Partial actions of groups on profinite spaces
aa r X i v : . [ m a t h . GN ] J a n Partial actions of groups on profinite spaces
Luis Mart´ınez, H´ector Pinedo and Andr´es Villamizar
Escuela de Matem´aticasUniversidad Industrial de SantanderCra. 27 calle 9, Bucaramanga, Colombiae-mail: [email protected], [email protected], [email protected]
January 14, 2021
Abstract
We show that for a partial action η with closed domain of a compact group G on aprofinite space X the space of orbits X/ ∼ G is profinite, this leads to the fact that when G is profinite then the enveloping space X G is also profinite. Moreover, we provide conditionsfor the induced quotient map π G : X → X/ ∼ G of η to have a continuous section. To provethis result we use enveloping actions to construct partial actions of quotient groups on orbitspaces. Relations with continuous sections of the quotient map induced by the envelopingaction of η are also considered. At the end of this work we treat with Exel semigroup andthe action groupoid, in particular we give conditions for them to be profinite spaces. Primary 54H15. Secondary 54H11, 54E99.
Key Words:
Partial actions, profinite space, orbit equivalence relation, globalization, contin-uous section, Borel section.
A topological space X is called profinite if there exists an inverse system of finite discretespaces for which its inverse limit is homeomorphic to X, equivalently by [11, Proposition 2.4] X is profinite if it is compact Hausdorff and zero-dimensional. A topological group is profiniteif it is profinite as a topological space. Important examples of profinite groups and spacesare the group of p -adic integers, p prime, the Galois group on an arbitrary Galois extension,fundamental groups of connected schemes and the set of connected components of a compactHausdorff space. For details in profinite spaces and profinite groups the interested reader mayconsult [16] or [17]. 1n the other hand, partial actions of groups appeared in the context of C ∗ -algebras in [6],in which C ∗ -algebraic crossed products by partial automorphism were introduced and studiedby analyzing view of their internal structure. Since then, partial group actions have appearedin many different context, such as the theory of operator algebras, Galois cohomology, Hopfalgebras, Polish spaces, the theory of R -trees and model theory (see [5] for a detail accountin recent developments on partial actions). It is worth to mention that a deep application ofpartial actions is given in [3] where it was provided a counter-example for a conjecture of M.Rørdam and A. Sierakovski related the Banach-Tarski paradox.A natural question is whether a partial action obtained as restriction of a correspondingcollection of total maps on some superspace. In the topological context, this problem wasstudied in [1] and [10]. They showed that for any partial action η = { η g } g ∈ G of a topologicalgroup G on a topological space X, there is a topological space Y and a continuous action µ of G on Y such that X is embedded in Y and η is the restriction of µ to X. Such a space Y is called a globalization of X. They also show that there is a minimal globalization X G calledthe enveloping space of X (see Definition 2.9). However, structural properties of X are not ingeneral inherited by X G , for instance in [9] it is shown that the enveloping space of a partialaction of a countable group on compact metric spaces is Hausdorff iff the domain of each η g isclopen for all g ∈ G, while in [13] there were established conditions for which X G is a Polishspace provided that X and G are.The present work is structured as follows. After the introduction, in Section 2 we provide thenecessary background and notations on (set theoretic) partial and topological partial actions,and their corresponding globalization. Also we give some preliminary results that will be neededin the work. In Section 3 we work with partial actions on a profinite space X and present inTheorem 3.2 a sufficient condition for the space X/ ∼ G to be profinite, where ∼ G is the orbitequivalence relation determined by the partial action (see equation (3.1)). Later we treat theproblem of finding a continuous section to the quotient map π G : X → X/ ∼ G and show theexistence of such a section when the group G is finite (see Proposition 3.4) this result is extendedin Theorem 3.10 to the case G profinite, to do so one needs to work with partial actions ofquotient groups, which are not possible to define in a natural way as classical (global) actions.Thus in Lemma 3.6 and Lemma 3.7 we use globalizations and induced partial actions to providea partial action of a quotient group of G on a certain quotient space. Some applications toquotient maps induced by enveloping actions and enveloping spaces are presented in Subsection3.2, in particular having a continuous section of π G we show how to find a continuous sectionof Π G (the corresponding quotient map induced by globalization), at this point it is importantto remark that the converse does not seem to be true, that is having a continuous section ofΠ G does not seem to imply that π G has a continuous section, items 4) and 5) of Proposition2.13 deal with this problem (see also Proposition 3.14). We end this work with Section 4 byconsidering the Exel semigroup, and the groupoid action in the context of profinite spaces. In this section we state our conventions on partial actions, and prove some results that will beuseful throughout the work. We start with the following.
