Representing a profinite group as the homeomorphism group of a continuum

Representing a Profinite Group as the Homeomorphism Group of a Contin uum b y Karl H. Hofmann and Sidney A. Morri s Abstract . W e contribute some information to wards finding a general algorithm f or constructing, for a given profinite group G , a c om pact connected space X suc h that the full homeomorphism group H ( X ) with the c om pac t -op en topol ogy is isomorphic to G as a topological group. It is proposed that one should find a c omp act top olo gi c al oriented graph Γ such that G ∼ = Aut(Γ) . The replaceme n t of the edges of Γ b y rigid contin ua should wo rk as i s exemplifie d in v arious instances where discrete graphs were used. It is sho wn here t hat the strategy can b e impleme n te d for profinite monothetic groups G . Mathematics Sub j e ct Cl assification 2010: 22C05, 22F50, 54H15, 57S10 . Key W ords and P hrases: Homeomorphism group, compact group, profinite group, slice, G -space. 1. Intr oduction One knows that the compact homeomorphism gr o up H ( X ) of a Tyc honoff space has to b e profinite ([16] , [17] ). In the con vers e direction Gar tside and Gl yn [ 8 ] ha ve establi shed that ev ery metri c profinite group is the homeomor phism group of a contin uum ( i.e. a compact connected metric space ). F or the goal of represen ting a given group as the homeomorphism gro up of a space, authors ha v e pursued the following strategy: Step (1): find some connected graph Γ , usuall y orien ted, and find a n isomo r- phic represen tation π : G → Aut(Γ); the standard attempt is to use some form of Ca yley g raphs (see [9], [8], [4] ) Step (2): find a rigid con tin uum C , that is, a contin uum, t hat is, compact connected metric space, whose onl y contin uous selfmaps are the iden ti t y and the constan t function (see [7], [11]) and replace each of the directed edges of Γ b y C or a v ari a nt obtai ning a connected space X ; finally obtain an isomorphism γ : Aut(Γ) → H ( X ) (see [ 9], [11] , [4]). Obtain an isomorphism γ ◦ π : G → H ( X ). All known v ariations of the strategy are highly tec hnical , and differen t v aria- tions lead to rather different phase spaces X . It w ould b e nice to find a c onstruc- tion which is i n some w a y canonical, perhaps eve n functorial. How ev er, one of t he ma jor obstructions for represen tatio ns of a profinite group in a com binatio n wit h graph t heoreti cal met ho ds is that homeomorphism groups, like all automorphism groups in a category are, in no visible w a y , functorial. 1 W e prop ose, that in Step (1) one should in fact go more than halfw ay and construct a compact connected directed graph Γ and then apply Step (2) to ac hiev e the final goal. In the foll o wing we sho w that such constructions are p o ssible in principle a nd yield for ev ery profinite monothetic group G a con tin uum X suc h that H ( X ) ∼ = G . while not all compact monothetic groups are metric, the profinite ones among them are. Th us, in the v ein of a general existence result, our construction yi el ds nothing new b eyond what Gar tside and Gl yn ha v e sho wn i n [8]. How ev er, the con tinua w e construct are completely differen t from those pro duced in [ 8] and the prop osed construction may turn out to b e useful in the future. 