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Separation and Poincaré Profiles Corentin Le Coz Separation and Poincaré profiles Corentin Le Coz To cite this version: Corentin Le Coz. Separation and Poincaré profiles. Group Theory [math.GR]. Université Paris-Saclay, 2020. English. NNT : 2020UPASM014. tel-03192929 HAL Id: tel-03192929 https://tel.archives-ouvertes.fr/tel-03192929 Submitted on 8 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Prols de séparation et de Poincaré Separation and Poincaré proles Thèse de doctorat de l'Université Paris-Saclay Ecole Doctorale de Mathématiques Hadamard (EDMH, ED 574) Spécialité de doctorat : Mathématiques fondamentales Unité de recherche : Université Paris-Saclay, CNRS, Laboratoire de mathématiques d'Orsay, 91405, Orsay, France. Référent : Faculté des sciences d'Orsay Thèse présentée et soutenue le 23 septembre 2020, à l'Institut de Mathématique d'Orsay, par Corentin LE COZ Composition du jury : Pierre PANSU Président Professeur, Université Paris-Saclay Gábor PETE Rapporteur et examinateur Chercheur sénior, Institut Alfréd Rényi Laurent SALOFF-COSTE Rapporteur et examinateur Professeur, Université Cornell Goulnara ARZHANTSEVA Examinatrice Professeur, Université de Vienne Romain TESSERA Directeur Directeur de recherche, CNRS Jérémie BRIEUSSEL Codirecteur Maître de conférences, Université de Monpellier NNT: 2020UPASM014 Thèse de doctorat de Thèse Ce travail a été réalisé au Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay. Il a été soutenu par l'Agence Nationale de la Recherche dans le cadre des projets ANR-14-CE25-0004 GAMME et ANR-16-CE40-0022-01 AGIRA. Remerciements Je tiens `aremercier mes directeurs de th`esepour leur accompagnement durant ces trois ann´ees de recherche. Depuis mon stage de Master, Romain Tessera a toujours su me communiquer son entousiasthme d´ebordant. Sa vivacit´ed'esprit et son go^utde transmettre m'ont permis de travailler d'une mani`ereagr´eable.Je lui suis tr`esreconnaissant d'avoir cru en moi, malgr´eles difficult´esque j'ai pu manifester au d´epart,et de m'avoir incit´e`apouruivre ma voie dans la recherche en math´ematiques. J'ai eu la chance d'avoir J´er´emieBrieussel comme codirecteur de th`ese.Il a su me partager son go^utdu travail bien fait, en m'incitant `ad´evelopper au maximum mes id´ees. Je veux le remercier d'avoir favoris´eune atmosph`erede confiance, qui m'a aid´e`atravailler sereinement. Je remercie vivement les membres du jury de ma soutenance de th`ese: Pierre Pansu, Goulnara Arzhantseva, et plus particuli`erement G´abor Pete et Laurent Saloff-Coste qui ont accept´ed'^etrerapporteurs et m'ont fourni des rapports tr`esd´etaill´es. A Orsay, j'ai b´en´efici´ed'un environnement d'exception. J'ai bien travaill´eavec Gabriel et Suraj, puis avec Timoth´ee,Pierre-Louis et Irving. A la piscine ou ailleurs, j'ai pu me changer les id´eesavec Guillaume et Camille. Il y a aussi eu les parties de tennis avec Salim, et tout simplement les d´ejeunersconviviaux entre doctorants. Je remercie M´elaniepour la confiance qu'elle m'a accord´eedurant notre collaboration en Master MEEF, et de m'avoir donn´ego^ut`ala vulgarisation des math´ematiques. J'ai aussi une pens´eepour Mme Santoni, qui m'a donn´ego^utaux math´ematiques en classe pr´eparatoire.J'ai compris avec elle que la beaut´edes math´ematiquesse r´ev`eleau sein m^eme de leur rigueur. Cela a ´et´epour moi un tournant majeur, dans ma vie ´etudiante ainsi que dans ma vie personnelle. Je souhaite remercier mes beaux-parents de nous avoir accueilli pendant le confinement du printemps dernier. Leur aide pr´ecieusem'a permis de r´edigerce manuscrit sereinement. Je remercie enfin mes parents pour tout ce qu'ils m'ont apport´edepuis mon enfance. Avec le recul des ann´eeset mon r^olede jeune Papa, je me rends compte de la g´en´erosit´eet de la patience dont j'ai b´en´efici´ejusqu’`apr´esent, et b´en´eficieencore aujourd'hui. Cela fait presque trois ans que je partage ma vie avec Bertille. Trois ann´eesde complicit´e et de joie profonde. J'admire que tu me suives dans mes ´elansmath´ematiques,sans rechigner lorsque notre d^ınerse transforme en un expos´einterminable. Enfin, il y a Cl´ement et Marie qui me donnent du bonheur chaque jour. \Il est plus beau d'´eclairer que de briller seulement" Saint Thomas d'Aquin R´esum´e(en fran¸cais) Ce manuscrit de th`eser´ecapitulemes travaux de recherche sur le profil de s´eparationet les profils de Poincar´e. Le profil de s´eparationest apparu en 2012 dans un article fondateur de Benjamini, Schramm et Tim´ar. La d´efinitiondonn´ee