Definition 2.1. [10, p. 87-88] Let G be a group with identity element 1 and X be a set. Apartially defined function η : G × X X , ( g, x ) g · x is called a (set theoretic) partial action of G on X if for each g, h ∈ G and x ∈ X the following assertions hold:(PA1) If ∃ g · x , then ∃ g − · ( g · x ) and g − · ( g · x ) = x ,(PA2) If ∃ g · ( h · x ), then ∃ ( gh ) · x and g · ( h · x ) = ( gh ) · x ,(PA3) ∃ · x and 1 · x = x, where ∃ g · x means that g · x is defined. We say that η acts (globally) on X if ∃ g · x, for all( g, x ) ∈ G × X. Remark 2.2.
In [10] a partial action is only required to satisfy (PA1) and (PA2), and it iscalled unital if also (PA3) holds, so for us partial actions are always unital.Let η be a partial action of G on X, g ∈ G, x ∈ X and U a subset of X. We fix the followingnotations: • G ∗ U = { ( g, u ) ∈ G × U | ∃ g · u } . In particular G ∗ X is the domain of η. • G x = { g ∈ G | ∃ g · x } , G U = S u ∈ U G u and G U · U = { g · u | u ∈ U, g ∈ G u } . • X g = { x ∈ X | ∃ g − · x } . Then η induces a family of bijections { η g : X g − ∋ x g · x ∈ X g } g ∈ G , such that η e isthe identity of X and η g − = η − g . We also denoted this family by η. The following resultcharacterizes partial actions in terms of a family of bijections.
Proposition 2.3. [15, Lemma 1.2] A partial action η of G on X is a family η = { η g : X g − → X g } g ∈ G , where X g ⊆ X, η g : X g − → X g is bijective, for all g ∈ G, and such that:(i) X = X and η = id X ;(ii) η g ( X g − ∩ X h ) = X g ∩ X gh ;(iii) η g η h : X h − ∩ X h − g − → X g ∩ X gh , and η g η h = η gh in X h − ∩ X h − g − ;3or all g, h ∈ G. Example 2.4. [13, Example 2.3] Let η be a partial action of G on X . Consider the family ofbijections ˆ η = { ˆ η g : ( G × X ) g − → ( G × X ) g } g ∈ G , where ( G × X ) g = G × X g and ˆ η g ( h, x ) =( hg − , η g ( x )) for each g ∈ G and ( h, x ) ∈ G × X g − . Then ˆ η is a partial action of G on G × X. The partial action ˆ η defined above, will be useful for our purposes. Definition 2.5.
Let η : G ∗ X → X be a partial action of G on X and U ⊆ X, then U is G -invariant if G U · U ⊆ U, or equivalently, G U · U = U. We have the next.
Lemma 2.6.
Let η be a partial action of G on X and U a nonempty subset of X . Then thefollowing statements are true:(i) η ( G ∗ U ) = G U · U .(ii) G U · U and its complement are G -invariant. Proof.
Statement ( i ) is straightforward. To show ( ii ) denote A := G U · U and B = X \ A . Theassumption G A · A ⊆ A follows from (PA2). Now we check G B · B ⊆ B . Take y ∈ G B · B ,then there are x ∈ B and g ∈ G x such that η g ( x ) = y . Suppose y ∈ A and let m ∈ U, h ∈ G m such that η h ( m ) = y . Note that η h ( m ) ∈ X h and η h ( m ) = y ∈ X g , then m ∈ η − h ( X h ∩ X g ) = X h − ∩ X h − g and by (iii) of Proposition 2.3 follows that η g − h ( m ) = x , this implies that x ∈ A ,which is a contradiction, therefore y ∈ B .From now on in this work G will denote a topological group and X a topological space, weendow G × X with the product topology and G ∗ X with the induced topology of subspace.Moreover η : G ∗ X → X will denote a partial action. We say that η is a topological partialaction if every X g is open and η g is a homeomorphism, g ∈ G, if moreover η is continuous, wesay that η is a continuous partial action .Respect the partial action ˆ η given in Example 2.4 we have the next. Proposition 2.7.