2. Directed top ological graphs. In order to construct top ol ogical spaces with prescribed homeomorphism groups we first construct directed top ologi cal gra phs wit h prescrib ed automo r- phism groups. Definit ion 2.1. A dir e cte d (to p olo gic al) gr aph is a triple Γ = ( V , E , η ) consist- ing of t op ological spaces V and E and a con t in uous function η : E → V × V suc h that ( † ) V = im(pr 1 ◦ η ) ∪ im(pr 2 ◦ η ) . The set V is call ed the space of vertic es and E is called the space o f (oriente d) e dges . If e ∈ E we write η ( e ) = ( e 1 , e 2 ) ∈ V × V , t hen e 1 is the origi n a nd e 2 is the tar get of e . Conditi o n ( † ) sa ys ( ‡ ) V = [ e ∈ E { e 1 , e 2 } , and this means that there is no v ertex that is not an endp oin t of a n edge. Note that we a l lo w a whole space η − 1 ( v 1 , v 2 ) of (di r ected) edges from v 1 to v 2 . W e shall, ho w ever, not use this fact i n the se q uel. If t he spaces V and E of a directed graph Γ = ( V , E , η ) are discrete, w e reco ver the more classical concept of a directed graph. Example 2.2. (i) Let n b e natural n um b er n > 2 and let Z ( n ) = Z /n · Z b e the cyclic group of n elemen ts. Define V = Z ( n ) = { m + Z ∈ Z /n Z : m = 0 , 1 , . . . , n − 1 } , E = V , η ( m + n · Z ) = ( m + n · Z , m + 1 + n · Z ) ∈ V × V . 2 The directed graph C ( Z ( n ) ) def = ( V , E , η ) is the n - cycle . It is the Cayley-graph of the pair ( Z ( n ) , { 1 + n Z } ) consisting of the cyclic g roup of order n and the singleton generating set con taining the elemen t 1 + n Z . (ii) More generally , let G b e a top olog i cal group and let g ∈ G . W e set V = G, E = V , η ( x ) = ( x, xg ) ∈ V × V . The directed graph C ( G ) def = ( V , E , η ) is the top olo gi c al Cayley gr aph of ( G, { g } ) . (iii) T aking Z with the discrete top ol ogy we obtain the Cayley gr aph C ( Z ) o f ( Z , { 1 } ), the ch ai n Z with its natural order-orien tati on. ⊓ ⊔ Definit ion 2.3. A d oubly poi n ted c onnected top ological space L = ( L, b 1 , b 2 ) with b 1 6 = b 2 is called a link . ⊓ ⊔ T ypicall y I = ([0 , 1] , 0 , 1) is a link: an in terv al joining its endp oints. If L = ( L, b 1 , b 2 ) i s a link and a T ychonoff space, then there is a contin uous function F : L → I , F : L → I , F ( b 1 ) = 0 and F ( b 2 ) = 1, called a morphism of links . Construction 2.4. Let Γ = ( V , E , η ) b e a top olo g ical directed graph and L = ( L, b 1 , b 2 ) a link. W e construct a space | Γ | L from these data b y “inserting in to eac h orien ted edge e = η ( v ) = ( e 1 , e 2 ) a cop y of { e } × L of the l ink L suc h that ( e, b 1 ) is iden t ified with e 1 and v while ( e 2 , b 2 ) is iden t ified with with e 2 and v .” Indeed w e let X = E × L and define an equiv alence relation ρ on X w i th the follo wing equiv alence classes: ρ ( e, x ) =      { ( e, x ) } if x 6 = b 1 , b 2 , { ( e ′ , b 1 ) : η ( e ′ ) = ( v , ( e ′ ) 2 ) } ∪ { v } if η ( e ) = ( v , e 2 ) , x = b 1 , { ( e ′′ , b 2 ) : η ( e ′′ ) = (( e ′′ ) 1 , v ) } ∪ { v } if η ( e ) = ( e 1 , v ) , x = b 2 , { ( e ∗ , x ) : η ( e ∗ ) = ( v , v ) } ∪ { v } if η ( e ) = ( v , v ) , x ∈ { b 1 , b 2 } . Then let | Γ | L def = X/ρ . The space | Γ | L is called the top olo gic al r e alis ation of Γ via the link L = ( L , b 1 , b 2 ). 3 If L is a Tyc honoff link and F : L → I a morphism o f links then our construc- tion obviously induces a morphism F ∗ : Γ L → Γ I of top ol o g ical realisatio ns. ⊓ ⊔ Notice t hat in the case of a Cayley graph of a group G w i th an elemen t g ∈ G , the q uotien t space (( E × L )) /ρ can b e expresse d in the form ( ∗ ) | C ( G ) | L = ( G × L ) /ρ . and that there is a morphism F ∗ : | C ( G ) | L → | C ( G ) | I of top ological realisat i ons give n b y F ∗ ( ρ L ( a, x )) = ρ I ( a, f ( x )). The verification of the detai ls of the follo wing examples is strai g h tforw ard. Example 2.5. | C ( Z ( n )) | I is a circle and | C ( Z ) | I is homeomorphic to R . ⊓ ⊔ F or the follo wing, let R be a compact conne cted space and pic k t w o differen t p oin ts b 1 , b 2 ∈ E . and R = ( R, b 1 , b 2 ) the corresponding link. Lemma 2.6. L et ( X , x 0 ) b e a c omp act c onne cte d p ointe d sp ac e and ( R, b 1 , b 2 ) a doubly p ointe d de Gr o ot-c ontinuum. Assume that X and R ar e di s joint with the exc eption of b 2 and x 0 which ar e assume d to b e e qual. Then a c ontinuous function f : R → X ∪ R i s exactly one of the fol lowing kind (i) f is c ons tant. (ii) f ( R ) = R and its c or estricti on R → f ( R ) is the identity map of R . (iii) f ( R ) ⊆ X and f ( b 2 ) = b 2 . (iv) f ( R ) ∩ R = ∅ . Pro of. The function π : R ∪ X → R defined by π ( X ) = { b 2 } is contin uous; hence the con tinuous self-map π ◦ f : R → R is either the iden tity or i s constan t with imag e { b 2 } . ⊓ ⊔ Example 2.7. L et R = ( R , b 1 , b 2 ) b e a doubly p oin ted de Gro ot-con tin uum and F : R → I a mor phisn of links. Then (i) | C ( Z ( n )) | R is a one-dimens ional contin uum X whose homeomorphism group H ( X ) is isomorphic to t he cyclic group Z /n · Z . Moreov er F ∗ : | C ( Z ( n )) | L → | C ( Z ( n )) | I is a morphism of reali sations onto, or “o ver”, a circle. (ii) | C ( Z ) | R is a one-dimensional connected lo cally compact space X whose homeomorphism space H ( X ) is isomorphic to the infinite cyclic g r o up Z . Moreov er F ∗ : | C ( Z ) | L → | C ( Z ) | I is a morph ism of realisations o v er a line. 4 Pro of. F rom ( ∗ ) a b o v e recall that for a cyclic group Z = Z /m Z , n = 0 , 1 , . . . w e ha ve | C ( Z ) | R = ( Z × R ) /ρ and note via Lemma 2.6, that the action ( n, ( k + m Z , x )) 7→ ( n + k + m Z , x ) : Z × Z × R → Z × R giv es a unique action ( n, ρ ( k + m Z , x )) 7→ ρ ( n + k + m Z , x ) : Z × | C ( Z ) | R → | C ( Z ) | R represen ting the action of H ( | C ( Z ) | R ) on | C ( Z ) | R . ⊓ ⊔ W e no w generali ze Examples 2.7(i ) to monotheti c compact groups by utilizing Example 2. 7(ii). Main Lemma 2.8. Let G b e a compact group wi th a noniden ti t y elemen t g . Let C ( G ) b e the top olo gical Ca yl ey gra ph of ( G, g ) . Let L = ( L, b 1 , b 2 ) b e a compact l ink. Let F : L → I b e a morphism of l i nks on to the in terv al. Then the follo wing conclusions hold: (i) Z acts freely with discrete orbits on G × | C ( Z ) | L via m · ( a, x ) = ( a m , − m · x ) and the compact orbit space ( G × | C ( Z ) | L ) / Z is naturall y i somorphic to | C ( G ) | L . It is local ly homeomorphic to G × | C ( Z ) | L under the orbit map o f the Z -actio n. (ii) If G i s monotheti c wit h generator g , then | C ( G ) | L is a compact connected Hausdorff space. If G is profinite monothetic, then | C ( G ) | I is homeomorphic to a solenoid ( p -adic if G = Z p , the additive group of p -adic in tegers), and F ∗ : | C ( G ) | L → | C ( G ) | I is a morph ism of realisations o v er a solenoid. (iii) The group G acts on G × | C ( Z ) | L b y the left regular action on the l eft factor. This act i on comm utes with the Z -action. It thus induces an action ( a, Z · ( b, x )) 7→ Z · ( a + b, x ) : G × ( G × | C ( Z ) | L ) / Z → G × | C ( Z ) | L ) / Z . The orbit space | C ( G ) | L /G i s homeomorphic to t he space L/ { b 1 , b 2 } arising from L b y iden tifyi ng the t w o p oi n ts { b 1 , b 2 } . Pro of. ( i ) The assertio ns on the action are straightforw ard. No w | C ( Z ) | L = ( Z × L ) /ρ has ˙ L def = ρ − 1 ( { 0 } × L ) / ρ ∼ = L as a fundamen tal domain for the Z action, i.e. eac h orbit meets ˙ L only once except the orbit of b 1 whic h meets ˙ L in { ˙ b 