g1 Universal Groups for Right-angled Buildings 1 g ANA C. SILVA Promotor: Prof. Dr. Tom De Medts Vakgroep Wiskunde Faculteit Wetenschappen Universiteit Gent Academiejaar 2016-2017 Proefschrift voorgelegd tot het behalen van de graad van Doctor in de wetenschappen: wiskunde. A F´elix Costa, que onde quer que esteja, continue orgulhoso de mim. Acknowledgement Four years of research are never possible to do alone. Therefore I cannot consider my dissertation to be complete without expressing my enormous gratitude to all the people that, in their own way, helped me and supported me during my PhD. To Tom, my supervisor, my many thanks for choosing me to be part of the department in Gent and for many discussions full of cre- ativity. To Koen Struyve, whom I had the immense pleasure to work with, an enormous thank you for always being full of patience to support me and to check my craziest ideas. I am endlessly thankful to Anne Thomas. The research visit that I did in Zurich to work with her confirmed the kind, inspiring and imaginative mathematician that she is. I am so pleased to have been able to work with her on a daily basis for some time. This enrichment experience and the support that I got from her in the rest of my PhD contributed greatly to my mathematical and personal life. I also want to express my deepest thanks to Pierre-Emmanuel Caprace, who always answered so kindly and with a smile to all my questions, no matter how deep (or obvious) they were. Our conver- sations gave rise to many results presented in this thesis. My many thanks to Hendrik Van Maldeghem for so many enthusi- astic conversations, for always being concerned about my future and for making me feel an active member of the department. I am thankful to my master supervisor, Csaba Schneider, who presented me the possibility of a PhD in Belgium and who pushed me to pursuit my mathematical interests on research. I also want to thank my officemates during the last 4 years. Thank you Jeroen Demeyer, my officemate in S25, for being so kind to answer to all my existential questions when I moved to Belgium and for introducing me to the routine of the department. Thank you Karsten, my officemate in the \nuclear office". It was such a pleasure to engage with you once in a while in pointless dis- cussions to distract us from the mathematical stuckness and to be your partner in runs around Flanders. I will never forget that you handed in your thesis but stayed around in the office to support my last push in the writing. To Manuel, my afilhado, words are not enough to express my gratitude. The colleague who never became an officemate but who became my best friend instead. Thank you for the huge amount of moments that we shared, for being my partner in so many adventures and for being a company that I will never prescind of. To Myriam, the Spanish woman of my life, who shares with me the \I am from the Capital" feeling. Thank you for all the fun, personal guidance, sharing and love during these last years. I am so proud to have you in my close circle of friends. To Malika and Torsten, my favorite Belgians in any country's soil, I am so grateful for all the fun, the experiences and the caring that I got from you. Thank you for all the support, even in almost impossible moments. To all the other friends I met in Gent and who made these 4 years the best time of my life, a huge hug of thanks: Cec´ılia, the Portuguese that brings me back to reality quite often and who supports me in every moment; Els, the best example of mixing perseverance and fun; Anneleen, the strength example that I aim to follow; John, the partner for any party; Lu´ıs, for many adventures in Gent (and now in Lisbon); Anurag, for all the fun discussing math and a lot of other things; Paul, the friend who is always present; Ruy, for all the sharing ideas about everything; Marian, my favorite Dutch teacher; Gon¸calo, the example of always pursuing your dreams; Alice and Matteo, for all the cheerful moments... and to all the others who had an influence on my growth and stability. To the friends that I met in Zurich and who made my stay full of mathematics, fun and sharing: Thank you Claire, Carlos and Nir! I want to thank all my friends from Lisbon who were always there to listen to my experiences and to give me a hug when I needed it. 2000km (or more) apart turned out not to change our connection: Thanks to C´atia, Beatriz, Joana, Ana Marta, Tatah and Bruno for all the guidance words, joy and deep moments. To Mafalda, my most frequent visitor in Gent, who I am infinitely proud of being friends with. To Joana, Filipe, Raquel, Marta and Mariana, for all the good moments and for such a good bound that comes from early university times. To Ricardo, for being my main motivation source to move to Belgium and for being an essential support during the first year of my PhD. To Maria Jo~ao,without a doubt still the best match to work together, my many thanks for all the support in the last almost ten years. It is so nice to see how we started as student-professor in my bachelor and we developed such a good friendship along the years. I am also thankful to the people that I associate to the Univer- sity on Lisbon and who always cared about how I was doing outside Portugal. Thank you prof. Gracinda, S~ao,Atle, Teresa (MF), Eliana and Filipe. To Evelyn, thank you, a bit for everything. For such a beautiful story and for proving me that it is indeed possible to look back in time and think: \ I did not have an O.K. life. I had a GREAT life!". E porque os ´ultimos s~aosempre os primeiros: Obrigada V´o,por todos estes anos a acreditares em mim e por todas as tuas s´abias palavras nos momentos em que mais precisei. Estou eternamente orgulhosa de ser tua neta e de poder partilhar contigo todas as minhas conquistas. Obrigada M~ae, por seres o meu melhor exemplo de for¸ca,dedica¸c~ao e amor incondicional. Obrigada por todo o apoio, orgulho e estima que sempre me fizeste sentir. O lugar que voc^esas duas t^em no meu cora¸c~aoe a gratid~aoque sinto pelas duas ´edemasiado grande para expressar em palavras. Ana 2013{2017: Four years in a right-angled mood After four years of research in a very specific topic, it is a very inter- esting exercise to come back to the beginning of the story, motivate the research done and highlight the most beautiful and productive moments of my PhD. The solution of that exercise is provided in this introductory chapter that, by lack of a better name, can also be called preface. The initial project of my PhD, and the reason why I left the beautiful weather of Lisbon in 2013 to move to Gent, was to study groups acting on locally finite trees from a combinatorial, geometri- cal, topological and algebraic point of view. So many points of view! The possibility of connecting so many areas of mathematics to study structures that at first glance seemed simple really attracted my at- tention and turned the trip from Lisbon to Gent way shorter than it actually is (and it is already really short). Reading the title of this thesis, one can think that I engaged in another research project during my PhD. That is really not the case. As soon as I arrived to the department of Mathematics in Gent, buildings were everywhere. And I do not have in mind the \beauti- ful" location of the department. I am considering the \mathematical structures" called buildings, that were made by gluing together apart- ments which, in turn, were constructed with chambers, walls, etc. I had never heard of anything like that before and in Gent everyone { v { 2013-2017: FOUR YEARS IN A RIGHT-ANGLED MOOD seemed to be more or less familiar with the notion and to have a few examples of those \buildings" ready at hand. If my curiosity was already piqued regarding these geometric struc- tures, my mind was blown when I heard the first time that \a tree is a building". What a beautiful and curious world Mathematics is! So this sentence, that I often use in the introduction of my talks as a catch up phrase, justifies that in fact my PhD involves groups acting on locally finite trees and also on a more general class of structures of which trees are examples. Framing the research Totally disconnected locally compact groups I started my PhD by studying a topology that one considers in the automorphism group of a locally finite tree, called the permutation topology. Endowed with that topology, the automorphism group of a locally finite tree (or of a locally finite graph in general) is a totally disconnected locally compact (t.d.l.c.) group. The study of locally compact groups G can naturally be split into the connected and totally disconnected cases. This is due to the fact that the connected component of the identity, G0, is a closed normal subgroup of G and G=G0 is a totally disconnected group. The connected case has found a satisfactory answer with the so- lution of Hilbert's fifth problem.