UC San Diego UC San Diego Electronic Theses and Dissertations Title Lattices of minimal covolume in real special linear groups Permalink https://escholarship.org/uc/item/3b65c9ww Author Thilmany, François Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SAN DIEGO Lattices of minimal covolume in real special linear groups A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by François Thilmany Committee in charge: Professor Alireza Salehi Golsefidy, Chair Professor Tara Javidi Professor Shachar Lovett Professor Amir Mohammadi Professor Efim Zelmanov 2019 Copyright François Thilmany, 2019 All rights reserved. The dissertation of François Thilmany is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California San Diego 2019 iii DEDICATION À Marie-Louise et Maurice De Coninck iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Tables . vi Acknowledgements . vii Vita................................................ ix Abstract of the Dissertation . .x Chapter 1 Introduction . .1 Chapter 2 Background material . .5 2.1 Affine algebraic groups . .5 2.2 The structure of absolutely simple groups . 16 2.3 Galois cohomology . 24 2.4 Bruhat-Tits theory . 27 2.5 Lattices in locally compact groups . 32 2.6 Arithmetic groups . 36 Chapter 3 Prasad’s volume formula . 40 3.1 Tamagawa numbers . 40 3.2 The volume formula . 42 Chapter 4 Lattices of minimal covolume in SLn(R) .................... 46 4.1 A brief outline of the proof . 47 4.2 The setting . 49 4.3 An upper bound on the index . 51 4.3.1 The inner case . 53 4.3.2 The outer case . 54 4.4 The field k is Q ............................... 57 4.4.1 The inner case . 58 4.4.2 The outer case . 58 4.5 G is an inner form of SLn ......................... 66 4.6 The parahorics Pv are hyperspecial and G splits at all places . 68 4.7 Conclusion . 70 4.8 Bounds for sections 4.4 through 4.6 . 71 Bibliography . 80 v LIST OF TABLES Table 2.2.3. Dynkin diagrams of the irreducible reduced root systems . 17 Table 2.4.1. Irreducible Affine Dynkin diagrams . 28 Table 4.4.1. Summary of the various discriminant bounds used . 66 Table 4.8.5. Values of M ..................................... 74 Table 4.8.7. Values of M 0 .................................... 76 Table 4.8.8. Values of C ..................................... 76 Table 4.8.9. Values of min Dk .................................. 76 Table 4.8.10. Values of H .................................... 77 Table 4.8.11. Bounds for Dl ................................... 77 vi ACKNOWLEDGEMENTS First and foremost, I would like to wholeheartedly thank Alireza Salehi Golsefidy for his invaluable guidance throughout these past five years, for his matchless perspicacity, for his endless knowledge, and for all these exciting afternoons spent discussing mathematics and/or playing chess. Alireza has had a substantial influence on my mathematical education; the standards of rigor and excellence he works by are now those I aspire to. Neither this venture nor those to come would have been possible without his advising. I am very grateful to Amir Mohammadi and Efim Zelmanov for their advice and instruc- tion over the last few years, as well as for serving on my thesis committee. I am grateful to Tara Javidi and Shachar Lovett for the time and effort they spent serving on my thesis committee. While being a graduate student at UC San Diego, I had the chance to be surrounded by wonderful officemates, roommates, classmates, friends. In memory of all the good times, I would like to thank Jake, Peter, Zach, Marino, Stephan, Pieter, Jacqueline, Iacopo, Justin, Daniël, Zonglin, Brian, Michelle, Taylor, ... Je tiens à remercier Artémis, Cédric, Roxane, Mitia, Sylvie, Hoan-Phung, Charel, Lau- rent, Julie, Christine, Keno, Thibaut, Patrick, Blaža, Thierry, Nicolas et Pierre-Alain pour les doux étés et hivers passés en leur compagnie à Bruxelles et ailleurs. Votre amitié m’est très chère. Finalement, je voudrais remercier ma famille pour son appui inconditionnel, et en par- ticulier Artémis pour avoir parcouru le monde avec moi. The author is grateful to have been supported by the Luxembourg National Research Fund (FNR) under AFR grant 11275005. vii Chapter 4 contains material from Lattices of minimal covolume in SLn(R), Proc. Lond. Math. Soc., 118 (2019), pp. 78–102. The dissertation author was the primary investigator and author of this paper. viii VITA 2012 Bachelier en Sciences Mathématiques, Université Libre de Bruxelles 2014 Maîtrise en Sciences Mathématiques, Université Libre de Bruxelles 2017 Candidate in Philosophy in Mathematics, University of California San Diego 2019 Doctor of Philosophy in Mathematics, University of California San Diego PUBLICATIONS François Thilmany, Lattices of minimal covolume in SLn(R), Proc. Lond. Math. Soc., 118 (2019), pp. 78–102, doi: 10.1112/plms.12161. ix ABSTRACT OF THE DISSERTATION Lattices of minimal covolume in real special linear groups by François Thilmany Doctor of Philosophy in Mathematics University of California San Diego, 2019 Professor Alireza Salehi Golsefidy, Chair The objective of the dissertation is to determine the lattices of minimal covolume in SLn(R), for n ≥ 3. Relying on Margulis’ arithmeticity, Prasad’s volume formula, and work of Borel and Prasad, the problem will be translated in number theoretical terms. A careful analysis of the number theoretical bounds involved then leads to the identification of the lattices of minimal covolume. The answer turns out to be the simplest one: SLn(Z) is, up to automorphism, the unique lattice of minimal covolume in SLn(R). In particular, lattices of minimal covolume in SLn(R) are non-uniform when n ≥ 3, contrasting with Siegel’s result for SL2(R). This answers for SLn(R) the question of Lubotzky: is a lattice of minimal covolume typically uniform or not? x Chapter 1 Introduction A brief history The study of lattices of minimal covolume in SLn (or for that matter, in Lie groups in general) originated with Siegel’s work [Sie45] on SL2(R), which in turn can be traced back to Hurwitz’s work [Hur92]. Siegel showed that in SL2(R), lattices of minimal covolume are given by the conjugates of the (2; 3; 7)-triangle group (the stabilizer in SL2(R) of a (2; 3; 7)-triangle tiling). He used the action of SL2(R) on the hyperbolic plane to relate the covolume of the lattice to the area of its Poincaré fundamental domain, which tiles the plane. The result then follows from the fact that the (2; 3; 7)-triangle is the (ideal) polygon with smallest area which tiles the hyperbolic plane. In particular, lattices of minimal covolume in SL2(R) are arithmetic and uniform. Siegel raised the question of determining which lattices attain minimum covolume in groups of isometries of higher-dimensional hyperbolic spaces. For SL2(C), which acts on hyperbolic 3-space, the minimum among non-uniform lattices was established by Meyerhoff [Mey85]; among all lattices in SL2(C), the minimum was exhibited 1 more recently by Gehring, Marshall and Martin [GM09, MM12] using geometric methods, and is attained by a (unique up to conjugacy) uniform lattice. Using the action of SL2 Fq((t)) on its Bruhat-Tits tree and Bass-Serre theory, Lubotzky −1 established the analogous result [Lub90] for SL2 Fq((t)) , where this time SL2 Fq[t ] attains the smallest covolume. Lubotzky observed that in this case, as opposed to the (2; 3; 7)-triangle group in SL2(R), the lattice of minimal covolume is not uniform; he then asked whether, for a lattice of minimal covolume in a semi-simple Lie group, the typical situation is to be uniform, or not. Progress has been made on this question. By proving a quantitative version of the Kazhdan–Margulis theorem, Salehi Golsefidy showed [SG09] that for most Chevalley groups G of −1 rank at least 2, G Fq[t ] is the unique (up to automorphism) lattice of minimal covolume in G Fq((t)) . Using Prasad’s formula, Salehi Golsefidy also obtained [SG13] that for most simply connected simple groups over Fq((t)), a lattice of minimal covolume will be non-uniform (provided Weil’s conjecture on Tamagawa numbers holds). On the other side of the picture, when the rank is 1, Belolipetsky and Emery [Bel04, BE12] determined the lattices of minimal covolume among arithmetic lattices in SO(n; 1)(R) for n ≥ 5, and showed that they are non-uniform. For SU(n; 1)(R), Emery and Stover [ES14] determined the lattices of minimal covolume among the non-uniform arithmetic ones, but to the best of the author’s knowledge, this has not been compared to the uniform arithmetic ones in this case. Unfortunately, in the rank 1 case, it is not known whether a lattice of minimal covolume is necessarily arithmetic. A summary of the advances made in this case (along with references) can be found in [Bel14]. The most recent results concerning lattices of minimal covolume are due to Emery and Kim [EK18], who determined the lattices of minimal covolume in Sp(n; 1)(R) for n ≥ 2, both 2 among uniform lattices and among non-uniform lattices. For n ≥ 3, lattices of minimal covolume in Sp(n; 1)(R) are again non-uniform. The above results give a partial answer to the question of Lubotzky. In this document, we work out the case of SLn(R). We show that for n ≥ 3, up to automorphism, the non-uniform lattice SLn(Z) is the unique lattice of minimal covolume in SLn(R). This is in sharp contrast with Siegel’s lattice in SL2(R). Some open questions and further directions Describing the lattices of minimal covolume in an arbitrary (real) Lie group is likely an unreasonably difficult question, namely because of the groups of rank 1.