AN INTRODUCTION TO TOTALLY DISCONNECTED LOCALLY

AN INTRODUCTION TO TOTALLY DISCONNECTED LOCALLYCOMPACT POLISH GROUPSPHILLIP R. WESOLEKi

iiPHILLIP R. WESOLEKContentsPart 1.Introduction1Part1.2.3.4.5.6.2. Topological structure and basic techniquesvan Dantzig’s theoremIsomorphism theoremsBuilding new groups from oldA first example: Aut(Tn )A first class of examples: Locally elliptic groupsExercises33457810Part1.2.3.4.3. Haar MeasurePreliminariesExistence and uniquenessThe modular functionExercises1212121617Part1.2.3.4.4. Geometric StructureGraphsCayley-Abels graphsFirst applicationsExercises1818192327Part1.2.3.4.5. Essentially Chief SeriesPreliminariesExistence of essentially chief seriesUniqueness of essentially chief seriesExercises2929292929Part1.2.3.4.5.6. The Scale FunctionPreliminariesSemi-tidy and tidy subgroupsProperties of the scale functionApplication: Halmos’ questionExercises303031434848Part1.2.3.4.7. Elementary GroupsPreliminariesElementary groups and well-founded treesExamples and non-examples of elementary groupsExercises5050526062Part 8.IndexChief factors6465

TOTALLY DISCONNECTED LOCALLY COMPACT POLISH GROUPSReferencesiii66

TOTALLY DISCONNECTED LOCALLY COMPACT POLISH GROUPS1Part 1. IntroductionA topological group is a group endowed with a topology such that the group operationsare continuous. We consider topological groups such that the topology is Polish:Definition 1.1. A topological space is Polish if it is separable and admits a compatiblecomplete metric.This book considers topological groups such that the underlying topology is locally compactand Polish. In the setting of locally compact topological groups, Polish topologies arise easily.Theorem 1.2 (cf. [12] (5.3)). 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.The Polish assumption on locally compact groups is thus quite mild. Indeed, locally compactPolish groups are pervasive in mathematics. For example, GLn (C), Aut(Γ) for Γ a locallyfinite connected graph, and G(k) for k a local field and G a k-algebraic group are all locallycompact and Polish.The study of locally compact Polish groups reduces to studying connected locally compactgroups and totally disconnected locally compact (t.d.l.c.) groups. Indeed, let G be a locallycompact Polish group and G the connected component of the identity. It is easy to verifyG is a closed normal subgroup of G. We thus obtain a short exact sequence of topologicalgroups:{1} G G G/G {1}where G/G is the group of left cosets given the quotient topology. By classical results [10],G/G is a t.d.l.c. group and G is a connected locally compact group. The theory of locallycompact groups thus splits into the theory of connected groups and the theory of totallydisconnected groups.Many deep, general results have been discovered for connected locally compact groups. Forexample, connected locally compact groups are inverse limits of Lie groups by the celebratedsolution to Hilbert’s fifth problem. The t.d.l.c. groups, on the other hand, are just beginningto be studied, and today it seems these groups admit a rich, general, and tractable theory.To get a sense of the modern theory of t.d.l.c. groups, let us philosophize metamathematically for a moment. What are we studying when we study connected locally compact groups?The only discrete connected group is the trivial group, so all groups under considerationought to be non-discrete. To put this positively, the connected groups under considerationhave non-trivial topological structure. The solution to Hilbert’s fifth problem further tells usthat connected groups are well-behaved up to compact groups. Since compact groups havetrivial geometric structure, we assert the most detailed general theorems concern connectedgroups with non-trivial geometric structure. We thus deduce that the theory of connectedlocally compact groups is most powerful for groups which have both non-trivial local structureand non-trivial geometric structure.One then naturally asks: What is it that the makes the presence of both large and smallscale structure so useful in the study of connected groups? Reviewing the arguments fromthe theory of connected groups, one concludes that it is not merely the presence of largeand small scale structure but the interactions between these structures. We thus assertthat many of the most general results for connected locally compact groups follow from

