View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Lancaster E-Prints The rigidity of countable frameworks in normed spaces Sean Dewar Department of Mathematics and Statistics Lancaster University This dissertation is submitted for the degree of Doctor of Philosophy October 2019 To my family, friends, and all who helped me get to this point. Declaration Hereby I declare that the present thesis was prepared by me and none of its contents was obtained by means that are against the law. I also declare that the present thesis is a part of a PhD Programme at Lancaster University. The thesis has never before been a subject of any procedure of obtaining an academic degree. Sean Dewar October 2019 Acknowledgements I’ve got a lot of awesome people to thank, so buckle up. I’d first like to thank my supervisor Derek Kitson for introducing me to the wonderful world of rigidity. You taught me to calmly take a step back and look at a problem from all the angles, and not just rush in head first like I am prone to. Your advice and teachings have been priceless. I’d like to also thank Tony Nixon, Bernd Schulze, Stephen Power, John Hewetson, Lefty Kastis, Viktória Kaszanitzky and Hattie Serocold of the Lancaster geometric rigidity research group. I always felt included, and you really can’t ask for more than that. The Lancaster Mathematics department as a whole has been brilliant, and to list you all would add another hundred pages to an already rambling thesis, but I would like to especially thank Yemon Choi, David Pauksztello and Julia Tawn for always being up for a chat. I’d like to thank my fellow PhD students at Lancaster for all the help they have given me over the years. The opportunity to speak at the Post Graduate Forum has been a great boon to my public speaking, though I would like to apologise for anyone who witnessed a couple of the dud talks I have given over the years. I would like to especially thank Konrad Królicki for being the other half of our double act, and Chris Menez for being my mathematical senpai through out all the seven years we attended Lancaster University together. viii Big shout out to all my friends from school and college who I still bother to this day; Roberts, Dan, TC, Tosh, Holt, Scott and Chris. You are some of the weirdest people I have ever met and I wouldn’t have it any other way. Without sounding like I am bragging (even though I most certainly am), I have had too many good friends over all my years at Lancaster University to name you all. You have all helped me grow from the shy introverted fresher to what I am today (so I guess you can all collectively take the blame for that). I would like to give special thanks to all the good friends I met through LURACS, keep up the good pubbing! At the risk of getting too emotional, I would like to give a special acknowledgement to Kirstie Ferriday. For the last few years you have always been there to calm me when there was some hiccup along the road, and I cherish you dearly for it. I feel like a better person when I am with you, and I hope you feel the same. Lastly, I’d like to thank my family; Mum, Dad, my brother Ben, Nan, Fred, Grandma, Grandad (I wish you could have been here for this), and all the rest of you. You have all been a solid rock for me, even if what I do sounds like a load of nonsense to all of you. I hope to one day repay all the help I have received off you, though I know that what you have given me is priceless. Abstract We present a rigorous study of framework rigidity in finite dimensional normed spaces using a wide array of tools to attack these problems, including differential and discrete geometry, matroid theory, convex analysis and graph theory. We shall first focus on giving a good grounding of the area of rigidity theory from a more general view point to allow us to deal with a variety of normed spaces. By observing orbits of placements from the perspective of Lie group actions on smooth manifolds, we obtain upper bounds for the dimension of the space of trivial motions for a framework. Utilising aspects of differential geometry, we prove an extension of Asimow and Roth’s 1978/9 result establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a d-dimensional Euclidean space. Further, we establish the independence of all graphs with d + 1 vertices d-dimensional normed space, and also prove they will be flexible if the normed space is non-Euclidean. Next, we prove that a graph has an