Decomposing locally compact groups into simple pieces

DECOMPOSING LOCALL Y COMP A CT GR OUPS INTO SIMPLE PIECES PIERRE-EMMANUEL CAPRACE* AND NICOLAS MONO D ‡ Abstra ct. W e present a con tribution to the structure theory of lo cally compact groups. The emphasis is on compactly generated locally compact groups which admit no inﬁnite discrete quotient. It is sho wn that such a group p ossesses a characteris tic cocompact subgroup whic h is either connected or admits a non-compact non-discrete top ologically simple quotient. W e also provide a description of characteri stically simple group s and of groups all of whose prop er quotients are compact. W e sho w that No etherian locally compact groups without inﬁnite discrete quotient admit a subnormal series with all sub quotients compact, compactly generated Ab elian, or compactly generated t opologically simple. Tw o app endices introduce results and examples around the concept of quasi-pr o duct . Contents 1. In tro duction 2 On structur e theory 2 Characteristically simp le group s and quasi-pro ducts 4 Groups with ev er y p rop er qu otien t compact 4 Ac knowledgeme n ts 5 2. Basic to ols 5 Generalities on lo cally compact groups 5 Filters of closed normal su bgroups 7 Minimal n ormal subgrou p s 7 Just-non-compact grou p s 8 3. T op ologica l BN-pairs 9 4. Discrete qu otien ts 10 Residually discrete groups 10 The discrete residu al 11 Quasi-discrete grou p s 11 5. On structure th eory 12 Quasi-simple qu otien ts 12 Maximal normal sub groups 13 Upp er and lo wer structure 14 6. Comp osition series with top ologically simp le sub quotien ts 15 App end ix I. T he adjoint closure and asym p toticall y cen tral sequences 18 On the Braconnier top ology 18 Adjoin t representat ion 19 Asymptotically central sequences 20 *F.N.R.S. R esearc h A ssociate. ‡ Supp orted in part by the Swiss National Science F oundation. 1 2 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D The adjoint closure 21 Lo cally ﬁn itely generated groups 21 The adjoint closure of d iscr ete groups 22 App end ix I I. Quasi-pro d ucts and dense normal subgroup s 24 Deﬁnitions and the Galois connection 24 On the non-Hausdorﬀ q u otien ts of a qu asi-pr o duct 26 Quasi-pro du cts with Ad-closed quasi-factors 27 Non-direct qu asi-pro ducts and dense analytic n ormal sub groups 28 Op en p roblems 31 On dense normal su bgroups of top ologically simple groups 31 References 31 Index 34 1. Introduction On structure theory. The stru cture of ﬁnite groups can to a large extent b e reduced to ﬁnite simp le group s and th e latter ha v e famously b een classiﬁed (see e.g. [GLS94] sqq. ). F or general lo cally compact group s, b oth th e r eduction to simp le group s and the study of the latter still constitute ma j or c hallenges. T he connected case h as foun d a satisfactory answ er: I n deed, th e solution to Hilb ert’s ﬁfth p r oblem (see [MZ55, 4.6]) r educes the question to Lie theory up on discarding compact k ernels. L ie group s are then describ ed by inv estigating separately the solub le groups and the simp le factors, which are classiﬁed since the time of ´ E. Cartan. The con temp orary structure problem therefore regards totally disconnected groups; there is not yet ev en a conjectural p icture of a stru ctur e theory . In fact, until recen tly , the only structure theorem on totally disconn ected lo cally compact groups w as this single sentence in v an Dant zig’s 1931 th esis: “ Een (gesloten) Cantorsche gr o ep b e v at wil leke u rig kleine op en onder gr o ep en. ” (I I I § 1, TG 38 on page 18 in [vD31]) Recen t progress, including statemen ts on simple group s, is p ro vid ed by the w ork of G. Willis [Wil94, Wil07]. New examples of simp le groups ha ve app eared in the geometric con text of trees and of general buildings. As results and examples for simple groups are b eing discov ered, it b ecomes d esirable to ha ve a reduction step to simple groups — in parallel to the k n o wn cases of ﬁn ite and connected groups. How ev er, any reasonable attempt at classiﬁcation must in one w ay or another exclude discr e te group s: the latter are widely considered to b e un classiﬁable; this opinion can b e giv en a mathematic al con ten t as e.g. in [TV99]. The discrete case also illus tr ates that th er e ma y b e n o simple (inﬁnite) quotien t, or ev en su b quotient , at all. The follo wing tric hotomy shows th at, a w ay from the una voidable discrete situation, there is a comp elling ﬁrst answ er. Theorem A. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup. Then exactly one of the fol lowing holds. (i) G has an inﬁnite discr ete quotient. DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 3 (ii) G has a c o c omp act normal sub gr oup that is c onne c te d and soluble. (iii) G has a c o c omp act normal sub gr oup that admits exactly n non-c omp act simple quo- tients (and no non-trivial discr ete quotient), wher e 0 < n < ∞ . By a simple group, we mean a top olo gic al ly simple group , i.e. a group all of whose Hausdorﬀ quotien ts are trivial. Since a co compact closed su bgroup of a compactly generated lo cally compact group is itself compactly generated [M ´ S59], it follo ws that the n simple quotien ts app earing in (iii) of Theorem A are compactly generated. (W e alw a ys imp licitly endo w quotien t groups with the quotient top ology .) The ab o v e theorem describ es the upp er structur e of G . T he ﬁrst alternativ e can b e m ad e more precise in combination with the well-kno wn (and easy to establish) f act that an inﬁ nite ﬁnitely generated group either admits an inﬁ nite r esidually ﬁnite quotien t or has a ﬁnite index subgroup which admits an in ﬁ nite simple qu otien t. In some sense, Th eorem A pla ys the rˆ ole of a non-discrete analogue of the latter fact; n otice how ev er that th e ﬁ niteness of the n um b er n of s im p le sub-quotien ts in case (iii) ab ov e is particular to n on-discrete grou p s. It tur n s out that the ab o ve theorem is supplemented b y the follo wing description of th e lower structur e of G , wh ich do es n ot seem to h a ve any analogue in the discrete case. Theorem B. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup. Then one of the fol lowing holds. (i) G has an inﬁnite discr ete normal sub gr oup. (ii) G has a non-trivial c lose d normal sub g r oup which is c omp act-by- { c onne cte d soluble } . (iii) G has exactly n non-trivial minimal close d normal sub gr oups, wher e 0 < n < ∞ . The norm al sub groups app earing in the ab o ve hav e no reason to b e compactly generated in general. On another hand, since an y (Hausdorﬀ ) quotien t of a compactly generated group is itself compactly generated, it follo ws that Th eorem B ma y b e app lied rep eatedly to the successiv e quotien ts that it pro vides. Suc h a p ro cess will of course not terminate after ﬁnitely man y steps in general. Ho wev er, if G satisﬁes additional ﬁniteness cond itions, this recursiv e pro cess ma y in deed reac h an end in ﬁn ite time. In order to mak e this precise, we introduce the follo wing terminology . W e call a top ologic al group G No et he ria n if it satisﬁes the ascending c hain condition on op en sub groups. Ob vious examples are pro vided by compact ( e.g. ﬁ nite) groups, connected group s and p olycyclic groups. If G is lo cally compact, then G is Noetherian if and only if ev ery op en sub group is compactly generated. (W arning: the n otion introd uced in [Gui73, § I I I] is more restrictiv e as it p osits compact generation of all close d subgroups.) Theorem C. L et G b e a lo c al ly c omp act No etherian gr oup. Then G p ossesses an op en normal sub gr oup G k and a ﬁnite series of close d subnormal sub gr oups 1 = G 0 ✁ G 1 ✁ G 2 ✁ · · · ✁ G k ✁ G such that, for e ach i ≤ k , the sub quotient G i /G i − 1 is either c omp act, or isomorphic to Z or R , or top olo g i c al ly simple non-discr ete and c omp actly gener ate d. In the sp ecial case of connected groups, the existence of the ab o ve d ecomp osition follo ws easily from the solution to Hilb ert’s ﬁfth problem. In that case , the simp le sub quotien ts are connected non-compact adj oint sim p le Lie group, while the presence of discrete free Ab elian groups acco un ts for p ossible central extensions of simple Lie groups suc h as ^ SL 2 ( R ) (an example going bac k to S c h r eier [S c h25, § 5 Beispiel 2]). No analogue of Theorem C seems to b e k n o wn for discr ete No etherian groups. 4 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Characteristically simple groups and quasi-pro ducts. By a characteristic subgroup of a top ological group G , w e mean a closed sub group whic h is preserved by ev ery top olog- ical group automorph ism of G . A grou p admitting no non-trivial suc h su bgroup is called c haracteristically simple . This prop erty is satisﬁed by any minimal norm al su bgroup, for instance th ose o ccurr in g in Theorem B. In fact, our ab o ve results lea d also to a description of c haracteristically simple groups, as follo ws. Corollary D. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup. If G is char acteristic al ly simple, then one of the fol lowing hold s. (i) G is discr ete. (ii) G is c omp act. (iii) G ∼ = R n for some n . (iv) G is a quasi-pr o duct with top olo gic al ly simple p airwise isomorph ic quasi-factors. By deﬁ n ition, a top ological group G is call ed a quasi-pro duct with quasi-factors N 1 , . . . , N p if N i are closed normal subgroups suc h that the multiplicatio n m ap N 1 × · · · × N p − → G is injectiv e with dense image. Usual direct pro du cts are ob vious examples, bu t the situation is m uc h more complicated f or general tota lly disconnected groups. The abov e deﬁnition degenerates in th e commutativ e case; for in stance, R is a quasi-pro d uct with quasi-factors Z and √ 2 Z . Seve ral c entr efr e e examp les of quasi-pro ducts, including c haracteristically simp le ones, w ill b e constructed in App end ix I I. Ho wev er, as of to day we are n ot aw are of an y c omp actly gener ate d c haracteristically simple group whic h falls int o Case (iv) of Corollary D without b eing a gen uin e direct pro d uct. As we shall see in App endix I I, the existence of suc h an example is equiv alen t to the existence of a compactly generated top ologic ally simple lo cally compact group admitting a pr op er dens e normal subgroup . Groups wit h every prop er quotien t compact. The ﬁrst goal that we shall p ursue in this article is to describ e the compactly generated lo cally compact groups wh ic h admit only compact p rop er quotient s. The n on-compact group s satisfying this condition are sometimes called just-non-compact . In th e discrete case, the corresp ond ing notion is that of just- inﬁnite groups , n amely discrete groups all of w hose pr op er qu otien ts are ﬁn ite. A d escription of these was giv en b y J. S. Wilson in a classical article ([Wil71], Prop osition 1). Antic ipating on th e terminology introduced b elo w, we can epitomise our con trib ution to this question as follo ws: A just-non-c omp act gr oup is either discr ete or monolithic. The most ob vious case where a top ological group has only compact quotien ts is w hen it is quasi-simple , which means that it p ossesses a co compact normal sub group which is top ologica lly simp le and con tained in ev ery n on-trivial closed n ormal subgroup . This situation extends readily to the follo wing. W e say that a top ological group is mono- lithic with monolith L if the in tersectio n of all n on -trivial closed normal subgroup s is itself a non-trivial group L . Non-qu asi-simple examples are p r o vided by the standard wreath pro du ct of a top ologica lly simple group by a ﬁnite transitiv e p erm utation group (see Construction 1 in [Wil71]). Notice that th e monolith is necessarily characte ristically simp le. DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 5 In the discrete case, groups with only ﬁnite p rop er q u otien ts can b e very far from mono- lithic, indeed residu ally ﬁnite: examples are p r o vided b y all lattic es in connected ce n tre- free simple Lie groups of rank at least t wo in view of a fundamental theorem of G. Mar- gulis [Mar91]; for instance, PSL 3 ( Z ) (this particular case was already kno wn to J . Men- nic ke [Men65]). Th e follo wing result sho ws that such examples do not exist in the non-discrete case. Theorem E. L et G b e a c omp actly gener ate d non-c omp act lo c al ly c omp act g r oup such that every non-trivial close d normal su b gr oup is c o c omp act. Then one of the fol lowing holds. (i) G is monolithic and its monolith is a quasi-pr o duct with ﬁnitely many isomorph ic top olo gic al ly simple gr oups as quasi- f actors. (ii) G i s monolithic with monolith L ∼ = R n . Mor e over G/L is isomorphic to a close d irr e duci ble sub g r oup of O ( n ) . In p articular G is an almost c onne cte d Lie gr oup. (iii) G i s discr ete and r esidual ly ﬁnite. W e shall see in § 3 b elo w that this theorem yields a top ological simplicit y corollary for lo cally co mpact groups endow ed with a B N -pair wh ic h supplements classical results by J. Tits [Tit64]. The p ro of of T heorem E relies on an analysis of ﬁltering families of close d normal subgroup s in totally d isconnected lo cally compact groups , whic h is carried out in Prop osition 2.5 b elo w. As a by-pro duct, it yields in p articular a c haracterisation of residu ally discrete group s (see Corollary 4.1) and the follo wing easier compan ion to T heorem E, where Res( G ) d enotes the discrete residual of G , namely the in tersection of all op en n orm al su bgroups. Theorem F. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup al l of whose discr ete quotients ar e ﬁnite. Then Res( G ) i s a c o c omp act char acteristic close d sub g r oup of G witho ut non-trivial discr ete quotient. The co compact su bgroup Res( G ) is compactly generated (see [M ´ S59] or Lemm a 2.1 b e- lo w). Since any co mpact totally disconnected group is a p roﬁnite group and, hence, admits n umerous discrete quotien ts, it follo w s that, un der th e h yp otheses of Theorem F, any c omp act quotient of the discr ete r esidual is c onne cte d . Lo osely sp eaking, the discrete residu al is th us a sort of c o c omp act c or e of the group G . As a consequence, we obtain the follo wing. Corollary G. L et G b e a c omp actly gener ate d total ly disc onne cte d lo c al ly c omp act gr oup al l of whose discr ete qu otients ar e ﬁnite. Then the discr ete r esidual of G is c o c omp act and admits no non-trivial discr e te or c omp act quotient.  Theorem A f ollo ws easily by com b ining Theorem E with the fact, d ue to R. Grigorc h u k and G. Willis, that any n on -compact compactly generated lo cally compact group adm its a just-non-compact qu otien t (u n publish ed, see Prop osition 5.2 b elo w ). Ac knowledgemen ts. W e are very grateful for the detailed commen ts pro vided by th e anony- mous r eferee. 2. Basic tools Generalities on lo cally compact groups. In this preliminary section, we collect a num b er of s ubsidiary facts on lo cally compact groups whic h will b e used rep eatedly in th e sequel. Unless sp eciﬁed otherwise, all top ological grou p s are assumed Hausdorﬀ. W e will frequently inv ok e the follo wing w ell-kno w n statemen t without explicit reference. 6 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Lemma 2.1. If a close d sub g r oup of a c omp actly gener ate d lo c al ly c omp act gr oup is c o c omp act, then it is itself c omp actly gener ate d. Pr o of. See [M ´ S59], Corollary 2.  W e say that a subgroup H of a top ological group G is top ologically lo cally ﬁnite if ev ery ﬁnite su bset of H is conta ined in a compact su bgroup of G . Any locally compact group G p ossesses a maximal normal top ologically lo cally ﬁnite sub grou p which is closed and called the LF-radical and denoted Rad LF ( G ); an other imp ortan t fact is that an y compact su bset of a lo cally compact top ologically lo cally ﬁ nite group is contai ned in a compact subgroup . W e r efer to [Pla65] and [Cap09, § 2] f or details. It is wel l kn o wn that the LF-radical is compact for conn ected groups: Lemma 2.2. Every c onne cte d lo c al ly c omp act gr oup ad mits a maximal c omp act norma l sub- gr oup. Mor e over, the c orr esp onding quotient is a c onne cte d Lie g r oup. Pr o of. The solution to Hilb ert’s ﬁfth problem [MZ55, Th eorem 4.6] provides a compact nor- mal subgroup suc h that the qu otien t is a Lie group; no w the statemen t follo ws from the corresp ondin g fact for connected Lie grou p s.  As a further element of terminology , the quasi-cen tre of a top ological group G is the subset Q Z ( G ) consisting of all those elemen ts p ossessing an op en centrali ser. The s u bgroup Q Z ( G ) is top ologic ally c h aracteristic in G , bu t n eed not b e closed. Since any elemen t with a discrete conjugacy class p ossesses an op en cen traliser, it follo ws that th e quasi-cen tre conta ins all discrete normal sub groups of G . W e shall use the follo wing result of U ˇ sak ov, for w h ic h w e recall that a top ological grou p is called top ologically F C if ev ery conju gacy class has compact closure. Theorem 2.3 (U ˇ sako v [U ˇ sa63]) . L et G b e a lo c al ly c omp act top olo gi c al ly F C-gr oup. Then the union of al l c omp act sub g r oups of G for ms a close d normal sub gr oup, which ther efor e c oincides with Rad LF ( G ) , and the c orr esp onding quotient G/ Rad LF ( G ) is Ab elian. Mor e over, if in addition G is c omp actly gener ate d, then G is c omp act-by-Ab e lian. Mor e pr e cise ly, Rad LF ( G ) is c omp act and G/ Rad LF ( G ) is isomorphic to R n × Z m for some n, m ≥ 0 .  The follo w ing consideration p ro v id es a necessary (and suﬃcien t) cond ition for the group considered in Theorem E to adm it a non-trivial discr ete normal subgroup . Prop osition 2.4. L et G b e a c omp actly g e ner ate d non-c omp act lo c al ly c omp act gr oup such that every non-trivial close d normal sub gr oup is c o c omp act. If G admits a non-trivial discr ete normal sub gr oup, then G is either discr ete or R n -by-ﬁnite. Pr o of. Let H ✁ G b e a non -trivial discrete n ormal su bgroup. Then H is co compact and, hence, it is a co compact lattice in the compactly generated group G . Therefore H is ﬁ nitely generated. W e deduce th at the normal subgroup Z G ( H ) ✁ G is op en, s ince every elemen t of H has a d iscr ete conjugacy class and , hence, an op en cen traliser. In particular, G is discrete if Z G ( H ) is trivial and w e can therefore assume that Z G ( H ) is co compact. The quotien t Z G ( H ) / Z ( H ) is compact sin ce it sits as op en (hence closed) subgroup in G/H . Hence, the centre of Z G ( H ) is co compact for it con tains Z ( H ). Th us Theorem 2.3 ap- plies to Z G ( H ). W e claim that the LF-radical of Z G ( H ) is tr ivial: otherw ise, b eing norm al in G , it w ould b e cocompact, hence compactly generated, thus compact (Lemma 2.3 in [Cap09]) DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 7 and now ﬁ nally trivial after all since G is non-compact b y h yp othesis. In conclusion, Z G ( H ) is isomorphic to R n × Z m . I n addition, n or m v anishes since the iden tit y comp onent of Z G ( H ) is normal in G and hence trivial or cocompact. W e conclude by recalling that Z G ( H ) is op en and co compact, th u s of ﬁn ite index in G .  Filters of closed normal subgroups. W e con tinue with another general fact on compactly generated groups, w hic h w as the starting p oin t of this w ork. The argument w as in spired by a reading of L emm a 1.4.1 in [BM00]; it also plays a k ey rˆ ole in the structur e theory of isometry groups of non-p ositiv ely curved spaces dev eloped in [CM09]. Prop osition 2.5. L et G b e a c omp actly gener ate d total ly disc onne cte d lo c al ly c omp act g r oup. Then any identity neighb ourho o d V c ontains a c omp act normal sub gr oup Q V such that any ﬁltering f amily of non-discr ete close d normal sub gr oups of the quotient G/Q V has non-trivial interse ction. Pr o of. Let g b e a Schreier graph for G associated to a co mpact o p en subgroup U < G con tained in the giv en iden tit y neigh b ourho o d V (whic h exists by a classica l result in D. v an Dantz ig’s 1931 thesis [vD31, I I I § 1, TG 38 p age 18], see [Bou71 , I I I § 4 No 6]). W e recall that g is obtained b y deﬁn ing the v ertex set as G/U and drawing edges according to a compact generating set whic h is a union of double cosets mo d ulo U ; see [Mon01, § 11.3 ]. Th e k ern el of the G action on g is nothin g bu t Q V = T g ∈ G g U g − 1 whic h is compact and con tained in V . Since w e are int erested in closed normal subgroups of the quotient G/Q V , th ere is no loss of generalit y in assum ing Q V trivial. In other w ords, we assume henceforth that G acts faithfully on g . L et v 0 b e a v ertex of g and denote by v ⊥ 0 the set of neighb ou r ing vertice s. Since G is v ertex-transitive on g , it follo ws that for an y normal subgroup N ✁ G , the N v 0 -action on v ⊥ 0 deﬁnes a ﬁnite p ermutation group F N < Sym( v ⊥ 0 ) w hic h , as an abstract p ermutatio n group, is ind ep endent of the choic e of v 0 . Therefore, if N is non-discrete, th is p erm utation group F N has to b e non-trivial since U is op en and g connected. No w a ﬁltering family F of non- discrete normal su bgroups yields a ﬁ ltering family of n on-trivial ﬁnite sub groups of Sym( v ⊥ 0 ). Th us the in tersection of these ﬁnite groups is non-trivial. Let g b e a non-trivial elemen t in this intersectio n. F or any N ∈ F , let N g b e the in v erse image of { g } in N v 0 . Thus N g is a non-empty compact subset of N for eac h N ∈ F . Since the family F is ﬁltering, so are { N v 0 | N ∈ F } and { N g | N ∈ F } . The resu lt follo ws, s ince a ﬁltering family of non-empt y closed su bsets of the compact s et G v 0 has a non-empty int ersection.  Minimal normal subgroups. With Prop osition 2.5 at hand , we deduce the follo w in g. An analogous result for the upp er s tructure of totally d isconnected group s will b e established in Section 5 b elo w (see Prop osition 5.4). Prop osition 2.6. L et G b e a c omp actly gener ate d total ly disc onne cte d lo c al ly c omp act gr oup which p ossesses no non-trivial c omp act or discr ete normal su b g r oup. Then e very non-trivial close d normal sub g r oup of G c ontains a minimal one, and the set M of non-trivial minimal close d normal sub gr oups is ﬁnite. F urthermor e, for any pr op er sub se t E ⊂ M , the su b gr oup h M | M ∈ E i i s pr op erly c ontaine d in G . Pr o of. In view of Prop osition 2.5, an y c hain of non-trivial closed normal sub groups of G has a min imal elemen t. Thus Zorn’s lemma ensu res that the set M of minim al n on-trivial closed normal subgroup s of G is non-empty , and that any non-trivial closed normal subgroup con tains an elemen t of M . 8 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D In order to establish that M is ﬁ nite, w e use the follo w ing notation. F or eac h subset E ⊆ M w e set M E = h M | M ∈ E i ; M E denotes its closure. The follo wing argument w as inspired by the pro of of Prop osition 1.5.1 in [BM00]. W e claim that if E is a p rop er su bset of M , then M E is a prop er subgroup of G . I n deed, for all M ∈ E and M ′ ∈ M \ E we ha v e [ M , M ′ ] ⊆ M ∩ M ′ = 1. Th u s M ′ cen tralises M E . In p articular, if M E = G , th en M ′ cen tralises G . Th us M ′ ≤ Z ( G ) is Ab elian, and any prop er sub group of M ′ is norm al in G . Since M ′ is a min im al normal sub group, it follo w s that M ′ has no prop er closed subgroup. Th e only total ly disconnected lo cally compact group s with this pr op ert y b eing the cyclic groups of p r ime order, we dedu ce th at M ′ is ﬁnite, wh ic h con tradicts th e hypotheses. The claim stands prov en. Consider now the f amily N = { M M \ F | F ⊆ M is ﬁnite } of closed normal s u bgroups of G . T his family is ﬁ ltering. F urthermore the ab o ve claim sho ws that M M \ F is pr op erly con tained in G wh enev er F is non-empty . Since T N = 1, it f ollo ws from Pr op osition 2.5 that N is ﬁn ite, and hence M is so, as desir ed .  Just-non-compact groups. Pr o of of The or em E. W e s hall us e rep eatedly the fact that ev ery n orm al sub group of G has trivial LF-radical, which is established as in the ab o v e pro of of Pr op osition 2.4. In particular, normal su b groups of G h av e no non-trivial compact n orm al subgroup s. W e b egin by treating the case where G is totally disconnected. Let L b e the int ersection of all non-trivial closed normal subgroups. W e distinguish t wo cases. If L is trivial, then Prop osition 2.5 sho ws that G admits a non-trivial d iscrete normal subgroup. Thus Prop osition 2.4 applies and G is discr ete; in that case, Wilso n’s result ([Wil71], Pr op osition 1) completes the pro of. Assume now that L is not trivial. Then it is cocompact when ce compactly generated since G is so. Notice that by deﬁ n ition L is c haracteristically simple. W e further d istinguish t wo cases. On the one hand, assume th at the quasi-cen tre Q Z ( L ) is non-trivial. Th en it is dense in L . Since L is compactly generate d, the argumen ts of th e pro of of Th eorem 4.8 in [BEW08] (see Prop osition 4.3 b elo w) sh o w that L p ossesses a compact op en norm al sub grou p . Since L has n o non-trivial compact normal sub group, we deduce that L is d iscrete. No w L is a non-trivial discrete n ormal subgroup and w e h a ve already seen ab o v e ho w to ﬁ n ish the pr o of in that situation. On the other h an d , assume th at the qu asi-cen tre Q Z ( L ) is trivial. In particular L p ossesses no non-trivial discrete n orm al subgroup and w e deduce from Prop osition 2.6 that th e set M of non-trivial minimal closed n ormal sub groups of L is ﬁnite and non-empty . Since L has no non-trivial compact n orm al subgroup , no element of M is compact. The group G acts on M by conjugation. Let E denote a G -orbit in M . Since M E = h M | M ∈ E i is normal in G , it is d ense in L . By P r op osition 2.6, w e infer that E = M . In other w ord s G acts transitiv ely on M . It now follo ws that L is a quasi-pro duct with the elemen ts of M as qu asi-factors. In particular, an y normal s u bgroup M ′ of any M ∈ M is norm alised by M and cen tralised by eac h N ∈ M diﬀerent from M . Since Q N ∈ M N is dens e in L , we infer that M ′ is normal in DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 9 L . Since M is a minimal normal subgroup of L , it follo ws that M is top ologically simple and (i) holds . No w we turn to the case where G is not totally disconn ected, hen ce its identi t y comp onen t G ◦ is cocompact. Since the maximal compact normal s ubgroup of Lemm a 2.2 is trivial, G ◦ is a connected Lie group. S ince its soluble radical is c h aracteristic, it is tr ivial or co compact. In the form er case, G ◦ is semi-simple w ithout compact factors. S ince its isot ypic factors are characte ristic, there is only one isot yp ic factor and we conclude that (i) holds. If on the other h and the radical of G ◦ is co compact, w e deduce that G admits a c h aracter- istic co compact connected soluble s u bgroup R ✁ G . Let T b e the last non-trivial term of the deriv ed series of R . If the identit y comp onent T ◦ is trivial, th en T is a non-trivial discrete normal su b group of G and we ma y conclude by means of Prop osition 2.4. Oth erwise, the group R p ossesses a c haracteristic connected Ab elian subgroup T ◦ , w hic h is th us co compact in G . Since T ◦ has no non-trivial compact su bgroup, we h a ve T ◦ ∼ = R d for some d . The k ernel of the homomorphism G → O ut( T ◦ ) = Aut( T ◦ ) is a co compact normal subgroup N con taining T ◦ in its cen tre, and suc h that N /T ◦ is compact. In particular N is a compactly generated lo- cally compact group in wh ic h eve ry conjugacy class is relativ ely compact. In view of U ˇ sak o v’s result (Theorem 2.3) and of the trivialit y of Rad LF ( N ), the group N is of the form R n × Z m . In conclusion, since T ◦ is cocompact in N , we ha ve T ◦ = N ∼ = R n . C onsidering once again the map G → Aut( T ◦ ) ∼ = GL n ( R ), w e deduce that G/T ◦ is isomorphic to a compact subgroup of GL n ( R ), whic h has to b e irreducible since otherwise T ◦ w ould conta in a non-co compact subgroup normalised b y G . W e conclude the pro of of Theorem E b y recalling that ev ery compact subgroup of GL n ( R ) is conjugated to a su b group of O ( n ).  