2PHILLIP R. WESOLEKanalyzing the interaction between local and geometric structure. Together local structureand geometric structure restrict how the groups can look like. We call the study of theinteraction between local and geometric structure the the large-scale topological theory ofconnected locally compact groups.The modern theory of t.d.l.c. groups is, in this author’s opinion, simply the large-scaletopological theory for t.d.l.c. groups. This book covers the central developments in the largescale topological theory of t.d.l.c. Polish group leading up to modern research results anddirections. We hope the results herein convince the reader that, just as with connectedgroups, the large-scale topological theory of t.d.l.c. groups is rich and tractable. We stress animportant caveat: the large-scale topological theory of t.d.l.c. groups takes discrete groups andcompact groups as basic objects. It is worth noting that this does not imply the large-scaletopological theory has nothing to say about discrete groups; see [17] or [21].Prerequisites. We assume knowledge of abstract algebra, point-set topology, and elementary descriptive set theory. The reader should have taken undergraduate courses in abstractalgebra, point-set topology, and functional analysis. The ideal reader will have taken coursesin abstract algebra, point-set topology, and a first graduate course in descriptive set theoryor functional analysis.Remark 1.3. Much of the theory we develop can be done without the Polish, equivalentlysecond countable, assumption. We restrict to Polish groups as Polish spaces and, therefore,Polish groups are considered to be the “correct” topological spaces and groups to study. Thatis to say, they capture almost all of the natural examples of “nice” topological spaces, admituseful and general theorems such as the Baire category theorem, and exclude pathologicalexamples.0.1. Notations. All topological groups and spaces are taken to be Hausdorff and are typicallywritten multiplicatively. We use “t.d.”, “l.c.”, and “s.c.” for “totally disconnected”, “locallycompact”, and “second countable”, respectively.For a topological group G, all subgroups are taken to be closed unless otherwise stated.For H a closed subgroup, G/H denotes the space of left cosets. All quotient spaces of cosetsare given the quotient topology. We write H 6o G and H 6cc G to indicate H is an opensubgroup and cocompact subgroup of G, respectively. Recall H 6 G is cocompact if thequotient space G/H is compact.For any subset K G, CG (K) is the collection of elements of G that centralize everyelement of K. We denote the collection of elements of G that normalize K by NG (K). Thetopological closure of K in G is denoted by K. For A, B G, we put AB : bab 1 a A and b B ,[A, B] : haba 1 b 1 a A and b Bi, andAn : {a1 . . . an ai A}For k 1, A k denotes the k-th Cartesian power. For a, b G, [a, b] : aba 1 b 1 .We make use of ordinal numbers later in this work. For the moment, we merely note thefirst transfinite ordinal is denoted ω and is equal to N as linear orders. We thus often use “ω”and “N” interchangeable.

TOTALLY DISCONNECTED LOCALLY COMPACT POLISH GROUPS3Part 2. Topological structure and basic techniques1. van Dantzig’s theoremThe topology of a t.d.l.c. Polish group admits a well-behaved basis at the identity by a resultof D. van Dantzig. Note that by translating this basis by left multiplication, one recovers abasis for the topology of the group.Theorem 2.1 (van Dantzig, cf. [10, (7.7)]). A t.d.l.c. group admits a basis at 1 of compactopen subgroups.Proof. Let V be a neighborhood of 1 in G. By classical results [10, (3.5)], G admits a basisof clopen sets at 1. We may thus find U V a compact open neighborhood of 1.We claim there is L U a symmetric open neighborhood of 1 such that U L U . Indeed,for each x U there is WSx 3 1 open with xWx U and an open symmetric set Lx 3 1 withL2x Wx . Plainly, U x U xLx . So U x1 Lx1 · · · xk Lxk for some x1 , . . . , xk since UTis compact. Putting L ki 1 Lxi , we haveUL k[i 1xi Lxi L k[i 1xi L2xi k[x i W xi Ui 1It now follows that Ln U for all n S 0: if Ln U , then Ln 1 Ln L U L U bythe previous. We conclude that W : n 0 Ln U . Since L is symmetric, W is a opensubgroup of the compact open set U . As the compliment of W is open, W is indeed clopenand, therefore, a compact open subgroup. The compact open subgroups need not be normal, e.g. there are many non-discrete t.d.l.c.groups which are topologically simple. Further, the presence of such subgroups indicates nondiscrete t.d.l.c. groups are wildly different from almost connected locally compact groups.Indeed, the general linear group over the p-adics, GLn (Qp ), is a t.d.l.c. group and, therefore,admits a basis at 1 of compact open subgroups. On the contrary, GLn (R), having a finiteindex connected subgroup, does not admit such a basis!Notation 2.2. For a t.d.l.c. Polish group G, we denote the collection of compact opensubgroups by U(G). We denote the collection of closed subgroups by S(G). The space S(G)comes with a canonical topology called the Chabauty topology. Under this topology, S(G)is a compact Polish space and is called the Chabauty space of G.The compact open subgroups do allow us to realize all t.d.l.c. Polish groups as permutationgroups, a fact which makes the study of t.d.l.c. Polish groups interesting to mathematicallogicians. Let Sym(N) be the collection of permutations of N. Considering N as a discretespace, we endow Sym(N) with the topology of pointwise convergence. Under this topology,Sym(N) is a non-locally compact Polish group; cf. [12, Chapter 9].Corollary 2.3 (folklore). Every t.d.l.c. Polish group is isomorphic to a closed subgroup ofSym(N).Proof. Let G be a t.d.l.c. Polish group and (Ui )i ω a countable basis at 1 of compact opensubgroups as given by Theorem 2.1. Setting K to be the collection of left cosets of the Ui , wehave that G acts on K continuously and faithfully by left multiplication.Let ψ : G Sym(N) be the continuous homomorphism induced via the above action. Itsuffices to show ψ is a closed map. To that end, say A G is closed and (ψ(ai ))i ω ψ(A)