infinitesimally rigid placement in a non- Euclidean normed plane if and only if it contains a (2, 2)-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph K4 by considering smoothness and strict convexity properties of the unit ball. Finally, we carry our previous results to countably infinite frameworks where this is possible, and otherwise identify when such results cannot be brought forward. We first establish matroidal methods for identifying rigidity and flexibility, andapply these methods to a large class of normed spaces. We characterise a necessary and x sufficient condition for countably infinite graphs to have sequentially infinitesimally rigid placements in a general normed plane, and further stengthen the result for a large class normed planes. Finally, we prove that infinitesimal rigidity for countably infinite generic frameworks implies a weaker (but possibly equivalent) form of continuous rigidity, and infinitesimal rigidity for countably infinite algebraically generic frameworks implies continuous rigidity. Table of contents 1 Introduction to geometric rigidity theory in normed spaces 11 1.1 Normed space geometry . 11 1.1.1 Euclidean and non-Euclidean normed spaces . 12 1.1.2 Differentiation in normed space . 15 1.1.3 Support functionals . 17 1.1.4 Lie groups and Lie group actions . 31 1.1.5 The group of isometries of a normed space . 32 1.2 Frameworks and placements . 39 1.2.1 Notation and graph sparsity . 39 1.2.2 Definitions for frameworks and placements . 40 1.2.3 The rigidity map and the rigidity matrix . 41 1.2.4 Orbits of placements . 45 1.2.5 Full placements and isometrically full placements . 50 1.3 Rigidity and independence . 55 1.3.1 Local, continuous and infinitesimal rigidity . 55 1.3.2 Regular placements and independence for finite graphs . 57 1.3.3 Pseudo-rigidity matrices . 61 1.3.4 Necessary conditions for rigidity of frameworks and graphs . 62 1.3.5 Rigidity in the Euclidean spaces . 66 xii Table of contents 2 Framework rigidity in general normed spaces 71 2.1 Equivalence of local, continuous and infinitesimal rigidity . 72 2.1.1 Properties of constant and regular placements . 72 2.1.2 Equivalence of types of rigidity for constant frameworks . 73 2.2 Small frameworks and non-spanning placements . 78 2.2.1 Orbits of non-spanning placements . 78 2.2.2 Infinitesimal flexibility and independence of small frameworks and non-full frameworks . 81 2.3 Composition and substitution of rigid graphs and frameworks . 88 2.3.1 Composition of rigid frameworks . 88 2.3.2 Composition and substitution of rigid graphs in normed spaces with a finite number of linear isometries . 90 3 Graph rigidity in the normed plane 95 3.1 Frameworks in normed planes . 96 3.1.1 Isometries of a normed plane and full placements . 96 3.1.2 Necessary conditions for graph rigidity in the normed plane . 97 3.2 Rigidity of K4 in all normed planes . 99 3.2.1 The rigidity of K4 in not strictly convex normed planes . 99 3.2.2 The rigidity of K4 in strictly convex but not smooth normed planes106 3.2.3 The rigidity of K4 in strictly convex and smooth normed planes 113 3.3 Graph operations for the normed plane . 123 3.3.1 0-extensions . 123 3.3.2 1-extensions . 126 3.3.3 Vertex splitting . 128 3.3.4 Vertex-to-K4 extensions . 130 3.4 Graph sparsity and connectivity conditions for rigidity . 131 Table of contents xiii 3.4.1 A characterisation of rigid graphs in normed planes . 131 3.4.2 Analogues of Lovász & Yemini’s theorem for non-Euclidean normed planes . 132 4 Rigidity for countable frameworks 137 4.1 Preliminaries on countably infinite frameworks . 137 4.1.1 Well-positioned and completely well-positioned frameworks . 137 4.1.2 Towers of frameworks . 140 4.1.3 Independence for countably infinite frameworks . 143 4.1.4 The closure operator . 145 4.2 Countably infinite frameworks in generic spaces . 152 4.2.1 Generic placements, spaces and properties . 152 4.2.2 Generic rigidity for infinite graphs . 159 4.3 Combinatorial rigidity of countable graphs . 162 4.3.1 Rigidity and independence in normed planes . 162 4.3.2 Rigidity and independence in generic normed planes . 170 4.4 Continuous rigidity for countable frameworks . 173 4.4.1 Continuous rigidity for countable generic frameworks in generic spaces . 173 4.4.2 Continuous rigidity for countable algebraically generic frame- works in Euclidean spaces . 177 5 Further research and open problems 181 5.1 Expanding Theorem 2.1.5 to a larger class of frameworks .