Let η be a topological partial action of G on X . Then η is continuous if andonly if ˆ η is continuous.Proof. Suppose that η is continuous. It is clear that ˆ η is a topological partial action. To showthat ˆ η is continuous set α : G ∗ ( G × X ) ∋ ( g, ( h, x )) hg − ∈ G and β : G ∗ ( G × X ) ∋ ( g, ( h, x )) η g ( x ) ∈ X. (2.1)We show that α and β are continuous, for this consider U ⊆ G an open set for which α ( g, ( h, x )) ∈ U for some ( g, ( h, x )) ∈ G ∗ ( G × X ). Consider V and W open subsets of G g, h ) ∈ V × W and W V − ⊆ U and define Z := ( V × G × W ) ∩ G ∗ ( G × X ).It is easy to see that ( g, ( h, x )) ∈ Z and α ( Z ) ⊆ W V − ⊆ U . This shows that α is con-tinuous. On the other hand, let U ⊆ X be a open in X such that β ( g, ( h, x )) ∈ U for some( g, ( h, x )). Since η is continuous there are open sets T ⊆ G , H ⊆ X such that ( g, x ) ∈ T × H and η (( T × H ) ∩ G ∗ X ) ⊆ U. Let Z := ( T × G × H ) ∩ G ∗ ( G × X ) then ( g, ( h, x )) ∈ Z and β ( Z ) ⊆ η (( T × X ) ∩ G ∗ X ) ⊆ U . Thus we conclude that ˆ η is continuous.Conversely, suppose that ˆ η is continuous. Let U ⊆ X be an open set such that η ( g, x ) ∈ U for some ( g, x ) ∈ G ∗ X . Note that ( g, (1 , x )) ∈ G ∗ ( G × X ) and β ( g, (1 , x )) = η ( g, x ) ∈ U .Since the map β defined in (2.1) satisfies β = π ◦ ˆ η, where π : G × X ∋ ( g, x ) x ∈ X then it iscontinuous and there are open subsets M, N ⊆ G and V ⊆ X such that ( g, (1 , x )) ∈ M × N × V and β ( Z ) ⊆ U , where Z := ( M × N × V ) ∩ G ∗ ( G × X ). Define Z ′ := ( M × V ) ∩ G ∗ X . Notethat ( g, x ) ∈ Z ′ , moreover η ( Z ′ ) ⊆ β ( Z ) ⊆ U, and η is continuous as desired. Lemma 2.8.
Let η be a partial action of G on X. Then the following assertions hold:(i) If U is an open subset of X, G U · U is open.(ii) If G ∗ X is clopen then X g is clopen for all g ∈ G. Proof.
The first assertion follows from the fact that for every g ∈ G the set X g is open and G U · U = S g ∈ G U η g ( U ∩ X g − ) . We check the second one, take g ∈ G , first let us prove that X g is closed. Let x ∈ X \ X g , since G ∗ X is closed there open sets U ⊆ G and V ⊆ X such that( g − , x ) ∈ U × V ⊆ ( G ∗ X ) c . Moreover, if y ∈ V then ( g − , y ) / ∈ G ∗ X and y / ∈ X g . This showsthat X g is closed. To prove that X g is open, take x ∈ X g then ( g − , x ) ∈ G ∗ X and there areopen sets U ⊆ G and V ⊆ X such that ( g − , x ) ∈ U × V ⊆ G ∗ X from this we get x ∈ V ⊆ X g and X g is open. Let u : G × Y → Y be a continuous action of G on a topological space Y and X ⊆ Y an openset. For g ∈ G, set X g = X ∩ u g ( X ) and let η g = u g ↾ X g − . (2.2)Then η : G ∗ X ∋ ( g, x ) η g ( x ) ∈ X is a topological partial action of G on X. In this case wesay that η is induced by u. An important question in the study of partial actions is whether they can be induced byglobal actions. In the topological sense, this turns out to be affirmative and a proof was givenin [1, Theorem 1.1] and independently in [10, Section 3.1]. We recall their construction. Let η be a topological partial action of G on X. Define an equivalence relation on G × X as follows:( g, x ) R ( h, y ) ⇐⇒ x ∈ X g − h and η h − g ( x ) = y, (2.3)5nd denote by [ g, x ] the equivalence class of the pair ( g, x ) . Consider the set X G = ( G × X ) /R endowed the quotient topology. Then by [1, Theorem 1.1] the action µ : G × X G ∋ ( g, [ h, x ]) → [ gh, x ] ∈ X G , (2.4)is continuous and the map ι : X ∋ x [1 , x ] ∈ X G (2.5)is a continuous injection such that G · ι ( X ) = X G and if G ∗ X is open then ι is open.We finish this section with the next. Definition 2.9.