3. Topological BN-p airs By a celebrated lemma of T its [Tit64], any group adm itting a B N -pair of irr ed ucible t yp e has the prop erty that a n orm al su bgroup acts either trivially or cham b er-transitively on the asso ciated build in g. Tits used his trans itivit y lemma to sho w in lo c. cit. th at if G is p erfect and p ossesses a B N -pair w ith B soluble, then an y non-trivial normal s ubgroup is cont ained in Z = T g ∈ G g B g − 1 . More generally , the same conclusion holds provided G is p erfect and B p ossesses a soluble norm al subgroup U w hose conjugates generate G . If G is endo wed with a group top ology , the same argumen ts show that if G is top ologically p erfect and U is pro-soluble, then G/ Z is top ologically simple. Th e follo w ing consequ en ce of Th eorem E d o es not r equire any assumption on the normal s ubgroup str u cture of B . Corollary 3.1. L et G b e a top olo g ic al gr oup endowe d with a B N -p air of irr e ducible typ e, such that B < G is c omp act and op en. Then G has a close d c o c omp act (top olo gic al ly) char acteristic sub gr oup H c ontaining Z = T g ∈ G g B g − 1 , such that H / Z is top olo gi c al ly simple. It follo ws in particular that G is top olo gic al ly c ommensur able to a top ologica lly simple group sin ce Z is compact and H co compact. Pr o of. The assumption that B is compact op en implies that G is lo cally compact and that the bu ilding X asso ciated with the giv en B N -pair is lo cally ﬁnite. S ince the kernel of this action coincides with Z = T g ∈ G g B g − 1 , we may and shall assume that G acts faithfu lly on X . Since G acts c hamb er-transitiv ely on X and since B is the s tabiliser of some base c hamber c 0 , it follo ws that G is generated by the union of B w ith a ﬁnite set of elemen ts mapping c 0 to eac h of its neigh b ours. Th u s G is compactly generate d. By Tits’ transitivit y lemma [Tit64, 10 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Prop. 2.5], for any non-trivial n ormal s ubgroup N of G , we ha ve G = N .B , wh ence N is co compact p ro vid ed it is closed. If X is ﬁn ite, th en G is compact and w e are done. O therwise G is non -compact and n on- discrete, b ecause B is then necessarily inﬁn ite. W e are thus in a p osition to app ly Theorem E. Since G is a subgroup of the totally disconn ected group Aut( X ), it follo w s that G is totally disconnected and we deduce that G is monolithic with a quasi-pr o duct of top ological ly simp le groups as a mon olith. The fact that the monolith has only one simple factor follo ws fr om the fact that G acts faithfully , minimally an d without ﬁxed p oin t at inﬁn it y on a C A T(0) realisation of X . Suc h a CA T(0) realisation is necessarily ir reducible as a CA T(0) space b y [CH06] and [CM09, Theorem 1.10] ensures that no abstract normal subgroup of G sp lits non-trivially as a direct pro d u ct.  4. Discrete quotients Residually discrete groups. Any to p ological group admits a ﬁ ltering family of closed normal subgroup s, consisting of all op en normal su bgroups. Sp ecialising Prop osition 2.5 to this family yields the follo wing f act. Corollary 4.1. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup. Then the fol lowing assertions ar e e quivalent. (i) G is r esidual ly discr ete. (ii) G is a total ly disc onne cte d SIN-gr oup. (iii) The c omp act op en normal su b g r oups form a b asis of identity neighb ourho o ds. Recall that a lo cally compact group is called a SI N-group if it p ossesses a basis of iden tit y neigh b ourho o ds whic h are inv ariant b y conjugation. Equiv alen tly , SIN-group s are th ose f or whic h the left and righ t u niform structur es coincide. A classical th eorem of F reuden thal and W eil ([F re36] and [W ei4 0 , § 32]; see also [Dix96, § 16.4.6]) states that a conn ected group is SIN if and only if it is of th e form K × R n with K compact (connected) and n ≥ 0. This complemen ts n icely the ab o ve corollary , imp lying easily that any lo cally compact SIN-group is an extension of a discrete grou p by a group K × R n ; th e latter consequence is Theorem 2.13(1) in [GM71]. Pr o of of Cor ol lary 4.1. The implications (iii) ⇒ (i) and (iii) ⇒ (ii) are immediate. (ii) ⇒ (iii) Since G is totally disconnected, the compact op en subgroups form a basis of ident it y neigh b ourho o ds [Bou71, I I I § 4 No 6]. By assu mption, giv en any compact op en subgroup U < G , there is an identit y neigh b ourho o d V ⊆ U wh ich is in v ariant by conjugation. The subgroup generated by V is thus norm al in G , op en since it con tains V and compact since it is cont ained in U . Thus (iii) holds indeed. (i) ⇒ (iii) Assume that G is residu ally discrete. Then G is totally disconnected and the compact op en su bgroups form a basis of iden tit y neigh b ourho o ds. Let V < G b e compact and op en and Q V ⊆ V d enote the compact normal subgroup pro vid ed b y Prop osition 2.5. By assumption w e ha ve T F = 1, where F denotes the collection of all op en norm al su b groups of G . W e claim that the qu otien t G/Q V is residually discrete. Indeed, for any x ∈ G , the family { ( x · N ) ∩ Q V } N ∈ F is ﬁltering and its in tersection coincides with { x } ∩ Q V . Since Q V is compact and since op en subgroup s are closed, it follo ws that for eac h x 6∈ Q V there exist ﬁn itely many elemen ts N 1 , . . . , N k ∈ F such that ( T k i =1 x · N i ) ∩ Q V = ∅ . Sin ce N 0 = T k i =1 N i b elongs DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 11 to F , w e hav e thus foun d an op en normal su bgroup N 0 of G suc h that x 6∈ N 0 · Q V . The pro jection of N 0 · Q V in the quotien t G/Q V is th u s an op en n ormal subgroup whic h a v oids the pro jection of x . This prov es the claim. Since the collection of all op en normal subgroup s of G/Q V forms a ﬁ ltering family , Prop o- sition 2.5 no w implies that G/Q V p ossesses some discr ete op en normal subgroup . T herefore G/Q V is itself discrete and hence Q V is op en in G . This sho ws that any compact op en sub- group V con tains a compact op en normal sub group Q V . T he desired conclusion f ollo ws.  Remark 4.2. Notice that a proﬁnite extension of a d iscr ete group is not necessarily residually discrete, as illustrated by the unr estricted wreath pro d uct ( Q Z Z / 2) ⋊ Z , w here Z acts by shifting indices. Ho w ever, if a totally disconn ected group G p ossesses a compact op en normal subgroup Q w hic h is top ologica lly ﬁ nitely generated, then G is r esidually discrete. Indeed, the proﬁnite group Q has then ﬁnitely man y s u bgroups of an y given ﬁ nite index and, hence, p ossesses a basis of identit y neighbour ho o ds consisting of char acteristic subgroup s . The discrete residual. W e recall that the discrete residual of a top ologica l group is the in tersectio n of all op en norm al subgroup s. Notice that the quotien t of a group by its discrete residual is resid ually discrete. Pr o of of The or em F. Let R 0 denote the discrete r esidual of G . Since G/R 0 is residually discrete, it follo ws from Corollary 4.1 that R 0 is conta ined co compactly in some op en normal subgroup of G . In view of th e hyp otheses, this sho ws that R 0 is cocompact, whence compactly generated. Let R 1 denote the discrete residual of R 0 . W e h a ve to sho w that R 0 = R 1 . S ince R 1 is c haracteristic in R 0 , it is normal in G . Observe that all s ub qu otien ts of G/R 1 considered b elo w are totally disconnected since the latter is an extension of the r esidually discrete groups R 0 /R 1 and G/R 0 . W e consider the canonical pro jection π : G → G/R 1 and deﬁ ne an intermediate c haracteristic s ubgroup R 1 ≤ L ≤ R 0 b y L = π − 1 (Rad LF ( R 0 /R 1 )) . Since the group R 0 /R 1 is resid u ally d iscrete, it follo ws from Corollary 4.1 that its LF-radical is op en. I n other words the sub quotien t R 0 /L is discrete. S ince it is co compact in G/L , it is moreov er ﬁn itely generated. It follo ws that the normal su bgroup Z := Z G/L ( R 0 /L ) is op en. By hyp othesis, it has ﬁnite index in G/L and con tains R 0 /L . In particular, it h as co compact cen tre and thus Z is compact-b y- Z m for some m , as is c heck ed e.g . as an easy case of T heorem 2.3, recalling that Z is totally disconnected. In conclusion, Z has a compact op en charac teristic sub group; the latter has ﬁnite index in G/L by assumption. Thus L is co compact in G , whence compactly generated. The top ologically lo cally ﬁnite group L/R 1 is thus compact (Lemma 2.3 in [Cap 09 ]). In particular R 1 itself is co compact in G . No w G/R 1 is a p roﬁnite group, thus residually discrete. This ﬁnally implies that R 0 = R 1 , as desired.  Quasi-discrete groups. W e end this section with an additional remark regarding the quasi- cen tre. W e shall sa y th at a top ologica l group is quasi-discrete if its quasi-cen tre is d en se. Examples of q u asi-discrete groups in clude discrete group s as wel l as proﬁnite group s wh ic h are direct pr o ducts of ﬁ nite groups. A connected group is qu asi-discrete if and only if it is Ab elian. The follo wing fact can b e extracted from the p r o of of Theorem 4.8 in [BEW08]; since th e argument is short we include it for the sak e of completeness. 12 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Prop osition 4.3. In any quasi-discr ete c omp actly ge ner ate d total ly disc onne cte d lo c al ly c om- p act gr oup, the c omp act op en normal sub gr oups form a b asis of identity neig hb ourho o ds. Th us a compactly generated totally disconnected lo cally compact group whic h is quasi- discrete satisﬁes the equiv alen t cond itions of Corollary 4.1. Pr o of. Let G b e as in the statemen t and U < G b e any compact op en subgroup . By compact generation, there is a ﬁn ite set { g 1 , . . . , g n } that, together with U , generates G . S in ce G is quasi-discrete, G = Q Z ( G ) · U and th u s we can assume that eac h g i b elongs to Q Z ( G ). Th e subgroup T n i =1 Z U ( g i ) < U is op en and hence cont ains a ﬁnite in dex op en su bgroup V whic h is normalised by U . T h us V is a compact op en normal subgroup of G con tained in U .  W e ﬁn ish this subsection b y recording t w o consequences of the latter for the sake of futur e reference. Corollary 4.4. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup without non-trivial c omp act normal sub gr oup. If G is quasi-discr ete, then the i dentity c omp onent G ◦ is op en, c entr al and isomorp hic to R n for some n . M or e over, we have G = Q Z ( G ) . Pr o of. W e ﬁrst obs erv e that G ◦ is cen tral; indeed, it is cen tralised by the dense sub group Q Z ( G ) since G ◦ is con tained in ev ery op en subgroup. By L emm a 2.2, the LF-radical of G ◦ is a compact norm al subgroup of G , and is th us trivial by h yp othesis; moreo ver, it follo ws that G ◦ is an Ab elian Lie group w ith ou t p erio d ic elemen t. T hus G ◦ ∼ = R n for some n . Since an y quotient of a quasi-discrete group remains quasi-discrete, Pr op osition 4.3 implies that the group of comp onent s G/G ◦ admits some compact op en n ormal sub group V . It now suﬃces to prov e V = 1 to establish the r emaining s tatement s. Denote b y N ✁ G the G ◦ -b y- V extension; it is compactly generated and has only compact conjugacy classes since G ◦ is cent ral. In particular, Theorem 2.3 guaran tees that Rad LF ( N ) is compact and that N / Rad LF ( N ) is Ab elian without compact s ubgroup. Since N is normal in G , it follo ws that Rad LF ( N ) is trivial and th u s ind eed N = G ◦ as r equired.  