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Properties of Groups Acting on Twin-Trees and Chabauty Space

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Properties of Groups Acting on Twin-Trees and Chabauty Space PROPERTIES OF GROUPS ACTING ON TWIN-TREES AND CHABAUTY SPACE Robert Kelvey 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 December 2016 Committee: Rieuwert Blok, Advisor Lee Nickoson, Graduate Faculty Representative Mihai Staic Xiangdong Xie ii ABSTRACT Rieuwert Blok, Advisor In this dissertation, we study groups that act on twin trees. A twin tree consists of a pair of (infinite) simplicial trees (T +;T −) that are “twinned” by means of a co-distance function 훿*, which assigns a non-negative integer to pairs of vertices from each tree. If n = 훿*(x+; y−) for vertices x+ in T + and y− in T −, then we think of x+ and y− as having distance 1 − n. An example of a twin tree is T = (T +; T −; 훿*), where T + and T − are the associated Bruhat-Tits trees arising from two different discrete valuations on the field k(t). A group G acts on a twin tree T = (T +;T −; 훿*) if it acts on each tree T ± and preserves the co-distance function. For the twin tree T arising from discrete valuations on k(t), the group −1 GL2(k[t; t ]) naturally acts on the twinning. The subgroup GL2(k[t]) stabilizes a vertex of the the + − tree T . The action of GL2(k[t]) on T yields a fundamental domain an infinite ray, and from this action one obtains Nagao’s Theorem: ∼ GL2(k[t]) = GL2(k) *B(k) B(k[t]); where note B(k) and B(k[t]) are subgroups of upper triangular matrices in GL2(k) and GL2(k[t]), −1 respectively. In this work, we investigate the fundamental domains for subgroups G < GL2(k[t; t ]) that stabilize subtrees of the tree T +. For a general group G acting on a twin-tree, we consider its space of closed subgroups C(G), called the Chabauty space. By constructing a left-invariant metric on the underlying automorphism group of the twin-tree, one can endow C(G) with a metric as well. Using this, we study the distance between vertex stabilizer subgroups in G. This will hopeful lead to future work generalizing the −1 special case of T and GL2(k[t; t ]). iii ACKNOWLEDGMENTS There are many, many people for whom, without their direct impact on my life, I would not be where I am today. Certainly I would not be writing an acknowledgements section to a dissertation! So I am going to take this opportunity to get personal. I’m going to start somewhat near the beginning, so my apologies but - this might take while. Firstly, I would like to thank Mr. Klima, my AP Calculus teacher in High School. My current opinion on AP Calculus withstanding, I truly would not have even thought to continue with math- ematics in College were it not for Mr. Klima. At the time, I assumed I would major in computer science. I knew that would require some more mathematics, but had no idea of how much or really of what kind (all I knew was that I was scared of “proving things”). Mr Klima pointed out that, most likely one would be close to a minor in math by the end of a computer science degree, so why not just go ahead and do a double major? Until that moment, such a thought had never entered my mind. But as soon as I was made aware of such a possibility, the thought stayed with me. That is what I wanted to do. So thank you Mr. Klima, for helping to get me started. Secondly, I would like to thank all of my teachers at Carroll Community College. I know now that all of them were teachers because they love teaching. They love the students and the subject. Perhaps it is from all of you that I was instilled with my first desires to teach. In particular, I would like to thank Rob Brown (a fellow McDaniel alum!) for giving me the chance to take a math class my first semester, despite there being no actual class to take. And of course, Dr. Raza! Although you may, on occasion, say some outlandish things, your enthusiasm for Chemistry and obvious care for students had a great impact on me. I must also thank all those with whom I worked with at the Academic Center at Carroll Community College, especially for giving me the opportunity to tutor students, even after I left and was a student at McDaniel College. The tutoring experiences I gained at Carroll gave me the foundation I needed to become a teacher. Thank you all! None of what follows would be at all possible were it not for the amazing Mathematics and Computer Science department at McDaniel College. You challenged and pushed me academically, further than I thought I could go. I certainly would have never thought to take Topology before iv having Abstract Algebra or Analysis without your motivations and encouragement; nor would I have taken Cryptography that first Spring semester. It was certainly difficult (at times, very difficult). So thank you Sara, Spencer, Italo, Pavel, and (may you rest in peace) Harry. You were the giants that helped me first touch the sky. So much has happened in the last six years at Bowling Green State University, and there are many people to thank. I will be forever grateful that I got to spend my first year as a graduate assistant in the Learning Commons, since who knows if I would even be writing this without that first opportunity. Almost every faculty and staff member at the BGSU Math and Stats department deserves thanks for assisting me, even if it was in some minute way on a random day. So if I don’t mention you by name below, my apologies! But thank you all. I would like to thank Dr. Neal Carothers for letting me share in his last year of teaching at BGSU. You taught me to stop worrying and enjoy Analysis. And thank you to Dr. Steven Seubert for powering me through my comprehensive exams, and for teaching me to be confident and know when my words are true. And thank you to Dr. So-Hsiang Chou for teaching me mystical things, along with some applied math (which really isn’t so bad). I will forever emulate part of your teaching style as well! Thank you to Dr. Chan, Dr. Meel, and Dr. Chen for your service as department chair during my time at BGSU. And thank you to Dr. John Chen, Dr. Sun, Dr. Blok, and Dr. Zirbel for your time and efforts as Graduate Coordinators. Your work, effort and advice has certainly helped me considerably over the years. And I absolutely would not be writing this without the financial support of the department. Special thanks are certainly in order for the office staff: Mary, Marcia, and Barb, thank you all so much for the work that you do/did (congratulations on retirement!). I certainly must thank all of the Algebraists, as each of you has helped me in your own way. Thank you Dr. Xiangdong Xie for all your teaching, advising, and invitations. Dr. Mihai Staic deserves an MVP award for assisting me my last year of school. Your assistance with my job application process cannot be understated in its helpfulness. Thank you!! And of course, thank you Dr. Rieuwert Blok. Even across the pond, you found time to aid me through my dissertation, v job search, and life itself. Your humor and impressive ability to write quickly on the board will forever stay with me. Thank you for being my advisor and helping me get this document to its final destination. I have had many classmates and made many friends over the years. Even if I haven’t spoken to you in a long time, I would like to thank you for being my friend, even if just for a short time. My fellow McDaniel crew: John, Ben, Stephen, Feng, Shannon, Wesley, and David (both of you) - thank you; I’ll hopefully see you at a homecoming someday in the future! My first friends in Bowling Green were those of the Men’s and Women’s BGSU Cross Country team. All my teammates, I am proud to have run an incredible season with all of you. Thank you Ben for giving me a call and making me feel so welcome. Maybe I’ll scramble a mountain with you or we can run an ultra someday. Thank you Megan and Tara for M3 at El Zarape! Jesse and Aaron Smuda, you are two cool brothers. And of course, John Bernard: you are amazing and a wonderful friend. I am glad to have met you all! Thank you to all my fellow Math and Stat graduate students. We have laughed, cried, and drank at Stones Throw together many a good time. You all helped me keep my sanity. Special thanks to Jen, Rachelle, Chas, and Leo for helping me survive my first two years. Thank you to all past and future 470A-ers, especially Leo, Gokul, Chas, John, Nathan, Mark, and Jake. Having classmates, officemates and friends like you has been a blessing. Thank you! And best of luck to those of you still making your way through. I’ll see you at future JMMs! And a super special heartfelt thank you to Colleen. I am so very happy you walked over that day from the ASOR picnic to say hi to the Math nerds.
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