Let η be a topological partial action of G on X. The action µ provided by (2.4)is called the enveloping action of η and the space X G is the enveloping space or the globalizationof X. Given a topological partial action of G on X, one can define the orbit equivalence relation ∼ G on X as follows: x ∼ G y ⇐⇒ ∃ g ∈ G x such that g · x = y, (3.1)for each x, y ∈ X . The elements of X/ ∼ G are the orbits G x · x with x ∈ X and X/ ∼ G isendowed with the quotient topology, in particular by [13, Lemma 3.2] the induced quotient mapof η π G : X ∋ x G x · x ∈ X/ ∼ G (3.2)is continuous and open.Our next goal is to show that X/ ∼ G is profinite provided that X is profinite and G compact,but first for the reader’s convenience we give the next. Definition 3.1.
Two points u and v in X can be separated if there are disjoint open subsets U and V of X such that u ∈ U, v ∈ V and U ∪ V = X. It is known that any two different points in a compact space X can be separated, if and onlyif, X is Hausdorff and has a basis consisting of clopen sets (see for instance [11, Proposition2.3]). Theorem 3.2.
Let η : G ∗ X → X be a continuous partial action of a compact group G on aprofinite space X such that G ∗ X is closed, then X/ ∼ G is a profinite space.Proof. First of all note that X/ ∼ G is compact. Now we show that different points G x · x, G y · y ∈ X/ ∼ G can be separated. Let C := { U ⊆ X : U is clopen , x ∈ U } , then C 6 = ∅ . We claim that6here exist U ∈ C such that G U · U ∩ G y · y = ∅ . On the contrary, for each V ∈ C the set˜ F y ( V ) = { ( g, v ) ∈ G ∗ V : η g ( v ) = y } is nonempty. Since ˜ F y ( V ) = η − ( y ) ∩ G ∗ V , it is closedin G ∗ V and thus closed in G ∗ X . Now, if V , V ∈ C , then ˜ F y ( V ∩ V ) ⊆ ˜ F y ( V ) ∩ ˜ F y ( V ). Inthat sense, { ˜ F y ( V ) } V ∈C is a family in G ∗ X with the finite intersection property. Thus, thereexist ( g, v ) ∈ T V ∈C ˜ F y ( V ) which implies v = x and η g ( x ) = y and leads to a contradiction.Then there exist U ∈ C such that G U · U ∩ G y · y = ∅ . Now we check that G U · U is clopen,indeed it is open thanks to Lemma 2.8, moreover since G ∗ U is compact, X is Hausdorff and η ( G ∗ U ) = G U · U , we have that G U · U is closed. Let A = G U · U and B = X \ A by Lemma2.6 the sets A and B are G -invariant and clopen, then π − G ( π G ( A )) = A y π − G ( π G ( B )) = B and G x · x y G y · y are separated by the sets π G ( A ) and π G ( B ) , respectively. In this section we are interested in providing conditions for which the quotient map π G definedin (3.2) has a continuous section, for this we adapt some of the ideas presented in [11, Section2.4] to the context of partial actions. It is important to remark that the partial case is essentiallymore laborious than the classical one.First we give the next. Definition 3.3.