Corollary 4.5. L et G b e a c omp actly gener ate d total ly disc onne cte d lo c al ly c omp act gr oup admitting an op en quasi-discr ete sub gr oup. Then either G is c omp act or G p ossesses an inﬁnite discr ete quotient. Pr o of. Let H b e the giv en quasi-discrete op en subgrou p of G . Let h ∈ Q Z ( H ) b e an elemen t of the quasi-cen tre of H . Th en Z H ( h ) is op en in H , from whic h w e infer th at Z G ( h ) is op en in G and hence h ∈ Q Z ( G ). In particular, the closure Z = Q Z ( G ) is op en in G . If Z has inﬁnite index, then G/ Z is an inﬁn ite discrete quotien t of G . O therwise Z has ﬁnite ind ex and is th us compactly generated. By Prop osition 4.3, it follo ws th at Z p ossesses a compact op en normal subgroup. S ince Z has ﬁ nite in d ex in G , w e d educe that G itself p ossesses a compact op en normal su bgroup. The desired conclusion f ollo ws.  5. On structure the or y Quasi-simple quotien t s. Before u ndertaking the pro of of T heorems A and B , w e record one additional consequence of Pr op osition 2.5. Be fore stating it, we recall from the introd uction that a group is called quasi-simple if it p ossesses a cocompact normal subgroup which is top ologically simple and con tained in ev ery n on-trivial closed normal sub group of G ; in other w ords, a quasi-simple group is a monolithic group whose monolith is co compact and top ologica lly simp le. DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 13 Corollary 5.1. L et G b e a c omp actly gener ate d lo c al ly c omp act gr oup and { N v | v ∈ Σ } b e a c ol le ction of p airwise distinct close d normal sub gr oups of G such that for e ach v ∈ Σ , the quotient G/ N v is quasi-simple, non-discr ete and non-c omp act. Supp ose that T v ∈ Σ N v = 1 . Then Σ is ﬁnite. Pr o of. W e write H v := G/ N v . By hypothesis eac h H v is monolithic with simple co compact monolith, w hic h w e denote by S v . W e cl aim that G has no non-trivial compact normal subgroup. Let ind eed Q ✁ G b e a compact normal subgroup of G . By th e assu mptions made on H v , the image of Q in H v is trivial for eac h v ∈ Σ. Thus Q ⊆ T v ∈ Σ N v and hence Q is trivial. The same line of argument shows th at G h as no non-trivial soluble normal su bgroup. In particular, the iden tity comp onent G ◦ is a connected semi-simp le Lie group with trivial centre and no compact factor, see Lemma 2.2. Such a Lie group G ◦ is the direct pro d u ct of its simple factors. Moreo ve r, G has an op en charact eristic su b group of ﬁn ite ind ex which sp lits as a direct pro du ct of th e f orm G ◦ × D determining some compactly generated totally disconnected group D , see e.g. (the pro of of ) Th eorem 11.3.4 in [Mon01]. The id entit y comp onen t of eac h H v coincides with the image of G ◦ (since any quotien t of a totally disconnected group is totally d isconnected). Th us, w henev er H v is not totally disconnected, the hyp othesis implies S v = H ◦ v and H v is a proﬁnite extension of one of the simple factors of G ◦ , an d that eac h f actor app ears once. A t this p oint, w e can and s h all assu m e that G is totally disconnected. In view of Prop osition 2.4 and the assu mption made on H v , the group S v is n on-discrete for eac h v ∈ Σ . In p articular, it follo w s that H v has trivial qu asi-cen tre. Since the image of Q Z ( G ) in H v is con tained Q Z ( H v ), we d ed uce that Q Z ( G ) N v = N v for all v ∈ Σ. In other wo rds, w e ha ve Q Z ( G ) < T v ∈ Σ N v = 1 and w e conclude that G has trivial qu asi-cent re. Let now F b e the ﬁlter of closed normal subgroup s of G generated b y { N v | v ∈ Σ } . Since G has no compact non-trivial normal su bgroup and no non-trivial discrete n orm al subgroup (as Q Z ( G ) = 1), we deduce from Prop osition 2.5 that F is ﬁnite. Th us Σ is ﬁ nite as w ell, as d esired.  Maximal normal subgroups. W e sh all need the follo wing statemen t du e to R. Grigorc huk and G. Willis; since it is un published , w e p r o vide a pro of f or the reader’s con venience. Prop osition 5.2. L et G b e a total ly disc onne cte d c omp actly g e ner ate d non-c omp act lo c al ly c omp act gr oup. Then G admits a non-c omp act q u otient with every pr op er q u otient c omp act. Pr o of. By Zorn’s lemma, it suﬃces to prov e that for any chain H of non-cocompact closed normal subgroup s H ✁ G , th e group M = S H ∈ H H is still n on-cocompact. If not, then M is compactly generated. Th er efore, c ho osing a compact op en subgroup U < G , the c h ain { H .U } of groups is an op en co ve ring of M , whence there is H ∈ H with H .U ⊇ M . Then this H is co compact, w h ic h is absurd .  Remark 5.3. In view of the s tr ucture th eory of connected groups [MZ55], the ab ov e Prop o- sition h olds also true in the non-totally-disconnected case. The follo wing is a d u al companion to Prop osition 2.6. Additional information in th is direction w ill b e pro vided in P rop osition I I.1 in App end ix I I b elo w. 14 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Prop osition 5.4. L et G b e a c omp actly gener ate d total ly disc onne cte d lo c al ly c omp act gr oup which p ossesses no inﬁnite discr e te quotient, and let H = Res( G ) b e the discr ete r esidual of G . Then every pr op er close d normal sub g r oup of H i s c ontaine d in a maximal one, and the set N of pr op er maximal close d norma l sub gr oups is ﬁnite. Pr o of. By Theorem F , the discrete residual H is co compact in G , hen ce compactly generated. F urthermore it has no non-trivial ﬁnite qu otien t. Since H is total ly disconnected, an y compact quotien t wo uld b e pr oﬁnite, and w e inf er that H has no non-trivial compact quotien t. No w the same argumen t as in the pro of of Prop osition 5.2 using Zorn’s lemma sh o ws that ev ery prop er closed normal subgroup of H is conta ined in a maximal one. Let N denote the collectio n of all these. Any quotien t H / N b eing top ologically simple, hence quasi-simple, the ﬁniteness of N follo ws readily from Corollary 5.1.  Upp er and low er structure. Pr o of of The or em A. W e assum e throughout that G is non-compact since otherwise (ii) holds trivially . Assu me ﬁ rst that G is almost connected. In particular the n eutral comp on ent G ◦ coincides with the discrete resid ual of G . Let R denote the maximal connected soluble normal subgroup of G ; this soluble radical is ind eed w ell deﬁned ev en if G is not a Lie group as pro v ed b y K . I w asaw a (Theorem 15 in [Iwa4 9]; see also [P at88, (3.7)]. If R is co compact w e are in case (ii) of th e Theorem. Otherwise G is not amenable and using the stru cture theory of connected groups (n otably T heorem 4.6 in [MZ55]), we d educe that G ◦ /R p ossesses a non-compact (Lie-)simp le factor, so that all assertions of the case (iii) of th e Theorem are satisﬁed. W e no w assume that G is not almost connected. If G admits an inﬁnite d iscrete quotient w e are in case (i) of the Theorem. W e assume henceforth that G has no inﬁ nite d iscrete quotien t. In particular its discrete residual G + is co compact and admits neither n on-trivial discrete quotien ts nor d isconnected compact qu otients, see C orollary G. Moreo v er G + is compactly generated, n on-compact and con tains the identit y comp onent G ◦ . Let S b e the collection of all topologically simple quotient s of G + . Applyin g Corollary 5.1 to the quotient group G + /K , where K = T S ∈S Ker( G + → S ), w e deduce that S is ﬁn ite. Thus the assertion (iii) of Theorem A will b e established pro vided we sh o w th at S is non-empt y . T o this end , it suﬃces to p ro ve that the group of comp onen ts G + /G ◦ admits some n on- compact top ologically simple quotien t. But th is follo ws from Prop osition 5.4 since any quo- tien t of G + /G ◦ is non-compact and since eac h top ologica lly simple quotien t is aﬀorded by a maximal closed normal sub grou p .  Pr o of of The or em B. W e assum e that assertions (i) and (ii) of th e Theorem fail. Note that if G is totally disconnected, then P rop osition 2.6 ﬁn ishes the p r o of. As is w ell kno wn (see the pro of of Corollary 5.1), the non-existence of n on-trivial compact (resp. conn ected soluble) norm al su bgroups imp lies that G p ossesses a c haracteristic op en subgroup of ﬁ nite in dex G + < G w hic h splits as a direct pro du ct of the form G + = G ◦ × D , where G ◦ is a semi-simple Lie group and D is totally disconnected. Notice that G/G ◦ is a totally disconnected lo cally compact group whic h migh t p ossess non-trivial ﬁnite normal sub groups. In order to remedy this situation, w e s hall now exhib it a closed n orm al sub grou p G 1 ≤ G conta ining G ◦ as a ﬁnite index subgroup and su c h that G/G 1 has no non-trivial compact or d iscrete norm al subgroup . Let N be a closed normal su bgroup of G con taining G ◦ . Then N + = N ∩ G + is a ﬁnite index subgroup whic h d ecomp oses as a direct pro duct of the f orm N + ∼ = G ◦ × ( D ∩ N ). If DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 15 the image of N in G/G ◦ is compact (resp. discrete), then D ∩ N is a compact (r esp . d iscrete) normal su b group of G , and must therefore b e trivial, since otherwise assertion (ii) (resp. (i)) w ould hold true. W e deduce that an y compact (resp. discrete) normal subgroup of G/G ◦ is ﬁnite with order b oun ded ab o ve b y [ G : G + ]. In particular, there is a maximal such normal subgroup, and we denote by G 1 its p re-image in G . Since G 1 ∩ D injects into G 1 /G ◦ , it is a ﬁ nite normal sub group of G and m us t therefore b e trivial. Moreo v er G + = G ◦ D is closed in G . Thus D has closed image in G/G ◦ , whence in G/G 1 since the canonical pro jection G/G ◦ → G/G 1 is prop er, as it has ﬁnite k ern el. Th is implies that D G 1 is closed in G . In other w ords h D ∪ G 1 i is a c h aracteristic closed subgrou p of ﬁ nite index in G wh ic h is isomorp hic to G 1 × D . It follo ws at once th at th ere is a canonical one-to -one corresp ondence b etw een the closed normal sub groups of G contai ned in D and the closed normal sub groups of G/G 1 con tained in D G 1 /G 1 . No w G/G 1 is a compactly generated totally disconn ected lo cally compact group without non-trivial compact or discrete normal su bgroup, and Prop osition 2.6 guaran tees that the set M 1 of its non-trivial m in imal closed n orm al su bgroups is ﬁnite and non-emp t y . Moreo v er, an elemen t of M 1 do es not p ossess an y non-trivial ﬁnite index closed normal sub group and m ust ther efore b e conta ined in D G 1 /G 1 . Similarly , an y minimal closed normal sub group of G must b e con tained in G + . Since any minimal closed normal su b group of G is either connected or totally disconn ected, and since the connected ones are nothing but (regrouping of ) simple factors of G ◦ , w e ﬁ nally obtain a canonical one-to-o ne corresp ondence b etw een M 1 and the set of non-trivial minimal closed normal sub groups of G w hic h are totally disconnected. The desired conclusion follo w s since, as observed ab o v e, th e set M 1 is ﬁnite and non-empty .  Pr o of of Cor ol lary D. Assume G is not discrete. The discrete residual Res ( G ) is c harac- teristic. I f Res( G ) = 1 then G is residu ally discrete and h ence, its LF-radical is op en by Corollary 4.1 . Since the LF-radical is c haracteristic and G is n ot discrete, we dedu ce that G is top ologically lo cally ﬁn ite, hence compact since it is compactly generated. W e assume henceforth that Res( G ) = G and that G is not compact. The ab o ve argument sho ws moreo v er that G has tr ivial LF-radical and trivial qu asi-cen tre. If G is not totally disconnected, th en it is connected. If this is the case, th e LF-radical of G is compact (see Lemma 2.2) h ence trivial, and w e deduce that G is a Lie group . In this case, the standard s tructure th eory of connected Lie groups allo ws one to show that either G ∼ = R n or G is a direct p ro duct of pairwise isomorphic sim p le Lie groups. Assume ﬁnally that G is totally d isconnected. Th en Prop osition 2.6 guaran tees that the set M o f non-trivial minimal clo sed normal su bgroups of G is ﬁ nite and non-empty . Moreo ver, since for an y p rop er sub set E ⊂ M , the su bgroup h M | M ∈ E i is prop erly con tained in G , it follo ws that Au t( G ) acts transitiv ely on M . No w we conclude as in the pro of of Theorem E that G is a quasi-pro duct with the elemen ts of M as top ologica lly simple quasi-factors.  6. Composition series with topologicall y simple subquotients W e start with an elemen tary decomp osition result on qu asi-pro ducts. 