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An Introduction to Totally Disconnected Locally Compact Groups

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An Introduction to Totally Disconnected Locally Compact Groups An introduction to totally disconnected locally compact groups Phillip R. Wesolek December 11, 2018 ii Contents 1 Topological Structure 7 1.1 van Dantzig's theorem . .7 1.2 Isomorphism theorems . 11 1.3 Graph automorphism groups . 14 1.4 Exercises . 20 2 Haar Measure 23 2.1 Functional analysis . 23 2.2 Existence . 25 2.3 Uniqueness . 33 2.4 The modular function . 37 2.5 Quotient integral formula . 40 2.6 Exercises . 44 3 Geometric Structure 49 3.1 The Cayley{Abels graph . 49 3.2 Cayley-Abels representations . 53 3.3 Uniqueness . 57 3.4 Compact presentation . 60 3.5 Exercises . 62 4 Essentially Chief Series 65 4.1 Graphs revisited . 66 4.2 Chain conditions . 72 4.3 Existence of essentially chief series . 75 4.4 Uniqueness of essentially chief series . 78 4.5 The refinement theorem . 80 4.6 Exercises . 87 1 2 CONTENTS Introduction Preface For G a locally compact group, the connected component which contains the identity, denoted by G◦, is a closed normal subgroup. This observation produces a short exact sequence of topological groups f1g ! G◦ ! G ! G=G◦ ! f1g where G=G◦ is the group of left cosets of G◦ endowed with the quotient topology. The group G◦ is a connected locally compact group, and the group of left cosets G=G◦ is a totally disconnected locally compact (t.d.l.c.) group. The study of locally compact groups therefore in principle, although not always in practice, reduces to studying connected locally compact groups and t.d.l.c. groups. The study of locally compact groups begins with the work [11] of S. Lie from the late 19th century. Since Lie's work, many deep and general results have been discovered for connected locally compact groups - i.e. the group G◦ appearing above. The quintessential example is, of course, the celebrated solution to Hilbert's fifth problem: Connected locally compact groups are inverse limits of Lie groups. The t.d.l.c. groups, on the other hand, long resisted a general theory. There were several early, promising results due to D. van Dantzig and H. Abels, but these results largely failed to ignite an active program of research. The indifference of the mathematical community seems to have arose from an inability to find a coherent metamathematical perspective via which to view the many disparate examples, which were long known to include profinite groups, discrete groups, algebraic groups over non-archimedian local fields, and graph automorphism groups. The insight, first due to G. Willis [15] and later refined in the work of M. Burger and S. Mozes [3] and P.-E. Caprace 3 4 CONTENTS and N. Monod [4], giving rise to a general theory is to consider t.d.l.c. groups as simultaneously geometric groups and topological groups and thus to inves- tigate the connections between the geometric structure and the topological structure. This perspective gives the profinite groups and the discrete groups a special status as basic building blocks, since the profinite groups are trivial as geometric groups and the discrete groups are trivial as topological groups. This book covers what I view as the fundamental results in the theory of t.d.l.c. groups. I aim to present in full and clear detail the basic theorems and techniques a graduate student or researcher will need to study t.d.l.c. groups. Prerequisites The reader should have the mathematical maturity of a second year grad- uate student. I assume a working knowledge of abstract algebra, point-set topology, and functional analysis. The ideal reader will have taken graduate courses in abstract algebra, point-set topology, and functional analysis. Acknowledgments These notes began as a long preliminary section of my thesis, and my thesis adviser Christian Rosendal deserves many thanks for inspiring this project. While a postdoc at Universit´ecatholique de Louvain, I gave a graduate course from these notes. David Hume, Fran¸coisLe Ma^ıtre,Nicolas Radu, and Thierry Stulemeijer all took the course and gave detailed feedback on this text. Fran¸coisLe Ma^ıtredeserves additional thanks for contributing to the chapter on Haar measures. During my time as a postdoc at Binghamton University, I again lectured from these notes. I happily thank Joshua Carey and Chance Rodriguez for their many suggestions. Notes Self-containment of text This text is largely self-contained, but proving every fundamental background result would take us too far a field. Such results will always be stated as facts, as opposed to the usual theorem, proposition, or