Let η : G ∗ X −→ X be a topological partial action of a group G on a set X .We say that η is free if for each ( g, x ) ∈ G ∗ X such that η ( g, x ) = x , we have g = 1. Proposition 3.4.
Let G be a finite and discrete group, η a continuous and free partial actionsuch that G ∗ X is clopen. Then π G : X −→ X/ ∼ G has a continuous section.Proof. Let x ∈ X . We have that η g ( x ) = η h ( x ) for each g, h ∈ G x such that g = h because η is free. Since G is finite and X profinite, for each g ∈ G x there is a clopen set U g suchthat η g ( x ) ∈ U g and U g ∩ U h = ∅ , if g = h . Since x ∈ η − g ( U g ∩ X g ) for each g ∈ G x , x ∈ V := T g ∈ G x η − g ( U g ∩ X g ), then V is clopen thanks to the second item of Lemma 2.8. Wewill show that π G ↾ V : V → π G ( V ) is a homeomorphism. To show the injectivity, note that if η g ( V ) ∩ V = ∅ for some g ∈ G x then g = 1 , indeed let y ∈ η g ( V ) ∩ V since V ⊆ η g − ( U g ∩ X g ),then y ∈ U g . Thus y ∈ U g ∩ U and g = 1. Now, let a, b ∈ V with π G ( a ) = π G ( b ) then there exists g ∈ G a such that η g ( a ) = b . Thus, b ∈ η g ( X g − ∩ V ) ∩ V and for the previous reasoning g = 1.This shows that π G ↾ V is injective. Moreover, since V is compact and π G ( V ) is Hausdorff, wehave that π G ↾ V is a homeomorphism. The inverse of π G ↾ V is a continuous section of π G over π G ( V ). To finish the proof, note that π G ( V ) is clopen since π − G ( π G ( V )) = S g ∈ G η g ( X g − ∩ V )is clopen. Thus, for each x ∈ X , there exist a clopen neighborhood V x = π G ( V ) of π G ( x )and a continuous section q ↾ V x : V x → X of π G ↾ V . Since X is compact there exist m ∈ N and V , V , · · · , V m such that X = m S i =1 V i , moreover by [11, Lemma 2.4] there is a refinement7 W j } nj =1 of { V i } mi =1 such that W i ∩ W j = ∅ , if i = j and using the family { q ↾ W j } nj =1 we obtaina continuous section of π G . Example 3.5. [8, p. 22]
Partial Bernoulli action
Let G be a discrete group and X := { , } G . There is a continuous global action β = { β g } , where for all ω ∈ X , β g ( ω ) = gω .The topological partial Bernoulli action η is obtained by restricting β to the open set Ω = { ω ∈ X : ω (1) = 1 } . Thus by (2.2) D g := Ω ∩ β g (Ω ) = { ω ∈ X : ω (1) = 1 = ω ( g ) } and η g = β g ↾ D − g , g ∈ G . Let us show that G ∗ Ω is clopen. Let { ( n i , x i ) } i ∈ I be a convergent netto ( n, x ), since G is discrete then ( n i ) i ∈ I is constant and for all i ∈ I , n i = n . On the otherhand, as x i −→ x then 1 = x i (1) −→ x (1) and from this x (1) = 1. Similarly it is obtainedthat x ( n − ) = 1 and from the above we conclude ( n, x ) ∈ G ∗ Ω . On the other hand, take( n, x ) ∈ G ∗ Ω . Then x ∈ V = ( π n − ↾ Ω ) − ( { } ) and ( n, x ) ∈ { n } × V ⊆ G ∗ Ω . This showsthat G ∗ Ω is clopen. Thus if G is finite Theorem 3.2 implies that X/ ∼ G is a profinite space.We shall extend Proposition 3.4 to the case when G is profinite, but first we use globalizationsto obtain partial actions of quotient groups on certain orbit spaces.Let η be a topological partial action of G on X . Then for any subgroup H of G η inducesby restriction a topological partial action η H of H on X . The corresponding orbit equivalencerelation of η H is denoted by ∼ H . On the other hand it is necessary to clarify at this point thatthe orbits in the space X G / ∼ H will be denoted by H [ g, x ] for an element [ g, x ] ∈ X G . Lemma 3.6.