16 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Lemma 6.1. L et G b e a lo c al ly c omp act gr oup which is a quasi-pr o duct with inﬁnite top olo gi- c al ly simple quasi-factors M 1 , . . . , M n . Then G admits a se quenc e of close d normal sub gr oups 1 = Z 0 < G 1 < Z 1 < G 2 < Z 2 < · · · < Z n − 1 < G n = G, wher e for e ach i = 1 , . . . , n , the sub gr oup G i is deﬁne d as G i = Z i − 1 M i , the sub quotient G i / Z i − 1 is top olo g i c al ly simple and Z i /G i = Z ( G/G i ) . Pr o of. The requested p rop erties of the normal series pro vide in fact a recursive deﬁnition for the closed normal subgroups Z i and G i . In particular all w e n eed to sh o w is that Z i − 1 is a maximal p r op er closed normal subgroup of G i . W e ﬁr st claim that M j ∩ G i − 1 = 1 for all 1 ≤ i < j , where it is understo o d that G 0 = 1. Since no M j is Ab elian, th is amounts to showing that G i − 1 ≤ Z G ( M i M i +1 . . . M n ). W e pro ceed b y induction on i , the base case i = 1 b eing trivial. No w w e n eed to s h o w that M i and Z i − 1 are b oth con tained in Z G ( M i +1 . . . M n ). This is clear for M i . By in d uction M i . . . M n maps on to a dense normal subgroup of G/G i − 1 . Sin ce Z i − 1 /G i − 1 = Z ( G/G i − 1 ) by deﬁnition, we infer that [ Z i − 1 , M j ] ≤ G i − 1 for all j ≥ i . Of course we ha ve also [ Z i − 1 , M j ] ≤ M j since M j is normal. The intersectio n G i − 1 ∩ M j b eing trivial by ind uction, we infer that [ Z i − 1 , M j ] is trivial as well. Th us M i and Z i − 1 are indeed b oth con tained in Z G ( M i +1 . . . M n ), and so is th us G i . The claim stands pr o ven. Let now N be a closed n ormal subgroup G suc h that Z i − 1 ≤ N < G i . By the claim we ha ve G i ∩ M j = 1 for all j > i , hence N ∩ M j = 1. Now if N ∩ M i 6 = 1, then M i < N sin ce M i is top ologica lly sim p le. In that case, we deduce that N con tains Z i − 1 M i , whic h con tr adicts that N is p rop erly con tained in G i . T h us we ha ve N ∩ M i = 1. In particular we d educe that N ≤ Z G ( M i M i +1 . . . M n ). S ince M i M i +1 . . . M n maps den s ely into G/G i − 1 , w e deduce that the image of N in G/G i − 1 is cen tral. By deﬁnition, this means that N is con tained in Z i − 1 , thereb y p ro v in g that Z i − 1 is ind eed maximal normal in G i .  Pr o of of The or em C. In view of th e stru cture theory of connected lo cally co mpact groups (see Lemma 2.2) and of connected Lie group s, the desired result holds in the connected case. Moreo v er, any homomorp hic image of a No etherian group is itself Noetherian. Th erefore, there is no loss of generalit y in replacing G by th e grou p of comp onents G/G ◦ . Equiv alent ly , w e sh all assume henceforth th at G is totally disconnected. W e ﬁrst claim that any closed normal subgroup of G is compactly generated. I n deed, giv en suc h a subgroup N < G , pic k an y compact op en subgroup U and consider the op en sub group N U < G . Since N is a co compact su bgroup of N U , wh ic h is compactly generated as G is No etherian, we in f er that N itself is compactly generate d, as claimed. Notice that the same prop erty is shared by closed normal sub groups of an y op en subgroup of G . Let no w Res( G ) denote the d iscrete residual of G . Thus G/ Res( G ) is residually discrete and Noetherian. Corollary 4.1 thus implies that the LF-radical of G/ Res( G ) is op en, while the ab o ve claim guaran tees that it is compact. W e denote b y O the pre-image in G of Rad LF ( G/ Res( G )). Thus O is an op en c haracteristic subgroup of G conta ining Res( G ) as a co compact subgroup . In particular, the discrete quotient s of O are all ﬁnite. No w Theorem F guaran tees th at Res( G ) = Res ( O ) has no non-trivial d iscrete quotient. Setting H = Res( G ), we ha ve thus far constru cted a series 1 < H < O < G of c haracteristic subgroups with O o p en and O /H compact. W e shall no w construct inductiv ely a ﬁnite increasing s equ ence 1 = H 0 < H 1 < H 2 < · · · < H l = H < O of n ormal sub groups of O satisfying the f ollo wing conditions for all i = 1 , . . . , l : DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 17 (a) If Rad LF ( H /H i − 1 ) is non-trivial, then Rad LF ( G/H i ) = 1. (b) H i /H i − 1 is either compact, or isomorphic to Z n for some n , or to a q u asi-pro duct with top ologic ally simple pairwise O -conjugate qu asi-factors. Let j > 0 and assume th at the ﬁ rst j − 1 terms H 0 , . . . , H j − 1 of the desired series ha v e already b een constructed, in such a wa y that prop erties (a) and (b) hold w ith i < j . W e pro ceed to deﬁn e H j as f ollo ws. If Rad LF ( H /H j − 1 ) is non-trivial, then we let H j b e the pre-image in H of Rad LF ( H /H j − 1 ). Prop erties (a) and (b) clearly hold for j = i in this case. Assume no w that Rad LF ( H /H j − 1 ) = 1 and that Q Z ( H /H j − 1 ) is non-trivial. Let M denote the closure of Q Z ( H /H j − 1 ) in H /H j − 1 . Thus M is charac teristic and q u asi-discrete. F urthermore, the fact that Rad LF ( H /H j − 1 ) = 1 implies that M has no non-trivial compact normal sub group. T h erefore Corollary 4.4 ensur es th at M = Q Z ( H /H j − 1 ) and that the iden tit y comp onen t M ◦ is op en and isomorphic to R n for some n . Since H is totally disconnected, it follo ws that M ◦ is trivial. Thus M is totally d isconnected as w ell, hen ce compact-b y-discrete in view of Pr op osition 4.3. But M h as n o non-trivial compact normal sub group since Rad LF ( H /H j − 1 ) is trivial and it follo ws th at M is d iscr ete. W e claim that M is Ab elian. Indeed, since M is discrete and ﬁnitely generated, its centraliser in H /H j − 1 is op en. By assu mption H h as no non-trivial discrete quotien t, and this p rop erty is inherited b y th e quotien t H /H j − 1 . W e dedu ce that Z H/H j − 1 ( M ) = H /H j − 1 ; in other wo rds M is cen tral in H /H j − 1 hence Ab elian, as claimed. Let M 0 denote the unique maximal fr ee Ab elian subgroup of M . Then M 0 is non-trivial since M is not compact. W e deﬁne H j to b e the pre-image of M 0 in H . Then H j is c h aracteristic and again, p rop erties (a) and (b) are b oth satisﬁed with i = j in th is case. It remains to deﬁne H j in the case where the LF-radical Rad LF ( H /H j − 1 ) and the quasi- cen tre Q Z ( H /H j − 1 ) are b oth trivial. In that case, Prop osition 2.6 guaran tees th at H /H j − 1 con tains some non -trivial minimal closed normal su bgroups of O/H j − 1 , sa y M , pro vided H /H j − 1 is non-trivial. C learly M is c h aracteristica lly simp le, so that, by Corollary D, it is a quasi-pro du ct with ﬁnitely many top ologically simple q u asi-factors. No w O acts transitivel y b y conju gation on these quasi-factors, otherwise M w ould cont ain a p r op er closed normal subgroup (see Prop osition 2.6), con tradicting minimalit y . It remains to d eﬁne H j as the pre-image of M in O . Hence (a) and (b) h old with i = j in all cases. W e hav e thus constructed an ascend in g c hain of subgroup s 1 = H 0 < H 1 < H 2 < · · · < H < O w hic h are all normal in O and w e pro ceed to show that H k = H for some large enough index k . Supp ose for a con tradiction that this is not the case and set H ∞ = S ∞ i =1 H i . Since H ∞ is normal in O , it is compactly generated (see the second paragraph of the presen t p ro of ab o ve). Let V < H ∞ b e a compact op en subgroup. Then the ascending c hain V · H 1 < V · H 2 < · · · yields a cov ering of H ∞ b y op en sub groups. The compact generation of H ∞ th us implies that V · H k = H ∞ for k large enough. In particular H k is co compact in H ∞ . Therefore Rad LF ( H /H k ) is non-trivial. By prop ert y (a), this implies that Rad LF ( H /H k +1 ) is trivial and h ence H ∞ ⊆ H k +1 . T his con tradiction establishes the claim. It only remains to show that the series of charact eristic subgrou p s 1 = H 0 < H 1 < H 2 < · · · < H l = H that we hav e constructed can b e reﬁn ed into a subn ormal series satisfying the desired conditions on the su b quotient s. By construction, it suﬃces to reﬁne the non-compact 18 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D non-Ab elian sub quotien ts H i /H i − 1 . S ince these are qu asi-pro ducts with ﬁnitely many top o- logica lly simple pairwise O -conjugate qu asi-factors, we may r eplace O b y an ap p ropriate closed normal subgroup of ﬁn ite index, say O ′ , in such a w ay that for all i , eac h top ologic ally simple quasi-factor of H i /H i − 1 is normal in O ′ /H i − 1 . Consider now the decomp osition of H i /H i − 1 pro vided by Lemma 6.1. E ac h term of this decomp osition is normal in O ′ /H i − 1 , and m ust therefore b e compactly generat ed. Therefore, the corresp ond in g su b quotients are compactly generated. In particular, the Ab elian su b quotients are compact- b y- Z k . In tro d ucing these in termediate term s in the series 1 = H 0 < H 1 < H 2 < · · · < H l = H < O ′ < O < G , we obtain a reﬁn emen t which has all the desired p rop erties.  Appendix I. The adjoint closure and asymptot icall y central sequen ces On t he Braconnier t op ology. Let G b e a lo cally compact grou p and Aut( G ) denote th e group of all homeomorphic automorph isms of G . There is a natur al top ology , s ometimes called the Braconnier top ology , turn ing Aut( G ) into a Hausd orﬀ top ologica l group; it is deﬁned by the sub-base of identit y neigh b ourh o o ds A ( K, U ) :=  α ∈ Aut( G ) | ∀ x ∈ K , α ( x ) x − 1 ∈ U and α − 1 ( x ) x − 1 ∈ U  , where K ⊆ G is compact and U ⊆ G is an id en tity neigh b ourho o d (see Chap. IV § 1 in [Bra48] or [HR79, Theorem 26.5]). In other wo rds, this top ology is the common reﬁnement of the compact-op en top ology for automorphisms and their in v ers es; r ecall in addition that a top ologic al group h as canonical uniform structures so that the compact-op en top ology coincides with the top ology of uniform con verge nce on compact sets ([Bra48] p. 59 or [Kel75] § 7.11). In f act, the Braconnier top ology coincides with the restriction of the g -top ology on the group of all homeomorphisms of G introdu ced b y Arens [Are46], itself hailing fr om Birkhoﬀ ’s C -con ve rgence [Bir34, § 11]. It can alternativ ely b e d eﬁned by restricting the compact-op en top ology for the Alexandroﬀ compactiﬁcation, an idea originating with v an Dan tzig and v an der W aerden [vDvdW28, § 6]. Braconnier shows by an example that the compact-open top ology itself is in general to o coarse to tu rn Aut( G ) in to a top ologic al group [Bra48 , pp. 57–58 ]. W e shall establish b elo w a basic disp ensation fr om this fact for the adjoin t representati on (Prop osition I.1). Meanwhile, w e recall that the Braconnier top ology coincides with the compact-op en top ology when G is compact (Lemma 1 in [Are46 ]) and w h en G is lo cally conn ected (Theorem 4 in [Are46]). T here are of course non-lo cally-connecte d connected groups: the solenoids of Vietoris [Vie27, II ] and v an Dantz ig [vD30, § 2 Satz 1]. Nev ertheless, u sing notably the solution to Hilb ert’s ﬁ fth problem, S.P . W ang show ed that the t wo top ologies still coincide for all connected and indeed almost co nnected locally compact g roups [W an69 , Corolla ry 4.2 ]. Finally , the top ologies coincide for G discrete and G = Q n p , s ee [Bra48, p. 58]. W e emphasise that the Braconnier top ology on Aut( G ) need not b e lo cally compact , see [HR79, § 26.18.k]. A criterion ensu ring that Au t( G ) is locally compact will b e presente d in Th eorem I.6 b elo w in the case of totally disconnected groups. Nev ertheless, Aut( G ) is a Poli sh (hence Baire) group when G is second countable. Ind eed, it is b y deﬁnition closed (ev en f or the w eak er p oint w ise top ology) in the group of homeomor- phisms of G endo wed with Arens’ g -top ology; the latter is second countable (see e.g. [GP57 , 5.4]) and complete for the bilater al un if orm structur e [Are46, Th eorem 6]. Notice that this DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 19 complete uniformisation is n ot the usual left or r igh t uniform str u cture, which is kno w n to b e sometimes incomplete at least for th e group of homeomorphisms (Arens, lo c. cit. ). The Baire pr op ert y im p lies f or instance that Aut( G ) is discrete wh en countable, whic h w as observ ed in [Pla77, Satz 2] for G itself discrete. Adjoin t representation. Given a closed n ormal subgroup N < G , the conjugation action of G on N yields a m ap G → Aut( N ) w hic h is con tinuous (see [HR79, Theorem 26.7]). In particular, th e natural map Ad : G → Aut( G ) ind uced by the conjugation action is a con tinuous h omomorp hism. W e endo w the group Ad( G ) < Aut( G ) with the Brac onnier top ology . Th u s, a sub -b ase of iden tit y n eigh b ourho o d s is giv en by the image in Ad( G ) of all subsets of G of the form B ( K, U ) :=  g ∈ G | [ g , K ] ⊆ U and [ g − 1 , K ] ⊆ U  , where ( K , U ) r uns ov er all pairs of compact subsets and identit y neigh b ourho o ds of G . As an ab s tract group , Ad ( G ) is isomorphic to G/ Z ( G ); we emphasise ho wev er that the latter is endow ed with the generally ﬁner quotien t topology . Prop osition I.1. L et G b e a lo c al ly c omp act gr oup such that the gr oup of c omp onents G/G ◦ is u nimo dular. Then the Br ac onnier top olo gy on Ad( G ) c oincides with the c omp act-op en top olo gy. Pr o of. Let { g α } α b e a n et in G suc h th at Ad( g α ) con v erges to the iden tit y in the compact - op en top ology . According to the result of S .P . W ang quoted earlier in this section, the automorphisms Ad( g α ) | G ◦ of th e identit y comp onen t G ◦ con verge to the iden tit y for the Braconnier top ology on Au t( G ◦ ). According to Pr op osition 2.3 in [W an69], it no w su ﬃces to pr o ve that th e ind uced automorphisms on G/G ◦ also co n v erge to the iden tit y for the Braconnier top ology on Aut( G/G ◦ ). Therefore, we can supp ose h enceforth that G is totally disconnected. By assump tion, { g α } ev en tually p enetrates ev ery set of the form B ′ ( K ′ , U ′ ) := { g ∈ G | [ g , K ′ ] ⊆ U ′ } , where K ′ ⊆ G is compact and U ′ ⊆ G is a neighbourh o o d of e ∈ G . Thus it suﬃces to sh o w that for all K ⊆ G compact and U ⊆ G identi t y neigh b ourho o d, th er e is K ′ and U ′ with B ′ ( K ′ , U ′ ) ⊆ B ( K, U ) . Since G is totally disconnected, there is a compact op en sub group U ′ < G con tained in U . Set K ′ = K ∪ U ′ and ﬁ x any g ∈ B ′ ( K ′ , U ′ ). W e need to sh o w that [ g − 1 , K ] ⊆ U . First, notice that [ g − 1 , K ] = g − 1 [ g , K ] − 1 g . Next, [ g , K ] and h ence also [ g , K ] − 1 is in U ′ . Finally , [ g , U ′ ] ⊆ U ′ means that g U ′ g − 1 ⊆ U ′ ; by un imo dularity , it follo ws that g normalises U ′ . W e conclude that [ g − 1 , K ] ⊆ U ′ ⊆ U , as was to b e sh own.  