Itzkowitzs problem for group of finite exponent

ITZK O WITZ’S PR OBLEM FOR G R OUPS O F FINITE EXPONENT A. BARECHE AND A. BOUZIAD Abstra ct. Itzko witz’s problem asks whether every top ological group G has equal left and right u niform structures provided that b ounded left uniformly continuous rea l-v alued function on G are right uniformly conti nuous. This pap er p ro vides a p ositiv e answer to th is problem if G is of b ounded exp onent or, more generally , if there ex ist an in teger p ≥ 2 and a nonempty op en set U ⊂ G such that the p ow er map U ∋ g → g p ∈ G is left (or right) uniformly conti nuous. This also resolv es the problem for p erio dic groups which are Baire spaces. 1. Intr oduction Let G b e a Hausdorff top ological group, e its un it and N ( e ) the set of all n eigh b orho o d s of e in G . The left uniformit y U l of G h as as a basis of en tour ages the sets of the f orm { ( x, y ) ∈ G × G : x − 1 y ∈ V } , where V is a mem b er of N ( e ) (we write the la w of G multi plicativ ely). The right unifor- mit y U r is obtained b y writing xy − 1 in place of x − 1 y (in the f orm ab ov e). The group G is said to b e b alanced, or S IN (for small inv a rian t n eigh b orho o d s), if the tw o uniformities U l and U r coincide. As it is well known, compact groups and (obvio usly) Ab elian top ological groups are SIN. A top ological group G is called functionally b alanced or FSIN, if eve ry b ound ed left uni- formly con tinuous real-v alued function on G is righ t u niformly con tinuous. Here, a fu nction f : G → R is left un iformly con tinuous if f is u niformly con tinuous when G is equipp ed with its left uniformit y a nd the real s w ith the usual metric. Right un iform con tin u it y is defin ed similarly . Since the inv er- sion on G switc h es the left and right uniformities, the alternativ e righ t-left definition in the FSI N prop erty leads to the same thing. Ob viously , ev ery SIN group is an FS IN group , but it is still unk n o wn if the con verse holds for ev ery top ological group . Th is problem, called Itzk o witz’s problem after the work [5], h as receiv ed sev eral p ositiv e answers in the p ast; the reader is r eferred to [1] and the references therein for more inform ation and related qu estions about th e problem. Let us ju st recall that it h as receiv ed the atten tion of many authors and has b een solv ed for t wo notably 2000 Mathematics Subje ct Classific ation. Primary 22A05; 54E15; S econdary 22A10. Key wor ds and phr ases. T opological group; Left (righ t ) u niform structure; Left (right) uniformly continuous bound ed real-v alued function; S IN-group; FSIN- group; Left (right) thin subset; Left (right) neut ral subset; Left ( righ t) u niformly discrete subset, Group of finite exp onent. 