Let η be a partial action of G on X with G ∗ X open and X G its envelopingspace. Then for each subgroup H the map ϕ : X/ ∼ H ∋ H x · x H [1 , x ] ∈ X G / ∼ H (3.3)is an embedding, that is continuous, open and injective. Proof.
First of all note that ϕ is well defined. In fact, let x, y ∈ X such that x ∼ H y and take h ∈ H x with η h ( x ) = y . Thus, [1 , y ] (2.3) = [ h, x ] (2.4) = µ h ([1 , x ]) and [1 , y ] ∼ H [1 , x ]. This shows that ϕ is well defined, it is easy to check that ϕ is injective. To prove that ϕ is continuous, consider π H : X → X/ ∼ H and Π H : X G → X G / ∼ H the corresponding projection functions. Since themap ι defined in (2.5) is continuous and ϕ ◦ π H = Π H ◦ ι , we conclude that ϕ is continuous. Itremains to check that ϕ is open, let U ⊆ X/ ∼ H open, then ϕ ( U ) = Π H ( ι ( π − H ( U ))) is openbecause π − H ( U ) is open in X and the functions ι and Π H are open.We proceed with the next. Lemma 3.7.
Let η be a free partial action of a profinite group G on a profinite space X , and H a closed normal subgroup of G . Then there is a continuous free partial action η G/H of G/H on X/ ∼ H , such that the spaces ( X/ ∼ H ) / ∼ G/H and X/ ∼ G are homeomorphic.8 roof. Let µ be the globalization of η . Then µ is continuous and induces a continuous action τ G/H on X G / ∼ H as follows: τ gH : X G / ∼ H ∋ Π H ([ t, x ]) Π H ([ gt, x ]) ∈ X G / ∼ H , for each gH ∈ G/H . Note that τ G/H is free. In fact, let g, t ∈ G and x ∈ X such thatΠ H ([ gt, x ]) = Π H ([ t, x ]) . Then [ hgt, x ] = [ t, x ] for some h ∈ H and by (2.3) we have x ∈ X ( hgt ) − t and η t − hgt ( x ) = x , moreover the fact that η is free implies then hg = 1 and g ∈ H , this showsthat gH = H and τ G/H is free. Now let ϕ be defined by (3.3) since ϕ is open, by restricting τ G/H to the open set Im ( ϕ ) to obtain a partial action η ′ G/H = { η ′ gH : X g − H → X gH } gH ∈ G/H of G/H on Im ( ϕ ) , where X gH = τ gH ( Im ( ϕ )) ∩ Im ( ϕ ) and η ′ gH = τ gH ↾ X g − H (see equation2.2), it is not difficult to see that η ′ G/H is continuous and free. On the other hand, using η ′ G/H and the embedding ϕ we construct the partial action η G/H of G/H on T := X/ ∼ H , where T gH = ϕ − ( X gH ) and η gH ( x ) = ϕ − ( η ′ gH ( ϕ ( x ))) = ϕ − ( τ gH ( ϕ ( x ))) , (3.4)for each x ∈ ϕ − ( X g − H ). It is easy to check that η G/H is free and continuous.Let ∼ G/H be the orbit relation in T induced by η G/H . We are going to show that
T / ∼ G/H and X/ ∼ G are homeomorphic. In that sense, consider the following diagram: X π H (cid:15) (cid:15) π G / / X/ ∼ G T π G/H / / T / ∼ G/Hψ O O where ψ is made such that the diagram commutes, that is ψ ( π G/H ( π H ( x ))) = π G ( x ) , (3.5)for each x ∈ X . Let us first prove that ψ is well defined, take z ∈ T / ∼ G/H and x, y ∈ X suchthat π G/H ( π H ( x )) = π G/H ( π H ( y )) = z . Then there is g ∈ G with η gH ( π H ( x )) = π H ( y ), thatis, ϕ − ( τ gH ( ϕ ( π H ( x )))) = π H ( y ) which implies H [ g, x ] = H [1 , y ] and there is h ∈ H such that[ hg, x ] = [1 , y ]. We deduce that η hg ( x ) = y and π G ( x ) = π G ( y ) which shows that ψ is welldefined. Moreover the map ψ is continuous and