A lo cally compact group G for which the map Ad : G → Ad( G ) is closed w ill b e called Ad-closed . In that case, Ad( G ) is isomorph ic to G/ Z ( G ) as a topological group and th us in p articular it is lo cally compact. The group G can fail to b e Ad -clo sed even w hen it is a connected Lie group (Example I.3 b elo w ; s ee also e.g. [L W7 0, Zer76]). P erhaps more strikingly , G can fail to b e Ad -closed ev en when counta ble, discrete and Z / 2 Z -b y-ab elian [W u71 , 4.5]. 20 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Asymptotically cen tra l sequences. Let G b e a lo cally compact group. A sequence { g n } of elemen ts of G is called asymptotically cen tra l if Ad( g n ) con verge s to the ident it y in Ad( G ). Ob vious examples are cent ral sequences or sequences conv erging to e ; w e shall in v estigate the existence of non-ob vious ones (for an admittedly limited analogy , compare the pr op erty Γ in tro duced f or I I 1 -factors by Murra y and v on Neumann, Deﬁn ition 6.1.1 in [MvN43]). The existence of s u itably n on-trivial asymptotic ally cen tral sequences is related to the question whether the Braconnier top ology on G (strictly sp eaking, on G/ Z ( G )) coincides with the initial top ology , as follo ws. Prop osition I.2. L et G b e a se c ond c ountable lo c al ly c omp act gr oup. The fol lowing c onditions ar e e quivalent. (i) Ad( G ) is lo c al ly c omp act. (ii) The c ontinuous homom orphism Ad : G → Ad( G ) is close d. (iii) The map G/ Z ( G ) → Ad( G ) is a top olo gic al gr oup isomo rphism. (iv) The image in G/ Z ( G ) of e very asymptotic al ly c entr al se quenc e is r elatively c omp act. A suﬃcient c ondition for th is is tha t G ad mits some c omp act op en sub gr oup U such that N G ( U ) is c omp act. Pr o of. (i) ⇒ (ii) This is a we ll-kno wn application of the Baire catego ry principle, going bac k at least to [P on39, T h eorem XI I I]. (ii) ⇒ (iii) and (iii) ⇒ (iv) follo w f r om the deﬁn itions. (iv) ⇒ (i) Let { K n } b e an increasing sequ ence of compact subsets of G whose un ion co vers G and let { U n } b e a decreasing family of sets providing a basis of neigh b our ho o ds of e ∈ G . Assuming for a con tradiction that Ad( G ) is not lo cally compact, n one of the sets B ( K n , U n ) can h a ve a relativ ely compact image in Ad( G ). Therefore, we can choose for eac h n an elemen t g n in B ( K n , U n ) b u t not in K n . Z ( G ). By construction, the sequence { g n } is un b ounded in G/ Z ( G ) but Ad( g n ) conv erges to the iden tit y , a con tradiction. Finally , notice that if U is a compact op en subgroup of G , then N G ( U ) = B ( U, U ). This sho ws that if N G ( U ) is compact, then Ad( G ) admits Ad( B ( U, U )) as a compact iden tit y neigh b ourho o d.  W e recall f rom [KK44] that a σ -compact lo cally compact group G alw ays p ossesses a compact normal subgroup Q s uc h that the quotient G/Q is metrisable. In p articular, an y compactly generated lo cally compact group without non-trivial compact norm al sub group satisﬁes th e hypotheses of Prop osition I.2 . The follo win g construction pro vides examples of Lie groups whic h are not Ad -closed. Example I.3 . Let L ∼ = R < R 2 b e a one-parameter subgroup with irrational s lop e and denote b y Z the image of L in the torus T 2 = R 2 / Z 2 . Thus Z is a conn ected dense subgroup of T 2 . Let us now choose a con tin u ous faithful r ep resen tatio n of T 2 in O (4) and consider the corresp ondin g semi-dir ect pro du ct H = T 2 ⋉ R 4 . W e deﬁn e G = Z ⋉ R 4 . Thus G is a connected su bgroup of the Lie group O (4) ⋉ R 4 . W e claim that G is n ot Ad -closed. Indeed, let ( z n ) b e an unb ounded sequence of elemen ts of Z whic h conv erge to 1 in the torus T 2 . O n e v eriﬁes easily that G is cen trefree and that the ab o v e sequence is asymptotically cen tral in G . This yields the desired claim in view of Prop osition I.2 . DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 21 An illustration of the relev ance of the notion of Ad -clo sed groups is pro vided by the follo w- ing. Lemma I.4. L et G b e a lo c al ly c omp act gr oup and H < G b e a close d sub gr oup. If H is Ad -close d, then H. Z G ( H ) is close d in G . Pr o of. Without loss of generalit y , w e may assu me that H . Z G ( H ) is d ense in G . Then H is normal and there is a contin uous conjugation action α : G → Aut( H ). Since H . Z G ( H ) is dens e, it follo ws that Ad( H ) is dense in α ( G ). No w Ad( H ) b eing closed in Aut( H ) b y h yp othesis, we infer that α ( G ) = Ad( H ). The r esult follo ws, since the pre-image of Ad( H ) in G is nothing but H . Z G ( H ).  The adj oint closure. The closure of Ad( G ) in Aut( G ) will b e called the adjoin t closure of G and will b e denoted by Ad( G ). W e think of an automorphism in Ad( G ) as “appro ximately inner”. W e p oin t out that Ad( G ) is n ormal in Aut( G ) and h ence in particular in Ad( G ). Basic prop er ties of the adjoint closur e are summ arised in the follo wing. Notice that Ad( G ) is not assumed lo cally compact except in the last item. Lemma I.5. L et G b e a lo c al ly c omp act gr oup. (i) If G is c entr efr e e, then so is Ad( G ) . (ii) If G is top olo gic al ly simple, then so is Ad( G ) . (iii) If G is total ly disc onne cte d, then so is Ad( G ) . (iv) Supp ose that Ad( G ) is lo c al ly c omp act. If G is c omp actly gener ate d, then so is Ad( G ) . Pr o of. (i) Giv en α ∈ Z Aut( G ) (Ad( G )), w e ha v e α ( g ) xα ( g ) − 1 = g xg − 1 for all g , x ∈ G . Th u s α ( g ) − 1 g b elongs to Z ( G ) and the result follo ws. (ii) Let H < Ad( G ) b e a closed norm al su bgroup and let H 0 = Ad − 1 ( H ) b e the pre-image of H in G . Then H 0 is a closed normal subgroup of G and is thus trivial of the whole group. If H 0 = G , then H con tains Ad( G ) wh ich is d ense, thus H = G as w ell. If H 0 = 1, then H ∩ Ad( G ) = 1. This implies that [ H , Ad( G )] ⊆ H ∩ Ad( G ) = 1. Thus H comm u tes with the d ense sub grou p Ad( G ) and is th us con tained in the cen tre of Ad( G ), whic h is trivial by the assertion (i). (iii) See [Bra48, IV § 2] or [HR79, Theorem 26.8]. (iv) Let U b e a compact neighbour ho o d of the iden tity in Ad( G ) and C ⊆ G a compact generating set. Th en U. Ad ( C ) generates Ad( G ).  Lo cally ﬁnitely generated groups. W e shall sa y that a totally d isconnected lo cally com- pact group G is lo cally ﬁnit ely generate d if G admits some compact op en su bgroup that is top ologica lly ﬁnitely generated, i.e. p ossesses a ﬁ nitely generated dense sub grou p . Since an y t wo compact op en subgroups of G are commensurable, it follo ws that G is lo cally ﬁnitely gen- erated if and only if any compact op en sub grou p is top ologically ﬁnitely generated. Examples of suc h include p -adic analytic groups (see e.g. [DdSMS99, Theorem 8.36]), man y complete Kac–Moo d y groups o ver ﬁnite ﬁelds [CER08, Theorem 6.4] as we ll as sev eral (but not all) lo cally compact group s acting prop erly on lo cally ﬁnite trees [Moz98 ]. An imp ortan t prop ert y of ﬁnitely generated proﬁnite groups is th at they admit a (coun t- able) basis of id en tity neighb ou r ho o ds consisting of charact eristic subgroup s, b ecause they ha ve only ﬁnitely man y closed subgroups of an y giv en index. Lo cally ﬁ nitely generated groups are thus co vered b y the follo w ing r esult. 22 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Theorem I.6. L et G b e a total ly disc onne cte d c omp actly gener ate d lo c al ly c omp act gr oup. Supp ose that G admits an op en sub gr oup U that has a b asis of identity nei ghb ourho o ds c on- sisting of char acteristic sub gr oups of U ( e.g. G is lo c al ly ﬁnitely gener ate d). Then Aut( G ) is lo c al ly c omp act. The pro of will use th e follo wing v ersion of the Ar zel` a–Ascoli Th eorem. Prop osition I.7. L et G b e a lo c al ly c omp act gr oup and V ⊆ Aut( G ) a subset such that (i) V = V − 1 ; (ii) G has arbitr arily smal l V -invariant identity neighb ourho o ds; (iii) V ( x ) is r elatively c omp act in G for e ach x ∈ G . Then V is r elatively c omp act in Aut( G ) . In the case w here V is a compact s u bgroup of Aut( G ), this is Th eorem 4.1 in [GM67 ]. Pr o of of Pr op osition I.7. P oin t (ii) implies that V is equicont in u ous (in fact, uniformly equicon- tin u ous). Therefore, we can apply Arzel` a– Ascoli (in the generalit y of [Bou74], X § 2 No. 5) and deduce th at V has compact closure in the space of con tin uous maps G → G endow ed with the compact-op en top ology (whic h, as m en tioned, coincides with the topology of com- pact con v ergence). The closure of V remains in the space of con tinuous endomorphism s since the latter is closed ev en p oin twise. In view of th e symmetry of the assump tions and of th e fact that comp osition is cont in uous in the compact-op en top ology [Dug66, XI I.2.2], the closure of V remains in Aut( G ) and is compact for the Braconnier top ology .  Pr o of of The or em I.6. Let U < G b e an op en subgroup admitting a basis of iden tit y neigh- b ourh o o ds { U α } α consisting of charact eristic subgroups of U . W e can assume U compact up on in tersecting with a compact op en subgroup. Let C ⊆ G b e a sym metric compact set generating G and con taining U . W e shall pr o ve that V := A ( C , U ) ⊆ Aut( G ) satisﬁes th e assumptions of Prop osition I.7; this then establishes the theorem. The ﬁ rst assum ption holds by deﬁnition. F or the second, notice ﬁrst that V norm alises U since U ⊆ C implies V ⊆ A ( U, U ) = N Aut( G ) ( U ) . Assumption (ii) holds since the identit y neighbour ho o ds U α are charact eristic, hence n or- malised by N Aut( G ) ( U ). In order to establish th e last assumption, c ho ose x ∈ G . S ince C is generating and sym- metric, there is an in teger d su c h that x ∈ C d . T he deﬁnition of V s ho w s that for an y automorphism n ∈ V , we ha ve n ( x ) ∈ ( U.C ) d ; this implies (iii).  The adj oint closure of discrete groups. A particularly simple illustration of th e concepts in tro duced ab ov e is pro vided by discrete groups. The Braconnier topology on Aut( G ) is then the top ology of p oint w ise conv ergence and coincides with p oin t wise con ve rgence of the in v erse. The adjoin t closure Ad( G ) coincides therefore with th e group Lin n( G ) of lo cally inner automorphisms , i. e . auto morphisms that coincide on ev ery ﬁnite set with some inner automorphism. This concept w as apparen tly ﬁrst in tro duced ( lok al~no vnutrennim ) by Gol’b erg [Gb46 , § 3 Opredelenie 5 ]. Here are a few element ary pr op erties of the resu lting corresp ond ance G 7→ Ad( G ) fr om abstract (resp . counta ble) groups to top ological (resp. P olish) groups. Prop osition I.8. L et G b e a discr e te gr oup and A = Ad( G ) = Linn( G ) its adjoint closur e. DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 23 (i) Ad( G ) ≤ Q Z ( A ) . In p articuylar A is qu asi-discr e te; it is discr ete if and only if ther e is a ﬁnite set F ⊆ G with Z G ( F ) = Z ( G ) . (ii) A is c omp act if and only if G is F C 1 . (iii) A is lo c al ly c omp act if and only if ther e is a ﬁnite set F ⊆ G such that the index [ Z G ( F ) : Z G ( F ′ )] is ﬁnite for every ﬁnite F ′ ⊇ F in G . (iv) A is lo c al ly c omp act and c omp actly ge ne r ate d if and only if ther e is F as in (iii) and F 0 ⊆ G ﬁnite such tha t F 0 ∪ Z G ( F ) gener ates G . Pr o of. All ve riﬁcations are straigh tforward. On e uses n otably an elemen tary v ersion of Prop o- sition I.7 stating that, for G discrete, a s ubset V ⊆ Aut G has compact closure if and only if V ( x ) is ﬁnite for all x ∈ G .  Discrete groups are a safe pla ygroun d to exp erimen t with in termediate top ologies inb et wee n the original top ology and the Braconnier top ology induced via the adjoint representa tion. The follo wing construction will lead to interesting examples, see Ap p endix I I and esp ecially Example I I.7. Let N b e a discrete group and U < N a sub group such that (i) Z U ( g ) has ﬁnite ind ex in U for all g ∈ N ; (ii) the intersect ion of all N -conjugates of U is trivial. In particular, the N -conjugates of U generate a completable group top ology [Bou60, TG I I I, § 3, No 4] and N injects into the resu lting complete totally disconn ected top ological group M . Prop osition I.9. (i) The gr oup M is lo c al ly c omp act; in fact U has c omp act-op en closur e in M . (ii) Ther e i s a (ne c essarily unique) c ontinuous i nj e c tive homomorph ism M / Z ( M ) → Ad( N ) c omp atible with the maps N → M and N → Ad( N ) . In p articular, the dense image of N in M i s normal and quasi-c entr al (thus M is quasi-discr ete). (iii) If N is c entr efr e e (r esp. simple), then M / Z ( M ) is c entr efr e e (r esp. top olo gic al ly simple). (iv) M / Z ( M ) = Ad( N ) if and only if Z N ( F ) ⊆ U for some ﬁnite F ⊆ N . Pr o of. The ﬁrst assertion is d ue to th e fact that the closur e of U in M is a quotien t of the pr oﬁnite completion of U . The second assertion follo ws from the fact that a system of neigh b ourho o ds of the iden tit y for the M -top ology on N is giv en by U ∩ V , where V ranges o ver the Ad( N )-neigh b ourho o d s of the iden tit y . The third assertion follo ws b y the same argumen t as in the pro of of Lemma I.5. F or the last assertion, observ e th at M = Ad( N ) if and only if U is op en in th e Braconnier top ology on N .  