1 , ˙ b 2 } . No w w e deduce that in this spirit G × ˙ L is a fundamen tal domain of the Z -actio n on G × | C ( Z ) | L Th us the orbit space ( G × | C ( Z ) | L ) / Z is a con tin uous image of G × ˙ L and therefore is compact. Next we denote b y σ t he equiv alence rel a tion on Z × L whic h iden tifies ( n, b 2 ) and ( n + 1 , b 1 ) for all n ∈ Z so that | C ( Z ) | L = ( Z × L ) /σ . W e let q : G × | C ( G ) | L → 5 ( G × | C ( G ) | L ) / Z b e t he orbit map of the Z action. If Z · ( g , m, x ) = { ( g n , m − n, x ) : n ∈ Z } we set α ( Z · ( g , m , x )) = { ( g n , σ ( m − n, x )) : n ∈ Z . F urther we let π : ( G × Z ) / Z → G b e the isomorphism given b y π ( g ) = ( g , z )∆ for ∆ = { ( g n , − n ) : n ∈ Z } . Let ρ on G × L be the equiv alence rela tion whic h coll apses ( ag , b 1 ) and ( a, b 2 ) for all a ∈ G . Also, we define β : G × Z Z × L → G × L Z b y β ( { ( ag n , m − n ) : n ∈ Z } , x ) = β ( π ( a ) , x ) = ρ ( a, x ) and γ : G × L ρ → G ×| C ( Z ) | L Z b y γ ( ρ ( a, x )) = { ( ag n , σ ( m − n, x )) : n ∈ Z } . W e then hav e a commutativ e diagra m G × Z × L Q − − → G × Z × L Z Γ ← − − G × Z Z × L π × id L ← − − − G × L id G × σ   y α   y   y β   y ρ G × | C ( Z ) | L − − → q G ×| C ( Z ) | L Z ← − − γ G × L ρ = | C ( G ) | L . Indeed the comm uting of the first rectangle is clear from the definition of the action of Z on G × | C ( Z ) | L , and the comm uting of the right rectangle is an immediate consequence of the definitions. The middle rectangle, ho wev er, comm utes since for all a ∈ G and m ∈ Z w e ha ve α ◦ Γ( { ( a g n , m − n ) : n ∈ Z } , b 2 ) = α ( { ag n , m − n, b 2 ) : n ∈ N } ) = { ( ag n , σ ( m − n, b 2 )) : n ∈ Z } = { ( ag n , σ ( m − n + 1 , b 1 )) : n ∈ Z } (1) while γ ◦ β ( { ( ag n , m − n ) : n ∈ Z } , b 2 ) = γ ρ ( a, b 2 ) = γ ( ρ ( a g , b 1 )) = { ( ag p +1 , σ ( m − p, b 1 )) : p ∈ Z } = { ( ag n , σ ( m − ( n − 1) , b 1 )) : n ∈ Z } . (2) Since (1) and ( 2) are ob vio usly equal , the commuting of the middle rectangle follo ws. W e see at once that γ is surjective since α is surjectiv e. W e notice t hat γ ( ρ ( a, x )) = γ ( ρ ( a ′ , x ′ )) iff { ( ag n , σ ( m − n, x )) : n ∈ Z } = { ( a ′ g n ′ , σ ( m − n ′ ) , x ′ ) : n ′ ∈ Z } If x / ∈ { b 1 , b 2 } these Z =orbits on agree iff and only if their in tersections with the fundamen tal domain G × ˙ L agree. But σ ( k , x ) ∈ ˙ L iff one of t he three p ossibilities apply: (i) k = 0, (ii ) k = 1 and x = b 1 , or (ii i) k = − 1 and x = b 2 . In the first case, n = m = n ′ and a = a ′ , x = x ′ follo w. In the second case n = m + 1 = n ′ + 1 is a p ossibil it y , whence ag m +1 = a ′ g m that is a ′ = a g , x ′ = b 1 and x = b 2 , which implies that ρ ( a, x ) = ρ ( a ′ , x ′ ). The other cases are discussed similarly and likewise y i eld ρ ( a, x ) = ρ ( a ′ , x ′ ). Therefore the contin uous function γ 6 b et wee n compact Hausdorff spaces is bijectiv e and t herefore is a homeomorphism. This completes the pro of of (i) (ii) First we ha ve to pro ve connec ti vity of | C ( G ) | L . The space A = | C ( Z ) | L is arcwise connected. Si nce g Z is dense in G , the i mage Z · | C ( Z ) | L / Z o f A is dense in ( G × | C ( Z ) | L ) / Z which is natural ly homeomorphic | C ( G ) | L . Hence the la tter space is con nected. No w let A b e a subgroup of the discrete gro up Q con taining Z suc h that T def = A / Z is infinite. Then the c haracter group S def = b A is a compact connected one-dimensional ab eli an group called a solenoid . The ch aracter group G = b T is profinite monot hetic. By 2.7(i i) and (i) we kno w t hat | C ( G ) | I is