tirait ses origines dans des travaux ant´erieurs,dans le domaine du calcul formel : principalement des ´etudes de Lipton et Tra- jan concernant les graphes planaires, et de Miller, Teng, Thurston et Vavasis concernant des graphes d'intersection. Le profil de s´eparationutile en th´eorieg´eom´etriquedes groupes, mon domaine de recherche, `acause de sa propri´et´ede monotonie par plongements grossiers. Il a ´et´e g´en´eralis´epar Hume, Mackay et Tessera en 2019 en une gamme continue de profils, appel´es profils de Poincar´e. La premi`erepartie de cette th`eseconstitue l'´etatde l'art concernant ces profils. J'y ai d´etaill´eles r´esultatsles plus marquants du point de vue de la th´eorieg´eom´etrique des groupes. J'ai souvent propos´edes esquisses de preuves, me passant de certains d´etailstechniques. Je renvoie aux articles originaux pour les d´emontrations compl`etes. A l'oppos´e,j'ai plusieurs fois donn´edes preuves de propri´et´esbien connues des sp´ecialistes,qui ne sont g´en´eralement pas d´emontr´eesdans les articles de recherche. J'esp`ereque cela pourra rendre service aux d´ebutants. Les deux parties suivantes sont des adaptations de pr´e-publications,fruits de mes travaux des trois ann´eespass´ees. Le contenu de la deuxi`emepartie a ´et´e´ecriten collaboration avec Gournay. L'objectif y est de comparer le profil de s´eparationavec d'autres quantit´es mieux connues : le profil isop´erim´etrique,la compression des plongements dans les espaces de Hilbert, et la croissance du volume des boules. La troisi`emepartie correspond `aune pr´e-publicationqui porte sur la prescription de grands profils de Poincar´e.Ceci est r´ealis´een utilisant une construction d´evelopp´eepar Brieussel et Zheng en 2015. La compression des plongements dans les espacs Lp, et la th´eoriespectrale des graphes en degr´enon born´esont les principaux ingr´edients de ce projet. iv Abstract (english language) The goal of this thesis report is to present my research concerning separation profile and Poincar´eprofiles. Separation profiles first appeared in 2012 in a seminal article written by Benjamini, Schramm and Tim´ar. This definition was based on preceding research, in the field of computer science, mainly work of Lipton and Trajan concerning planar graphs, and of Miller, Teng, Thurston and Vavasis concerning overlap graphs. Separation profiles play now a role in geometric group theory, where my personal interests lies, because of their property of monotonicity under coarse embeddings. These were generalized by Hume, Mackay and Tessera in 2019 into a spectrum of profiles, called Poincar´eprofiles. The first part of this thesis is a summary what we know on these profiles. I have given the most important results, in the point of view of geometric group theory. I often give sketches of proofs, and refer to references for complete proofs. On the other hand, we have several times detailed some basic facts that are likely to be easy exercises for specialists and are not usually proved in research articles. We hope that this will help novices. The two next parts are adapted from two preprints, fruit of my research of the past three years. The content of the second part was written in collaboration with Gournay. Its purpose is to compare the separation profile with other well-studied quantities: isoperimetric profile, compression of embeddings in Hilbert spaces, volume growth. The third part corresponds to a preprint concerning the prescription of high Poincar´epro- files. This is done using a construction developed by Brieussel and Zheng in 2015. Compression of embeddings into Lp spaces and spectral graph theory in unbounded degrees context are the two main ingredients of this project. v Contents R´esum´e(en fran¸cais) iv Abstract (English language) v Notations x Introduction (en fran¸cais) xii Introduction (English language) xviii I Separation profile and Poincar´eprofiles: a survey 1 1 Prehistory of the separation profile 3 1.1 The planar separator theorem . .3 1.2 The geometric separator theorem . .5 2 Geometric obstructions to a coarse embedding 7 2.1 Coarse embeddings . .7 2.1.1 Definition . .7 2.1.2 Coarse embeddings into a Hilbert space . .8 2.1.3 Quasi-isometric embeddings . .9 2.2 Monotone invariants . .9 2.2.1 Volume growth . .9 2.2.2 Asymptotic dimension . 11 3 Separation profile and Lp-Poincar´eprofiles 13 3.1 Definitions . 13 3.1.1 Separation profile . 13 3.1.2 Poincar´eprofiles . 14 3.1.3 Comparing profiles . 19 3.1.4 Regular and coarse regular maps .
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