Example I.10 . Let Ω b e a coun tably inﬁnite set, let N < Sym(Ω) b e an in ﬁnite (almost) simple group of alternating ﬁnitary p erm uta tions , i.e. p ermutat ions with ﬁnite sup p ort. Cho ose also an equiv alence relation ∼ on Ω all of w hose equiv alence classes ha v e ﬁ nite cardinalit y , and let U < N b e an inﬁ nite subgroup p reserving eac h equiv alence class. F or eac h g ∈ N , there is a ﬁnite index subgrou p U ′ < U suc h that g and U ′ ha ve d isjoin t supp ort. Therefore Z U ( g ) has ﬁ nite ind ex in U for all g ∈ N . Moreo ver the in tersection of all N -conjugates of U is trivial since N is almost simple. 1 Recall th at G is an FC-group if all its conjugacy classes are ﬁnite. 24 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D Concretely , one could d eﬁne N as the group of all alternating ﬁ nitary p erm utation, whic h is ind eed simp le, and deﬁne U as the subgroup preserving all equiv alence classes of a relation ∼ whic h is a partition into subsets of ﬁxed size k > 1. The group U is then isomorphic to a restricted direct sum of ﬁ nite alternating groups of d egree k and M is a totally disconnected lo cally compact grou p whic h is top ologic ally simple, top ologically lo cally ﬁnite and qu asi- discrete. (Here M is centrefree b ecause ev ery asymp totica lly cen tral sequence of N con verges p oint wise to the identi t y .) W e remark that the examples of top ologically simple lo cally compact groups adm itting a d ense n ormal s ubgroup which were constructed by G. Willis in [Wil07, § 3] all ﬁt in this set-up, and can all b e obtained by taking v arious sp ecializati ons of the groups N < S ym(Ω) and U < N . Remark I.11. Th e p revious example tak es adv antag e of the fact that the group of all ﬁ n itary p ermuatio ns of a counta bly in ﬁnite set Ω is not Ad -closed. Notice how ev er that its adjoin t closure, which incidentall y coincides with the group Sym(Ω) of al l p erm utations of Ω , is ho w- ev er not locally compact. Proposition I.9 and Example I.10 th us corresp ond to completions whic h are gen uinely intermediate b et w een Ad( N ) and Ad( N ). Th is is an instance of a general sc h eme that we shall present b elo w, see Prop osition I I.5. Appendix I I. Quasi-products a nd dense normal subgroups F or general lo cally compact group s, there is a naturally o ccurr ing stru cture that is weak er than direct p ro ducts. W e establish its basic p rop erties and giv e some examples. In order to a voi d some obvi ous d egeneracies, it is go o d to hav e in mind the cen trefree case. Deﬁnitions and the Galois connection. Let G b e a top ologica l group. W e call a closed normal su bgroup N ✁ G a quasi-factor of G if N . Z G ( N ) is dense in G . In other wo rds, this means that the G -a ction on N is “appro ximately in n er” in the sense that the image of G in Aut( N ) is con tained in the adjoin t closur e Ad( N ) . If N is a quasi-factor, then N ∩ Z G ( N ) is con tained in the cen tre of G . T h us, in the cen trefr ee case, quasi-factors pr ovide an example of the follo wing concept with p = 2: W e say th at G is the quasi-pro duct of th e closed normal su bgroups N 1 , . . . , N p if the m ultiplicatio n map N 1 × · · · × N p − → G is injectiv e with dense image. W e call th e groups N i the quasi-factors of this qu asi-pro duct; notice that N i and N j comm u te for all i 6 = j and therefore eac h N i is ind eed a quasi-factor in the earlier sense. Giv en a quasi-pro d uct, one h as a family of quotien ts G ։ S i deﬁned by S i = G/ Z G ( N i ). Notice that the image of N i in S i is a dense n ormal sub group; moreo ver, when G is centrefree, N i injects into S i . Therefore, we obtain an injection with dense image: G/ Z ( G ) − → S 1 × · · · × S p . (The relation b et ween quasi-pr o ducts an d d ense norm al subgroup s will b e further inv estigated b elo w ; s ee Example I.10.) The map N 7→ Z G ( N ) is an an titone Galois connection on the set of closed normal sub- groups of G and in particular also on the collectio n of quasi-factors. It turns out that this corresp ondence b eha ves p articularly w ell for certain group s app earing in the main r esu lts of DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 25 this article, as follo ws. Denote by Max (resp. Min ) the s et of all maximal (resp . minimal) closed n orm al su bgroups whic h are non-trivial. Prop osition I I.1. L et G b e a non-trivial c omp actly gener ate d tot al ly disc onne cte d lo c al ly c omp act gr oup. Assume that G is c e ntr efr e e and without non-trivial discr ete quotient. If T Max is trivial, then the fol lowing hol d. (i) Min and Max ar e ﬁnite and non-empty. (ii) The assignment N 7→ Z G ( N ) deﬁnes a bije ctive c orr esp ondenc e fr om Min to Max . (iii) Every element of Min ∪ Max is a quasi-f actor. (iv) G is the quasi-pr o duct of its minimal normal sub gr oups. This result provides in particular additional information on c haracteristical ly simple groups, whic h sup plemen ts Corollary D. Ind eed, the h yp otheses of the p rop osition are in p articular fulﬁlled by charac teristically simple groups falling in case (iv) of Corollary D (see also Prop o- sition 5.4). Pr o of of Pr op osition II.1. Notice th at G has no non-trivial compact quotient since ev ery dis- crete quotie n t is tr ivial. The set Max is non-empt y since it has trivial in tersection; its ﬁniteness follo w s from Theorem A. Moreo v er, G em b eds in th e pro du ct of simple groups Q K ∈ Max G/K , whic h implies that G has tr ivial quasi-cen tr e and no non-trivial compact n or- mal subgroup. Ind eed, otherwise some G/K would b e compact or quasi-discrete (as in the pro of of C orollary 5.1). In particular, Th eorem B implies that Min is ﬁn ite and non-empty; assertion (i) is established. Actually , we sh all use b elo w not only that Min is non-empt y , but that every non-trivial closed normal sub group of G conta ins a minimal one, see Prop osi- tion 2.6. F or the d uration of the pro of, den ote b y Max QF the subset of Max consisting of those elemen ts whic h are quasi-factors of G . We claim tha t the map N 7→ Z G ( N ) deﬁnes a one-to-one c orr e sp ondenc e of Min onto Max QF . Mor e over, every element of Min ∪ Max QF is a quasi-factor. Let N ∈ Min . By hyp othesis there is some K ∈ Max whic h d o es not con tain N . By minimalit y of N we deduce that N ∩ K is trivial and hence [ N , K ] = 1. T h erefore N .K is dense in G b y maximalit y of K . In particular, N and K are b oth quasi-factors. Moreo ver, since Z G ( N ) con tains K , maximalit y implies K = Z G ( N ) b ecause G is cent refree. In other w ords, N 7→ Z G ( N ) deﬁ n es a map Min → Max QF . Sin ce any minimal close d normal subgroup of G diﬀerent fr om N commutes with N , it is con tained in K . Therefore, the ab o v e map is an injection of Min in to Max QF . It remains to sh o w that it is su rjectiv e. T o this end, pic k K ∈ Max QF . Then Z G ( K ) is a non-trivial closed normal su bgroup of G . It therefore cont ains an elemen t of Min , sa y N . By d eﬁ nition K is con tained in Z G ( N ), whence K = Z G ( N ) by maximalit y . Th e claim s tands p r o ven. We claim that every element of Max \ Max QF c ontains eve ry element of Min . If K ∈ Max and N ∈ Min are suc h that N 6≤ K , th en N and K comm u te and , hence, K = Z G ( N ). Thus K ∈ Max QF b y the previous claim. We claim that T K ∈ Max QF K = 1 . Otherwise T K ∈ Max QF K would con tain some N ∈ Min , which is also con tained in ev ery L ∈ Max \ Max QF b y the p revious claim. This con tradicts the hyp othesis that T Max is trivial. 26 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D We claim that G = [ G, G ] . By h yp othesis G h as no non-trivial discrete quotient . This pr op ert y is clearly in h erited by an y quotien t of G . Therefore, the claim follo ws fr om the fact that the only tota lly d isconnected lo cally compact Ab elian group with that p rop ert y is the trivial group . We claim that G = h N | N ∈ Min i . Set H = h N | N ∈ Min i and A = G/H . It follo ws from the ﬁrst claim ab o ve that ev ery elemen t K ∈ Max QF has dense image in A . In view of the third claim ab o ve, we infer that A admits d ense normal subgroups L 1 , . . . , L p with trivial in tersection. In v iew of Lemma I I.2 b elo w , it follo ws that A is nilp otent , h ence trivial by the p revious claim. We claim that Max = Max QF . Indeed, ev ery element of Max \ Max QF con tains ev ery elemen t of Min . By the p revious claim, this implies that eve ry elemen t of Max \ Max QF coincides with G and th us fails to b e a n on-trivial sub group. No w assertions (ii), (iii) and (iv) follo w at once.  Lemma I I.2. L et A b e a Hausdorﬀ top olo gic al gr oup c ontaining p dense normal sub gr oups L 1 , . . . , L p such that T p i =1 L i = 1 . Then A is nilp otent of de gr e e ≤ p − 1 Pr o of. F or eac h j = 1 , . . . , p , we set M j = T p i = j L i . I n particular M 1 is trivial and M p = L p . Set A i = A/ M i for all i = 1 , . . . , p . W e h av e a c h ain of con tin u ous surjectiv e maps A ∼ = A 1 → A 2 → · · · → A p − 1 → A p ∼ = 1 . Let i < p . S ince M i = L i ∩ M i +1 , it follo ws that the resp ectiv e images of L i and M i +1 in A i comm u te. Since moreo v er L i is dens e in A , it m ap s densely in A i and we dedu ce th at the image of M i +1 in A i is cen tral. In particular A i is a cen tral extension of A i +1 . I t readily follo ws that the upp er central series of A term in ates after at most p − 1 steps .  On the non-Hausdorﬀ quotients of a quasi-product. Th e follo win g resu lt describ es the algebr aic structur e of the generally n on-Hausdorﬀ quotient G/ N 1 · · · N p (its top ological structure b eing tr ivial). It applies in p articular to the case of totally disconnected groups that are No etherian. Prop osition I I.3. L et G b e a tota l ly disc onne cte d lo c al ly c omp act gr oup that is a q uasi- pr o duct with quasi-factors N 1 , . . . , N p . (i) If N i p ossesses a maximal c omp act sub gr oup U i for some i ∈ { 1 , . . . , p } , then for e ach c omp act sub gr oup U < G c ontaining U i , the quotient U /U i . ( U ∩ Z G ( N i )) is Ab elian. (ii) If N i p ossesses a maximal c omp act sub gr oup U i for e ach i ∈ { 1 , . . . , p } , then the quotient G/ Z ( G ) .N 1 · · · N p is Ab elian. Pr o of. Let U i < N i b e a maximal compact subgroup and let U < G b e a compact op en subgroup con taining U i . S ince U ∩ N i is a compact sub group of N i con taining U i , we ha ve U ∩ N i = U i b y maximalit y . Let also Z i = Z G ( N i ). W e shall ﬁ rst show that U /U i . ( U ∩ Z i ) is Ab elian, wh ich is the assertion (i). In order to establish this, consider the op en su bgroup H i := U.N i . Sin ce U i is a maximal compact subgroup of N i , it follo ws that U is a maximal compact su bgroup of H i . Notice th at H i ∩ Z i is centralise d by a co compact sub group of H i , namely N i . Therefore H i ∩ Z i is a topologically F C -group, indeed th e H i -conjugacy class of ev ery element is relativ ely DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 27 compact. By U ˇ sako v’s result [U ˇ sa63] (see Theorem 2.3 ab ov e), the set Q of all its p erio dic elemen ts coincides with the LF-radical and the quotien t ( H i ∩ Z i ) /Q is torsion-free Ab elian (and discrete b y total disconn ectedness). Since H i ∩ Z i is n ormal in H i , it follo ws that Q is normalised b y U . Thus we can f orm the sub group U · Q of H i , w hic h is top ologically lo cally ﬁnite and hence compact since U was maximal compact in H i . This implies Q ≤ U a nd in particular w e hav e Q ≤ U ∩ Z i =: V i . Since V i is normalised by U and cen tralised by N i , it is a compact normal sub grou p of H i con tained in H i ∩ Z i . By deﬁ nition, this implies that V i is cont ained in Q . Th u s Q = V i and the quotien t ( H i ∩ Z i ) /V i is thus torsion-free Ab elian. Notice that Z i con tains N j for all j 6 = i . Ther efore N i .Z i is den se in G and, since H i is op en, it f ollo ws that H i ∩ ( N i .Z i ) = N i . ( H i ∩ Z i ) is den s e in H i . Therefore the Ab elian group ( H i ∩ Z i ) /V i maps densely to H i / N i .V i ∼ = U /U ∩ ( N i .V i ) = U / ( U ∩ N i ) . ( U ∩ Z i ). W e deduce that the latter is Ab elian, as claimed. Our next claim is that G/ N i .Z i is Ab elian. In deed, we hav e G = U.N i .Z i since N i .Z i is dense. W e deduce that G/ N i .Z i ∼ = U /U ∩ ( N i .Z i ), w h ic h m ay b e view ed as a quotien t of the group U / ( U ∩ N i ) . ( U ∩ Z i ). Th e latter is kno wn to b e Ab elian by (i), w hic h conﬁrms the present claim. Supp ose n o w that eac h N i con tains some maximal compact subgroup U i . The ab ov e dis- cussion s h o ws that the der ived group [ G, G ] is cont ained in the inte rsection N := T p i =1 N i .Z i . In other wo rds the qu otien t G/ N is Ab elian, and it on ly r emains to sho w that N = N 1 · · · N p . Z ( G ) . Let g ∈ N 1 .Z 1 ∩ N 2 .Z 2 and write g = n 1 z 1 = n 2 z 2 with n i ∈ N i and z i ∈ Z i . T hen n − 1 1 z 2 = z 1 n − 1 2 b elongs to Z 1 ∩ Z 2 since N i ⊆ Z j for all i 6 = j . Since Z 1 ∩ Z 2 = Z G ( N 1 .N 2 ), we d ed uce that g ∈ N 1 .N 2 . Z G ( N 1 .N 2 ). Th is sh ows that N 1 .Z 1 ∩ N 2 .Z 2 ⊆ N 1 .N 2 . Z G ( N 1 .N 2 ). S ince the opp osite inclusion ob viously holds tr ue, we h av e in fact N 1 .Z 1 ∩ N 2 .Z 2 = N 1 .N 2 . Z G ( N 1 .N 2 ). A straigh tforw ard indu ction no w shows that p \ i =1 N i .Z i = N 1 · · · N p . Z G ( N 1 · · · N p ) . Since N 1 · · · N p is dense in G , we hav e Z G ( N 1 · · · N p ) = Z ( G ), fr om whic h the assertion (ii) follo ws.  