homeomorophic to the quotien t group ( G × R ) / ∆ for the subgroup ∆ = { ( n , − n ) : n ∈ Z } . This quoti en t i s one of the w ell -k no wn represen tations of the solenoid S . (See e.g. [ 15], Exercise E1.11 ., Theorem 8.2 2.) (iii) W e i dentify | C ( G ) | L with ( G × | C ( Z ) | L ) / Z . The assertions are straigh t- forw ard. The orbit space | C ( G ) | L /G =  ( G × | C ( Z ) | L ) / Z  /G is isomorphic to  ( G × | C ( Z ) | L ) /G  / Z ∼ = | C ( Z ) | L / Z ∼ = L/ { b 1 , b 2 } . ⊓ ⊔ Theorem 2.9. F or any pr ofinite monothetic gr oup G ther e is a c omp act c on- ne cte d 1-dimensiona l sp ac e X such that H ( X ) ∼ = G . Pro of. Let g ∈ G b e a generator of G . If g has finite order, there is nothing to pro ve b ecause t he a ssertio n w as established i n Example 2. 7(i). W e therefore assume for t he rest of the pro of that g has infinite order. W e apply Main Lemma 2.8 wit h X = | C ( G ) | R for a doubly p oin ted de Gro ot con tin uum R . By the Main Lemma 2.8 we kno w that X is a compact connected space which is lo cally homeomorphic to the space G × | C ( Z ) | R whic h is one-dimensional since R is one- dimensional. Therefore X is one-dimensional. W e hav e to prov e that H ( X ) ∼ = G . This is the most delicate p orti o n of the pro of. W e iden ti fy X with ( G × | C ( Z ) | R ) / Z . F or h ∈ G let γ h : X → X b e defined b y γ h ( ξ ) = h · ξ for the G -action on P . Let ϕ be a homeomorphism of X . W e claim that ϕ is of the form γ h for some h ∈ G , that is γ h ( Z · ( a, x )) = Z . ( h + a , x ). The path comp onen ts of G × | C ( Z ) | R are the spaces { a } × | C ( Z ) | R since G is totally discon nected, and they are p ermuted b y the acti on of G on t he left factor b y the regular represen t a tion. W e may t herefore consider a = 0 without lo ss of generalit y . The orbit O def = Z · (0 , σ ( 0 , b 1 )) = { ( n · g , σ ( − n, b 1 )) : n ∈ Z } meets { 0 } × | C ( Z ) | R in an element ( n · g , σ ( − n, b 1 )) only if n · g = 0 iff n = 0 since g has 7 infinite order. It follows that the orbit map q : G × | C ( Z ) | R → X maps each set { a } × | C ( Z ) | R con tin uously and bijectiv ely on to Z a def = Z . ( { a } × | C ( Z ) | R ) Z = Z · a × | C ( Z ) | R Z for a ∈ G . In particular, each of the sets Z a is arcwise connected. W e claim t hat the Z a are t he arc comp onen ts. This is well-kno wn in the case of the solenoid S = | C ( G ) | I ∼ = ( G × R ) / ∆ ∼ = b A (cf. 2.8( ii) a nd its proof and [ 15], Theorem 8. 3 0). W e ha v e the morphism F ∗ : | C ( G ) | R → | C ( G ) | I = S mapping Z 0 to ( { 0 } × R )∆ / ∆ = ( Z × R ) / ∆ the iden ti ty arc compo nen t of the solenoid S . Differen t sets Z a and Z b are mapp ed to differen t arc comp onen ts in S and so there can b e no arc connecting a p oin t in Z a to a point in Z b . Hence the Z a are precisely the arc comp onents of | C ( G ) | R . Th us the homeomorphism ϕ permutes the sets Z a W e consider the part i cular arc comp onen t Z = Z 0 = Z . ( { 0 } × | C ( Z ) | R ) / Z . Assume that ϕ ( Z ) = Z a . Then γ − 1 a ϕ is a homeomorphism of X which maps the arc comp onen t Z in to it self. The function ε : | C ( Z ) | R → Z , ε ( x ) = Z · σ (0 , x ) is a con tin uous bijec ti on. W e shall now inv ok e t he arc comp o nen t to p ology att ac hed functoriall y to a top ologi cal space a s summarized in [15], App endix 4, p. 781. Since R and t hus | C ( Z ) | R are lo cally arcwise connected, the bijective function ε | C ( Z ) | R → Z is t he univ ersal map ε Z : Z α → Z of Lemma A4.1 of [15], p. 781 . T herefore, b y Lemma A4.1(iv ) every homeomorphism of Z lifts