lemma. References for such CONTENTS 5 facts will be given in the notes section of the relevant chapter. Theorems, propositions, and lemmas will always be proved, or the proofs will be left as exercises, when reasonable to do so. Second countability Most natural t.d.l.c. groups are second countable, but non-second countable t.d.l.c. groups unavoidably arise from time to time. One can usually deal with non-second countable groups by writing the group in question as a directed union of compactly generated open subgroups and reasoning about this di- rected system of subgroups. Compactly generated t.d.l.c. groups are second countable modulo a profinite normal subgroup, so as we consider profinite groups to be basic building blocks, these groups are effectively second count- able. The theory of t.d.l.c. groups can thereby be reduced to studying second countable groups. In this text, we will thus assume our groups are second countable whenever convenient. In the setting of locally compact groups, second countability admits a useful characterization. Definition. A topological space is Polish if it is separable and admits a complete metric which induces the topology. Fact. The following are equivalent for a locally compact group G: (1) G is Polish. (2) G is second countable. (3) G is metrizable and Kσ - i.e. has a countable exhaustion by compact sets. Locally compact second countable groups are thus exactly the locally compact members of the class of Polish groups, often studied in descriptive set theory, permutation group theory, and model theory. We may in par- ticular use the term \Polish" in place of \second countable" in the setting of locally compact groups, and we often do so for two reasons. First and most practically, we will from time to time require classical results for Polish groups, and this will let us avoid rehashing the above fact. Second, the study of t.d.l.c. groups seems naturally situated within the study of Polish groups, and we wish to draw attention to this fact. 6 CONTENTS Basic definitions A topological group is a group endowed with a topology such that the group operations are continuous. All topological groups and spaces are taken to be Hausdorff. For G a topological group acting on a topological space X, we say that the action is continuous if the action map G×X ! X is continuous, where G × X is given the product topology. Notations We use \t.d.", \l.c.", and \s.c." for \totally disconnected", \locally compact", and \second countable", respectively. We write 81 to stand for the phrase \for all but finitely many." We use C, Q, R, and Z for the complex numbers, rationals, reals, and the integers, respectively. We use the notation R≥0 to denote the non-negative real numbers. For H a closed subgroup of a topological group G, G=H denotes the space of left cosets, and HnG denotes the space of right cosets. We primarily consider left coset spaces. All quotients of topological groups are given the quotient topology. The center of G is denoted by Z(G). For any subset K ⊆ G, CG(K) is the collection of elements of G that centralize every element of K. We denote the collection of elements of G that normalize K by NG(K). The topological closure of K in G is denoted by K. For A; B ⊆ G, we put AB := fbab−1 j a 2 A and b 2 Bg ; [A; B] := haba−1b−1 j a 2 A and b 2 Bi and n A := fa1 : : : an j ai 2 Ag : For k ≥ 1, A×k denotes the k-th Cartesian power. For a; b 2 G,[a; b] := aba−1b−1. We denote a group G acting on a set X by G y X. Groups are always taken to act on the left. For a subset F ⊆ X, we denote the pointwise stabilizer of F in G by G(F ). The setwise stabilizer is denoted by GfF g. Chapter 1 Topological Structure Our study of t.d.l.c. groups begins with an exploration of the topology. This chapter aims to give an understanding of how these groups look from a topological perspective as well as to develop basic techniques to manipulate the topology. 1.1 van Dantzig's theorem In a topological group, the topology is determined by a neighborhood basis at the identity. Having a collection fUαgα2I of arbitrarily small neighborhoods of the identity, we obtain a collection of arbitrarily small neighborhoods of any other group element g by forming fgUαgα2I . We thus obtain a basis B := fgUα j g 2 G and α 2 Ig for the topology on G. We, somewhat abusively, call the collection fUαgα2I a basis of identity neighborhoods for G. The topology of a totally disconnected locally compact (t.d.l.c.) group admits a well-behaved basis of identity neighborhoods; there is no need for the Polish assumption here. Isolating this basis requires a couple of classical results from point-set topology.
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