Quasi-pro ducts with Ad-closed quasi-factors. The follo wing giv es a simple criterion for a qu asi-pro duct to b e direct. Lemma I I.4. L et G b e a lo c al ly c omp act g r oup that is a quasi-pr o duct with quasi-factors N 1 , . . . , N p . If N 1 , . . . , N p − 1 ar e Ad -close d and c entr efr e e, then G ∼ = N 1 × · · · × N p . Pr o of. W e work b y indu ction on p , starting by noticing that the statemen t is empty for p = 1. Since [ N 1 , N i ] ⊆ N 1 ∩ N i = 1 for i > 1, w e deduce that N 2 · · · N p ⊆ Z G ( N 1 ). In particular N 1 . Z G ( N 1 ) is dense in G . F rom Lemma I.4 and the fact that N 1 has trivial cen tre, it follo ws that G ∼ = N 1 × Z G ( N 1 ). By pro j ecting G on to Z G ( N 1 ), w e deduce that the pro duct N 2 · · · N p is dens e in Z G ( N 1 ). Thus Z G ( N 1 ) is the quasi-pro du ct of N 2 , . . . , N p . T h e desired result follo ws by indu ction.  28 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D It will b e shown in the n ext subs ection that, con versely , a group N whic h is n ot Ad -closed ma y often b e used to construct a non-trivial quasi-pr o duct havi ng N as a qu asi-factor, s ee Example I I.8 b elo w. Non-direct quasi-pro ducts and dense analyt ic normal subgroups. W e prop ose a general scheme to construct quasi-pro du cts out of a pair of top ological groups M , N together with a faithful con tinuous M -acti on by automorphisms on N . Th e intuiti on is that M pla ys the role of some adjoint completion of N app earing in a quasi-direct pro du ct with tw o quasi- factors isomorph ic to N . Th e precise set-up is as follo ws . Let M , N b e top ological groups and α : M ֒ → Aut( N ) an injectiv e cont in uous r epresen- tation. In complete generalit y , c ontinuity shall mean that the map M × N → N is join tly con tinuous; therefore, when considering lo cally compact groups, it suﬃces to assume that α is a con tin uous homomorphism for the Braconnier top ology on Aut( N ). In ord er to formalise the idea th at M is a generalisat ion of the Ad-closure Ad( N ), we assume thr oughout Ad( N ) ⊆ α ( M ) and α − 1 (Ad( N )) = M . Th us α ( M ) is indeed inte rmediate in Ad( N ) ⊆ α ( M ) ⊆ Ad( N ). Th e tr ivial case, i.e. direct pro du ct, of our constru ction will b e characte rised by α ( M ) = Ad( N ). On the other hand, already α ( M ) = Ad ( N ) will pro duce in teresting examples. W e d enote by α ∆ : M ֒ → Aut( N × N ) the diagonal action, whic h is still injectiv e and con tinuous. W e form the semi-direct p ro duct H := ( N × N ) ⋊ α ∆ M , whic h is a top ologica l group for the m u ltiplicatio n ( n 1 , n 2 , m )( n ′ 1 , n ′ 2 , m ′ ) = ( n 1 α ( m )( n ′ 1 ) , n 2 α ( m )( n ′ 2 ) , mm ′ ) . W e obs er ve that the set Z := n ( n, n , m ) : α ( m ) = Ad( n ) − 1 o is a s u bgroup of H and w e write G := H / Z . F or con v enience, we wr ite N 1 = N × 1 and N 2 = 1 × N , w hic h we view as sub groups of H . Prop osition I I .5. (i) Z is a close d normal sub gr oup of H (thus we c onsider G as a top olo gic al gr oup). (ii) The morph ism N i → G is a top olo gi c al isomorphism onto its image, which is close d and normal in G ; we thus identify N i and its image. The r esulting quotients G/ N i ar e top olo gic al ly isomorphic to M . (iii) The morphism N × N → G has dense image; the latter is pr op erly c ontaine d in G if and only if α ( M ) 6 = Ad( N ) . The kernel is the diagonal c opy of Z ( N ) (in p articular, if N is c entr e - fr e e, G is a quasi-pr o duct). (iv) Z G ( N i ) = N 3 − i ; in p articular, if N i s c entr e-fr e e, so is G . (v) If N is top olo gic al ly simple, then G is char acteristic al ly simple. Mor e over, G c annot b e written non-trivial ly as a dir e ct pr o duct u nless α ( M ) = Ad( N ) . Pr o of. (i) The fact that Z is closed follo ws from th e fact that the d iagonal in N × N is closed and that α is con tin u ous. A computation sho ws that N × N cen tr alises Z , w hilst M normalises it; hen ce Z is normal. DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 29 (ii) The morphism N 1 → G is con tinuous and injectiv e. Sup p ose that a net ( n β , 1 , 1) ∈ N 1 con verge s to some ( n, n ′ , m ) mo dulo Z . Then there are nets ν β ∈ N , µ β ∈ M w ith α ( µ β ) = Ad( ν β ) − 1 suc h that ( n β ν β , ν β , µ β ) tends to ( n, n ′ , m ). Thus n β con verge s in N (to nn ′ − 1 ) and hence th e morphism is indeed closed. Th e image is normal by deﬁnition and the case of N 2 is analogous. It is straightfo rw ard to sho w that H = N i .Z.M and that N i .Z ∩ M = 1. Therefore H / N i .Z ∼ = M , as claimed. (iii) The d ensit y is equiv alen t to th e density of Z . ( N × N ) in H , whic h follo ws fr om the con- ditions on α . The add itional statemen t on the image f ollo ws from the canonical identi ﬁcation of coset sets G/ ( N × N ) ∼ = M /α − 1 (Ad( N )) . The description of the k ern el is d ue to th e fact that ( N × N ) ∩ Z consists of those ( n, n, 1) with Ad( n ) = 1. (iv) Su pp ose ( n 1 , n 2 , m ) ∈ H comm utes w ith N 1 mo dulo Z . Th us, for ev ery x ∈ N 1 there is ( ν, ν, µ ) ∈ Z w ith ( n 1 α ( m )( x ) , n 2 , m ) = ( xn 1 α ( m )( ν ) , n 2 α ( m )( ν ) , mµ ) . The last t w o co ordinates show that µ and ν are trivial. It follo ws that α ( m )( x ) = n − 1 1 xn 1 . Th us α ( m ) = Ad( n 1 ) − 1 and hence ( n 1 , n 2 , m ) b elongs to N 2 .Z . The statemen t follo ws by symmetry and us in g the d escription of the k ernel of N × N → G obtained ab o v e. (v) W e can assume that N h as trivial cen tre. Notice that the inv olutory automorphism of N × N deﬁned by ( u, v ) 7→ ( v , u ) extend s to a well deﬁned automorphism of H whic h descends to an automorphism ζ of G swapping the t wo factors of the p ro duct N 1 .N 2 . Let n o w C < G b e a (topologically) characte ristic closed subgroup. Assume ﬁ r st that C ∩ N 1 = 1. Then 1 = ζ ( C ∩ N 1 ) = C ∩ ζ ( N 1 ) = C ∩ N 2 . Thus C cen tralises N 1 .N 2 and is th us cont ained in Z ( G ) since N 1 .N 2 is dens e. Therefore w e hav e C = 1 by (iv) in this case. Assume no w that C ∩ N 1 6 = 1. Then N 1 is con tained in C since N is top ologically simple b y hyp othesis. T ransforming by the inv olutory automorphism ζ shows that N 2 , and hence also N 1 .N 2 , is then conta ined in C , which implies that C = G sin ce C is closed and N 1 .N 2 is dense. Thus G is in deed charac teristically simple, as d esired. The ab o ve arguments sho w moreo ver that N i are min im al clo sed normal s u bgroups of G . This implies that if G splits as a d ir ect pro duct G ∼ = L 1 × L 2 of closed normal subgroups , then, up on renamin g the factors, we ha ve N i < L i for i = 1 , 2. I t follo ws that L i < Z G ( N 3 − i ) = N i b y (ii). Thus N i = L i . In view of (iii), this implies that G do es not split non-trivially as a direct p ro duct of closed subgroup s p ro vid ed α ( M ) 6 = Ad( N ).  Our goal is n o w to present some concrete situations with M and N lo cally compact. Example II.6 . Let M , N b e totally disconn ected lo cally compact groups and let ϕ : N → M 2 b e a con tin uous injectiv e homomorp hism w hose im age is dense and norm al in M . In particular, the conjugation action of M on ϕ ( N ) induces a homormorphism α : M → Aut( N ); ho wev er α need not b e contin uous in general. Ho w ev er α is indeed contin uous in the follo wing cases. • N is d iscrete and ϕ ( N ) ≤ Q Z ( M ). 2 Of course ϕ ( N ) is only analytic when N is metrisable, b ut this is the standard situation to which th e subsection title refers. 30 PIERRE-EMMANUEL CAPRACE AND NICOLAS MONO D • M = Ad( N ) and ϕ = Ad. Of cours e these t w o cases are not mutually exclusiv e. One is then in a p osition to in v oke Prop osition I I.5, whic h provides a totally disconnected lo cally compact group G that is a quasi-direct p ro duct with t w o copies of N as quasi-factors. It is no w easy to construct n on -trivial quasi-pro ducts of totally disconn ected groups by exhibiting a group N satisfying the required conditions. Example I I .7 . Let N b e one of the discrete group s d escrib ed in Example I.10. This example yields a locally compact completion M and a con tinuous homomorphism α : M → Aut( N ) suc h that Ad( N ) ≤ α ( M ) ≤ Ad( N ). If N is simp le, the group G pro vided b y Prop osition I I.5 is c haracteristicall y simple. In this w a y , w e obtain v arious examples of c haracteristicall y simple lo cally compact groups w hic h are quasi-pro du cts but do not split as direct pro d u cts. Notice h o wev er that in these examples N is n ot ﬁn itely generated and the corresp ond ing G is never compactly generated. Another wa y to satisfy the conditions of Examp le I I.6 is to start with a group N which is not Ad -closed, but wh ose adjoint closure M = Ad( N ) is lo cally compact. W e pr o ceed to describ e a concrete example. Part of the interest of the example is that N will b e compactly generated, whic h implies that the asso ciated groups M and G will b e b oth compactly generated. Indeed, consider a compact generating set Σ for N and U any compact op en sub group of M . Then U ∪ ϕ (Σ) generate s M , since h U ∪ ϕ (Σ) i is op en an d con tains a dense sub group. Th us M is compactly generated, and so is H as well as all its quotient s, including G . Example I I .8 . Cons id er the semi-direct p ro duct N = SL 3  F p ( ( t ) )  ⋊ Z , where the cyclic group Z is any in ﬁnite cyclic su b group of the Galois group Aut  F p ( ( t ) )  . Then N is n ot Ad -closed, but Aut( N ) is lo cally compact, as follo ws from Th eorem I.6. Mo reo ver, the cyclic group Z normalises eve ry c h aracteristic sub group of the compact op en subgroup SL 3  F p [ [ t ] ]  < S L 3  F p ( ( t ) )  , from w h ic h it easily follo ws that Z contai ns an unb ou n ded asymptotically central sequence. Notice fu rthermore that N is cen tr efree. The easiest wa y to see this is by noticing that N acts minim ally without ﬁxed point at inﬁnity on the Bruhat–Tits b uilding asso ciated with S L 3  F p ( ( t ) )  (see [CM09 , T heorem 1.10]). Thus Prop osition I I.5 ma y b e applied. In conclusion, we ﬁn d that the group  SL 3  F p ( ( t ) )  ⋊ Z  ×  SL 3  F p ( ( t ) )  ⋊ Z  ! ⋊ Ad  SL 3  F p ( ( t ) )  ⋊ Z  n ( z , z , Ad ( z ) − 1 ) : z ∈ SL 3  F p ( ( t ) )  ⋊ Z o pro vides an example of a compactly generated totally d isconnected lo cally compact group with trivial quasi-cen tre (in particular cen trefree) whic h is a non-trivial quasi-pro duct. Ho w ev er, this example is not charact eristically simp le. DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 31 Op en problems. Corollary D n aturally s u ggests the follo w in g question. Question I I.9. Is ther e a c omp actly gene r ate d c har acteristic al ly simple lo c al ly c omp act gr oup G that is a quasi-pr o duct with at le ast two simple quasi-factors, but which do es not split non- trivial ly as a dir e ct pr o duct? Finally , we men tion tw o related problems. Question I I.10. Is ther e a c omp actly gener ate d top olo gic al ly simple lo c al ly c omp act g r oup G which c ontains a pr op er dense normal sub gr oup? Question I I.11. Is ther e a c omp actly gener ate d top olo gic al ly simple lo c al ly c omp act g r oup G which i s not Ad -close d? As explained in Example I I.8, a p ositiv e answ er to the latter implies a p ositiv e answ er to b oth Questions I I.9 and I I.10. Moreo ver, Prop osition I I.1 implies that a p ositive answer to Question I I.9 also giv es a p ositiv e answ er to Question I I.10. On dense normal subgroups of top ologically simple groups. W e close this app endix with the follo wing result, du e to N. Nik olo v [Nik09, P r op osition 2], whic h pro vides how ev er sev ere restrictions on p ossib le non-Hausdorﬀ qu otien ts of top ologically simple groups. Prop osition I I .12. L et M b e a c omp actly g ener ate d total ly disc onne cte d lo c al ly c omp act gr oup which is lo c al ly ﬁnitely gener ate d. 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Index Ad-closed, 19 adjoin t closure, 21 asymptotically central sequence, 20 Braconnier topology , 18 chara cteristic subgroup, 4 chara cteristically simple, 4 discrete residual, 5, 11 FC -group, 23 top ological ly , 6 ﬁnitary p erm utation, 23 just-inﬁnite, 4 just-non-compact, 4 LF-radical, 6 locally ﬁn itely generated, 21 locally inner automorphism, 22 monolith, 4 monolithic, 4 No et h erian group, 3 quasi-centre, 6 quasi-discrete, 11 quasi-factor in a group, 24 of a quasi-pro duct, 4, 24 quasi-pro duct, 4, 24 quasi-simple, 4, 12 radical LF, 6 soluble, 14 Schreier graph, 7 SIN-group, 10 soluble radical, 14 top ological ly FC-group, 6 top ological ly lo cally ﬁnite, 6 34 DECOMPOSING LOCALL Y CO M P ACT GROUPS INTO SIMPLE PIECES 35 UCLouv ain, 1348 Louv ain-la-Neuve , Belgium E-mail addr ess : pe.caprace@uc louvain.b e EPFL, 1015 Lausanne, S witzerland E-mail addr ess : nicolas.monod @epfl.ch

Decomposing locally compact groups into simple pieces

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