uniquely to a homeomorphism of | C ( Z ) | R and thus is an act i on of m ∈ Z on | C ( Z ) | R b y E x ample 2.7( i i). By the definition of the Z -action on G × | C ( Z ) | R according to 2 .8(i), the action b y m on | C ( Z ) | R when pushed do wn to Z is i nduced b y the action of γ m : X → X , X = ( G × | C ( Z ) | R ) / Z . Therefore the homeomorphism γ m − a ϕ = γ m γ − 1 a ϕ fixes the arc comp onen t Z ele- men t wise. Ho w eve r, Z is dense in X . Hence γ m − a ϕ = id X . Th us ϕ = γ a − m and this had to b e sho wn. ⊓ ⊔ As we hav e noted in the pro of o f 2.8(ii) , a compact profinite group G is monothetic, iff its ch aract er group b G is isomorphic to a subgroup A/ Z of the group 8 Q / Z . The solenoid attached to this monothetic group is the ch aracter group b A of A ⊆ Q . Our f eeling is t hat t he o ccurrence of compact homeomorphism groups, giv en certain restrictions, i s not so rare as one might surmise initially ev en though t he construction of compact spaces having a given homeomorphism group requires w ork. Ideally , one should be able to prov e the fol lo wing Conjecture . L et G b e a c omp act gr oup. Then the fol lowing c onditions ar e e q ui v - alent: (1) Ther e is a c omp act c onne cte d sp ac e X s uch that G ∼ = H ( X ) . (2) Ther e is a c omp act sp ac e X s uch that G ∼ = H ( X ) . (3) G is pr ofinite. Note that we hav e (1) ⇒ (tri v ially) (2) ⇒ (3); the impli cation (3) ⇒ (1) w e hav e pro ved only for compact monothetic groups G and Gar ts ide and Gl yn hav e pro ved it for arbitrary metric profinite groups. Cantor groups Z (2) X (with arbi- trary exp onen t S ) Kees li ng [18] w as able to represe nt as homeomorphism groups of one-dimensional metric spaces. 3. References [1] A nderson, R. D. , The algebraic simplicity of certain groups of homeomor- phisms, Amer. J. Math. 80 ( 1958), 955-963 . [4] A rhangel’ skii, A . V, and M. Tk achen ko, T op ological Groups and Related Structures, A tlan tis Studies in Mathematics 1 , 2008, 781pp. [2] A rens, R. F., T op ologies for homeomorphism groups, Amer. J. Mat h. 68 (1946), 59 3 -610. [5] B ourbaki, N., T op ol o gie g´ en ´ erale, many publishers from Hermann, P ari s, ca 1950, t o Springer Berl in, etc., ca 2000. [6] B redon, G., Introduction to Compact T ransformation Groups, Academic Press, N ew Y ork, 197 2. 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[14] Elli s, R. , Lo cal ly compact transformation g roups, Duk e Math. J. 24 ( 1957), 119–126 [15] Hofmann, K. H., and S. A. Morris, The Structure of Compact Groups, V erlag W alt er De Gruyter Berlin, 1998, xvii+834pp. Second Revised and Augmen ted Edition 2 0 06, xvii i+858pp. [16] —, Compact Homeomorphism Groups are Profinite, P r eprint [17] Keesling, J., Lo cally compact full homeomorphism groups are zero dimen- sional, P ro c. Amer. Math. So c. 29 (1 971), 390–396. [18] —, The group of homeomorphisms of a solenoid, T rans. Amer. Math. So c. 172 (1972 ) , 390–396. [19] Rybicky , T., Comm utat ors of homeomorphisms of a manifold, Universitatis Jagellonicae Act a Math 23 (19 9 6), 153–1960 . [20] tom Diec k, T. , T ransformation Groups, V erlag W alter De Gruyter Berlin, 19 87, x+312pp. Karl H Hofmann F ac hbere ich Mathematik T echnisc he Universit¨ at D armstadt Schloss gartenstrasse 7 64289 Darmstadt, German y hofmann@mathematik. tu-darmstadt.de Sidney A. Morris School of Science, IT, and Enginee ring Universit y of Ball arat P .O. 663, Ballarat Victoria 3353, Australia, and School of Engineering and Mathematical Sciences La T robe University , Bundoora Victoria 3086, Australia morris.sidney@gmail .com 10