1 2 A. BARECHE AND A. BOUZIAD differen t classes includin g resp ectiv ely lo cally compact groups and lo cally connected groups (see for instance [2, 4, 6, 7, 9 ]). It is clear that if G is lo cally compact then for every p ositiv e int eger p , the p o wer map φ p , defined b y G ∋ x → x p ∈ G , is left uniformly con tin u ous “lo cally”, that is, when it is restricted to some neighborh o o d of the unit of G (just tak e a compact neigh b orho o d ) and G is equip p ed with the left uniformit y . In particular, the class of groups for whic h th ere is p ≥ 2 su c h that th e p ow er map φ p is “locally” left un if orm ly con tinuous, includ es all lo cally compact group s. It includes also all groups of fin ite exp onent ; where a group G is said to b e of fin ite exp onent if for some p ≥ 1, x p = e for all x ∈ G . In connection with this, it s hould b e noted, as it is easy to see, that a group G is SIN if and only if th e p o wer map φ 2 is left u niformly con tinuous. Th is sh ould b e compared to the w ell-kno wn fact that G is SIN if and only if the pro d uct map ( x, y ) → xy is left un iformly contin uous; see [11]. The main result of this note is as follo ws: The equalit y FSIN=S IN h olds in the class of top ological groups G for w hic h there are a neigh b orho o d V of the u nit and an in teger p ≥ 2 such that the map φ p : V → G is left uniformly conti n uous. This extends the lo cally compact case and allo ws to giv e an affirmativ e answ er to Itzk o vitz’s question for p erio dic top ological groups whic h are of the second category . 2. Tw o lemma s In what follo ws, G is a fixed Hausdorff top ological grou p . A subs et A of G is called left V -thin in G , where V ∈ N ( e ), if the set ∩ a ∈ A a − 1 V a is a n eigh b orho o d of e in G . If A is left V -thin in G for every V ∈ N ( e ), then A is said to b e left thin in G . T he concept of “righ t V -thin” is defined similarly . Note that the group G is SIN if and only if G is left th in in itself. The set A is said to b e left ( right ) V -discr ete in G if a = b w henev er a ∈ bV ( a ∈ V b ) and a, b ∈ A . T he set A is said to b e R o elcke-discr ete if there is V ∈ N ( e ) suc h that a = b whenever a, b ∈ A and a ∈ V bV . Note that this m eans that A is a uniform ly discrete subs et of G (with resp ect to V ) when G is equipp ed with the lo wer unif orm it y U l ∧ U r , sometimes called the Ro elc ke uniform it y . Finally , the set A is said to b e left ne u tr al if for ev ery V ∈ N ( e ), th er e is U ∈ N ( e ) such that U A ⊂ AV . It is well k n o wn that the group G is FSIN if and only if eve ry subs et of G is left n eutral, see [10]. The main result is obtained as a consequence of th e f ollo wing lemmas. Lemma 2.1. L et ( V 1 , . . . , V q ) ( q ≥ 2) b e a de cr e asing se q u enc e of ne i ghb or- ho o ds of e in G and M ⊂ G , with V 2 2 ⊂ V 1 , such that mV 2 i +1 ⊂ V i m for every 1 ≤ i < q and m ∈ M . Supp ose that ther e ar e a symmetric U ∈ N ( e ) and 2 ≤ p ≤ q such that U ⊂ V p and m − p ( mu ) p ∈ V p for every m ∈ M and u ∈ U . Then M is right V 1 -thin in G . ITZK OW ITZ’S PR OBLEM F OR GR OUPS OF FINITE EXPONEN T 3 Pr o of. Let u ∈ U and m ∈ M . Since m − p ( mu ) p ∈ V p , we ha v e m − ( p − 1) u ( mu ) p − 2 m = m − p ( mu ) p u − 1 ∈ V 2 p ⊂ m − 1 V p − 1 m, hence m − ( p − 2) u ( mu ) p − 2 ∈ V p − 1 . W e con tin ue this calculation until we get m − 1 u ( mu ) ∈ V 2 , which gi v es um ∈ mV 2 2 ⊂ mV 1 as claimed.  