyper- bolic, that these representations induce limit maps with good properties, provide examples, and draw connections to work of Kapovich–Leeb which also introduces higher-rank analogues of geometric finiteness. ix CHAPTER 1 Background for a Generalish Reader For most of human experience, “geometry” has meant, and for many continues to mean, Euclidean geometry in two or three dimensions—the geometry that we see on a piece of paper, or that we instinctively and viscerally move through. The historical roots of this thesis lie in developments from the 19th century which significantly broadened the scope of geometry and enriched humanity’s study of spatial structure. 1.1 Fundamental groups of manifolds In the late 19th century, the branch of topology was born, with the appearance of Poincare’s´ Analysis Situs being a particular historical landmark. Topology concerns geometric properties, such as connectedness, compactness,
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