Lemma 2.2. Supp ose that every R o elcke discr ete subset of G is left thin i n G . L et A ⊂ G . Then, for every V ∈ N ( e ) and n ≥ 2 , ther e ar e M ⊂ AV and a de cr e asing se quenc e ( V 1 , . . . , V n ) of arbitr ary smal l neighb orho o ds of e in G ( e.g. V 1 ⊂ V and V 2 2 ⊂ V 1 ) , such that (a) mV 2 k +1 ⊂ V k m for every m ∈ M and 1 ≤ k < n , (b) if M is right V -thin in G , then A is right V 3 -thin in G . Pr o of. In addition to ( V 1 , . . . , V n ), we will build a sequence U 1 , ..., U n − 1 of neigh b orho o ds of e and a sequen ce M 1 , ..., M n − 1 of subsets of G ; then w e tak e M = M n − 1 and u se the sequence ( U 1 , . . . , U n − 1 ) to ensure the prop erties (a) and (b). T o b egin, p ut V 1 = V and let U 1 b e a symmetric neighborh o o d of e su c h that U n 1 ⊂ V 1 and U 3 1 ⊂ V 1 (if n < 3). I t follo ws fr om Zorn ’s lemma th at there is a maximal set B 1 ⊂ G whic h is Ro elc ke-discrete w ith r esp ect to U 1 ; in p articular A ⊂ U 1 B 1 U 1 . F or eac h a ∈ A , c ho ose ( u ( a, 1) , b ( a, 1) , v ( a, 1) ) ∈ U 1 × B 1 × U 1 suc h that b ( a, 1) = u ( a, 1) av ( a, 1) and put M 1 = { av ( a, 1) : a ∈ A } . Since B 1 is left U 1 -thin in G , M 1 is left U 3 1 -thin in G , th u s left V 1 -thin in G . Cho ose V 2 ∈ N ( e ), with V 2 2 ⊂ V 1 and V 2 n 2 ⊂ m − 1 V 1 m for ev ery m ∈ M 1 . A t step 2, c h o ose a sym metric U 2 ∈ N ( e ) suc h that U 2 ⊂ U 1 and U 3 2 ⊂ V 2 . Zorn’s lemma again giv es a maximal B 2 ⊂ G whic h is Ro elc ke -discrete with resp ect to U 2 , f or which we obtain M 1 ⊂ U 2 B 2 U 2 . By the d efinition of M 1 , for eac h a ∈ A , there is ( u ( a, 2) , b ( a, 2) , v ( a, 2) ) ∈ U 2 × B 2 × U 2 suc h that b ( a, 2) = u ( a, 2) av ( a, 1) v ( a, 2) . P u t M 2 = { av ( a, 1) v ( a, 2) : a ∈ A } and note as ab o v e that M 2 is left V 2 -thin in G . Cho ose V 3 ∈ N ( e ), with V 3 ⊂ V 2 and V 2 n 3 ⊂ m − 1 V 2 m for ev ery m ∈ M 2 . This p ro cess allo ws us to obtain sequences ( V 1 , . . . , V n ), ( M 1 , . . . , M n − 1 ) and ( U 1 , . . . , U n − 1 ), with the follo wing: (1) ( U 1 , . . . , U n − 1 ) is a decreasing sequence of symmetric neigh b orho o d s of e , w ith U n 1 ⊂ V 1 ; (2) for eve ry 1 ≤ k < n , U 3 k ⊂ V k ; (3) for eve ry 1 ≤ k < n , M k = { av ( a, 1) · · · v ( a,k ) : a ∈ A } , wh er e ( v ( a, 1) , . . . , v ( a,k ) ) ∈ U 1 × . . . × U k ; (4) for eve ry 1 ≤ k < n , V 2 n k +1 ⊂ m − 1 V k m for eac h m ∈ M k . It remains to v erify that the prop erties (a) and (b) are satisfied b y the sequence ( V 1 , . . . , V n ) and the set M = M n − 1 . Note that from (3) (with k = n − 1), it follo ws that M ⊂ AV and A ⊂ M V , since U 1 · · · U n − 1 ⊂ U n 1 ⊂ V . 4 A. BARECHE AND A. BOUZIAD (a) Let m ∈ M and 1 ≤ k < n . There is a ∈ A such that m = av ( a, 1) · · · v ( a,n − 1) with ( v ( a, 1) , . . . , v ( a,n − 1) ) ∈ U 1 × . . . × U n − 1 . In case 1 ≤ k < n − 1, wr ite m = m k · v ( a,k +1) · · · v ( a,n − 1) with m k ∈ M k . It follo w s f rom (1), (2) and (4) that mV 2 k +1 m − 1 ⊂ m k V 2 n k +1 m − 1 k ⊂ V k . F or k = n − 1, th e inclusions mV 2 n m − 1 ⊂ V n − 1 for eac h m ∈ M , follo w fr om (4). (b) Th is follo ws immediately from the f act that A ⊂ M V .  3. FSIN versus SIN F ollo wing [11], a top ological group G is said to b e ASIN (for almost S IN), if there exists a neigh b orho o d of the unit in G whic h is left (or righ t) thin in G . Equiv alen tly , G is ASIN if there exists a nonemp t y op en sub set of G , whic h is left (or right) thin in G (indeed, if A, B ⊂ G are left thin in G , then the set AB is left thin in G ). Prop osition 3.1. Supp ose that ther e ar e p ≥ 2 and a nonempty op en set Ω ⊂ G such that the mapping Ω ∋ x → x p ∈ G i s left uniformly c ontinuous. If every R o elcke-discr ete subset of G is lef t thin i n G , then G is ASIN . Pr o of. As noted ab o ve, it suffices to sho w that G h as a n onempt y op en set whic h is righ t thin in G . Fix g ∈ Ω and c ho ose U ∈ N ( e ) such that gU 3 ⊂ Ω. Using Lemmas 2.1 an d 2.2, w e w ill pro v e that the op en set A = g U is righ t thin in G . Let V ∈ N ( e ) with V ⊂ U . Ap p lying Lemma 2.2 to A and V , we get a set M ⊂ g U V and a sequence ( V 1 = V , V 2 , . . . , V p ) of neighborh o o ds of e satisfying (a) and (b ) in this lemma. Clearly , the assumption of Lemma 2.1 is satisfied by ( V 1 = V , V 2 , . . . , V p ) and M . Cho ose a symmetric W ∈ N ( e ) with W ⊂ U su c h that x − p y p ∈ V p whenev er x, y ∈ Ω and x − 1 y ∈ W . Then, for all m ∈ M and w ∈ W , we ha ve m, mw ∈ Ω and m − 1 mw ∈ W , thus m − p ( mw ) p ∈ V p . It follo ws from Lemma 2.1 that M is righ t V -thin in G , hence, b y Lemma 2.2(b), A is right V 3 -thin in G .  The next t wo lemmas corresp ond resp ectiv ely to Prop osition 3.5 and Lemma 3.3 in [2]. Lemma 3.2. Supp ose that every R o elcke - discr ete subset of G is left thin i n G . If G i s ASIN, then G is SIN. Lemma 3.3. If G is FSIN, then every R o elcke-discr ete subse t of G is left thin in G . No w, we arrive at the main result of this n ote. ITZK OW ITZ’S PR OBLEM F OR GR OUPS OF FINITE EXPONEN T 5 Theorem 3.4. Su pp ose that for some p ≥ 2 , the p ower ma p G ∋ g → g p ∈ G i s left u ni f ormly c ontinuous when r estricte d to some nonempty op en subset of G . If e v ery R o elcke- discr ete subset of G is left thin in G , then G is SIN. Pr o of. By Pr op osition 3.1 , G is ASIN. Then, from Lemma 3.2, G is SIN.  Corollary 3.5. L et G b e an FSIN gr oup. If for some p ≥ 2 , the p ower map G ∋ g → g p ∈ G is left uniformly c ontinuous when r estricte d to some nonempty op e n subset of G , then G is SIN. Pr o of. This f ollo ws from Theorem 3.4 and Lemma 3.3.  It is customary to sa y that the top ological group G is top olo gic al ly torsion if f or any g ∈ G th e sequence ( g n ! ) n ∈ N con verge s to e in G . The reader is referred to [3] for useful generalizatio ns of this concept. W e do not know if Corollary 3.5 remains tru e if G is assumed to b e top ologica lly torsion. As a partial answer, we offer the follo w ing. Prop osition 3.6. Supp ose that G is FSIN and that ther e is p ≥ 2 such that for every g ∈ G , the se qu enc e ( g pn ) n ∈ N c onver ges to e in G . Then G is SIN. Pr o of. Let V ∈ N ( e ) a nd A = G . As in the pro of o f Prop osition 3.1, considering a sequence ( V 1 = V , . . . , V p ) in N ( e ) and M ⊂ G (b y Lemma 2.2), we can sho w that M is right V -thin in G . Then, w e conclud e from Lemma 2.2(b) that G is r igh t thin in itself. In deed, tak e a symmetric W ∈ N ( e ) w ith W 4 ⊂ V p . F or g ∈ G and u ∈ W , there is N ∈ N such that g − pn , g pn , ( g u ) pn and ( g u ) − pn b elong to W for all n ≥ N . T aking n ≥ N + 1, we on tain g − p ( g u ) p = g p ( n − 1) g − pn ( g u ) p ( n +1) ( g u ) − pn ∈ W 4 ⊂ V p . Therefore, by Lemma 2.1, M is righ t V -thin in G .  Corollary 3.7. L et G b e an FSIN gr oup which is of finite exp onent. Then G is SIN. Clearly , Corollary 3.7 follo ws from b oth Corollary 3.5 and Prop osition 3.6. It can b e also dedu ced f rom the follo wing r esult for whic h we will give a direct pro of (based on Lemma 2.2). F or g ∈ G , th e left tr an s lation l g : G → G is defined b y l g ( h ) = g h . F or a giv en A ⊂ G , it is w ell kn o wn (and easy to chec k) that A is left thin in G if and only if the set L ( A ) = { l g : g ∈ A } is equicon tinuous (at the u nit e ), as a set of m aps f rom th e space G to the un iform space ( G, U r ). In particular, G is S IN if and only if L ( G ) is equicon tin u ou s . If G is FSI N, it suffices to supp ose that for some p ≥ 1, the set L ( { g p : g ∈ G } ) is equicon tinuous, as it is clear from the next statemen t. Prop osition 3.8. Supp ose that ther e is p ≥ 1 such that the set { g p : g ∈ G } is left thin in G . If every R o elcke-discr ete subset of G is left thin in G , then G is SIN. 6 A. BARECHE AND A. BOUZIAD Pr o of. W e may s u pp ose th at p ≥ 2. F or V ∈ N ( e ) and A = G , c h o ose ( V 1 = V , V 2 , . . . , V p ) and M ⊂ G as in Lemma 2.2. In view of Lemma 2.2(b), to conclude, it s uffices to v er if y that M is right V -thin in G . T ak e U ∈ N ( e ) suc h that g − p U g p ⊂ V p for ev ery g ∈ G . Let u ∈ U and m ∈ M ; starting from m − ( p − 1) um p − 1 ∈ mV p m − 1 ⊂ V p − 1 and con tinuing th is pro cess, w e arriv e at m − 1 um ∈ V 1 , that is, um ∈ mV .  Recall th at th e grou p G is called p erio dic (or a torsion group) if eve ry elemen t of G is of finite order. Corollary 3.9. Supp ose that G is FSIN, p erio dic and a Bair e sp ac e. Then G is SIN. Pr o of. A standard Baire category argumen t giv es a nonempty op en subset O of G and p ≥ 2 such that x p = e for ev ery x ∈ O . Hence Corollary 3.5 applies.  R emark 3.10 . Corollary 3.7 remains true if Ro elc ke -discrete su bsets of G are left thin (without assum in g that G is FSIN). Th is is of course the case when the Roelc k e uniform it y of G is precompact. As a consequence, we obtain the follo wing statemen t: Ev ery top ologica l group which is Ro elc k e precompact and of fi nite exp onen t is p recompact (equiv alen tly , a SIN group). Recall that Ro elc ke precompact p erio dic groups need not b e precompact (consider the group of fi nitely supp orted p ermutat ions of an infi nite set). No w, for the conv enience of the r eader, w e cite an example of a top ologica l group of fi nite exp onent w hic h is not SIN. Let S and A b e t wo non trivial groups, with A infinite, and consider the group H = S A with the p oin t wise pro du ct. The map η : A → Aut( H ) defin ed by η ( a )( h )( b ) = h ( ba ), a, b ∈ A , h ∈ H is an homomorp h ism, wher e Aut( H ) stands for the automorphism s group of H with the comp osition la w ( f , g ) → f ◦ g . Let G = H × η A b e the semi-direct pr o duct group asso ciated to ( A, H , η ), top ologized as f ollo ws: A is discrete and H is equipp ed w ith the pro du ct top ology , the group S b eing discrete. Th en, G is n ot SIN; indeed, the set { e } × A is neither left nor righ t thin in G . If S = Z 2 and the group A is of exp onent 2, then G is of exp onent 4. A c oncluding c omment. The statemen ts of Theorem 3.4 and Pr op osition 3.8 are differen t, although they ha v e some co mmon consequ en ces (e.g. C orollary 3.7). In fac t, in a wa y , they are complementary and we prop ose the follo wing discussion to explain th at. In general, a map f : ( G, U l ) → ( X , U ) (where ( X, U ) is a un iform space) is uniformly con tin uou s if and only if the s et { f g : g ∈ G } of left tr an s lations of f is equicontin uous (at e ). T hus, the p o w er map φ p : ( G, U l ) → ( G, U l ) is uniform ly con tinuous if and only if the set { ( φ p ) g : g ∈ G } of all left translations of φ p is left equicon tinuous (i.e., when G is equ ipp ed with the left un iformit y). T aking in Lemma 2.1 (via Lemma 2.2) a sequence ( V 1 , . . . , V q ) with an appropriate length, it is quite p ossible to weak en th e assumption in Theorem 3.4 assuming only that the ITZK OW ITZ’S PR OBLEM F OR GR OUPS OF FINITE EXPONEN T 7 set { ( φ p ) g q : g ∈ G } is left equ icontin uous for some p ≥ 2 and q ≥ 1. On the other h an d , the left thinness of the set A in Prop osition 3.8 means that th e set { ( φ 1 ) g p : g ∈ G } is righ t equicon tinuous (i.e., when G is equipp ed with the righ t uniformity) . It is again p ossible here to assume only that the set { ( φ p ) g q : g ∈ G } is right equiconti n uous for some p ≥ 1 and q ≥ 1. Ac knowledgemen t. The authors w ould like to thank the referee for h er/his v aluable remarks and suggestions. Referen ces [1] A. Bouziad and J. P . T roallic, Pr oblems ab out the uniform structur es of top olo gic al gr oups , in: E. P earl (Ed.), Op en Problems in T opology I I , Elsevier, Amsterdam, 2007. [2] A. Bouziad and J. P . T roallic, Nonsep ar ability and uniformi ti es in top olo gic al gr oups , T op ology Proc. 28 (2004) 343–359. [3] D. D ikranjan, T op olo gi c al l y torsion elements of top olo gic al gr oups , T op ology Proc. 26 (2001–20 02) 505–532. [4] S. Hern´ andez, T op olo gic al char acterization of e quivalent uniformities in top olo gic al gr oups , T op ology Pro c. 25 (2000) 181-188. [5] G. 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Dierolof, U niform S tructures in T op ological Groups and Their Quotients, McGra w-Hill, New Y ork, 1981. Universit ´ e de Rouen, D ´ ep ar tement de Ma th ´ ema tiques, UMR C NRS 6085, A venue de l ’Universit ´ e, BP.12, F76801 Saint- ´ Etienne-du-Rouvra y, France E-mail addr ess : aicha.bare che@etu.u niv-rouen. fr Universit ´ e de Rouen, D ´ ep ar tement de Ma th ´ ema tiques, UMR C NRS 6085, A venue de l ’Universit ´ e, BP.12, F76801 Saint- ´ Etienne-du-Rouvra y, France E-mail addr ess : ahmed.bouz iad@univ- rouen.fr