surjective.Let us prove that ψ is injective. Let z , z ∈ T / ∼ G/H such that ψ ( z ) = ψ ( z ), andlet x, y ∈ X such that π G/H ( π H ( x )) = z and π G/H ( π H ( y )) = z . Since π G ( x ) = π G ( y ),there is g ∈ G x such that η g ( x ) = y . To prove that z = z , we need t ∈ G such that η tH ( π H ( x )) = π H ( y ). We claim that η gH ( π H ( x )) = π H ( y ) . In fact, note that η gH ( π H ( x )) = ϕ − ( τ gH ( ϕ ( π H ( x )))) = ϕ − ( H [ g, x ])and 9 ( π H ( y )) = H [1 , y ] = H [ g, x ] , then η gH ( π H ( x )) = π H ( y ). Thus ψ is injective.Finally let U ⊆ T / ∼ G/H be an open set. Since π G is open, π G ( π − H ( π − G/H ( U ))) ⊆ X/ ∼ G is open. Thus, ψ ( U ) is open and ψ is a homeomorphism.Let H , H be subgroups of G such that H ⊆ H . We define π H ,H : X/ ∼ H → X/ ∼ H asthe only map such that π H = π H ,H ◦ π H , (3.6)in particular for a subgroup H of G the map π H,H is the identity on X/ ∼ H . Remark 3.8.
According to the notations of Lemma 3.6 and Lemma 3.7 suppose that thereexists a continuous section λ of π G/H . Note that λ ◦ ψ − is a continuous section of π H,G . Indeed,take x ∈ X . We have ( λ ◦ ψ − )( π G ( x )) (3.5) = ( λ ◦ π G/H )( π H ( x )) = π H ( y ) , (3.7)where y ∈ X is chosen such that λ ( π G/H ( π H ( x ))) = π H ( y ). From this follows that π G/H ( π H ( x )) = π G/H ( π H ( y )) then there exists g ∈ G such that η gH ( π H ( x )) = π H ( y ), that is π H ( y ) (3.4) = ϕ − ( η ′ gH ( ϕ ( π H ( x ))) (3.3) = ϕ − ( τ gH (Π H ([1 , x ]))) = ϕ − (Π H ([ g, x ])),then Π H ([ g, x ]) = Π H ([1 , y ]) and there is h ∈ H such that η gh ( x ) = y . This shows that π G ( x ) = π G ( y ) . Moreover[ π H,G ◦ ( λ ◦ ψ − )]( π G ( x )) (3.7) = π H,G ( π H ( y )) (3.6) = π G ( y ) = π G ( x ) , as desired.In order to show the main result of this section we recall the following: Lemma 3.9.
The following statements hold. • [17, Proposition 1.1.6] Let { X i ; f ji , I } be an inverse system of compact Hausdorff topo-logical spaces over a directed set I , and ( X ; π i ) be its inverse limit. If Y ⊆ X is such that π i ( Y ) = X i for each i ∈ I , then Y is dense in X . • [11, Lemma 2.12] Let G be a profinite group and N a closed normal subgroup of G . If N = { } , then there is a proper subgroup M of N such that M is open in N and normalin G .We write N E cl G to indicate that N is a closed normal subgroup of G. We give the next.
Theorem 3.10.
Let η be a continuous and free partial action of a profinite group G on aprofinite space X such that G ∗ X is clopen. Then π G : X → X/ ∼ G has a continuous section. roof. Consider X := { ( F, r ) : F E cl G and r is a continuous section of π
F,G } .Note that X 6 = ∅ because ( G, id π G ( X ) ) ∈ X . We define a partial order in X as follows: For each( F, r ) and ( F ′ , r ′ ) ∈ X , we say that( F, r ) ≤ ( F ′ , r ′ ) if and only if F ′ ⊆ F and π F ′ ,F ◦ r ′ = r. Let C = { ( F i , r i ) : i ∈ I } be a chain in X . Then { X/ ∼ F i ; π F i ,F j ; C} is an inverse system oftopological spaces. Consider lim ←− X/ ∼ F i and for each i ∈ I let π i : lim ←− X/ ∼ F i → X/ ∼ F i the corresponding projection map, we are going to prove that lim ←− X/ ∼ F i and X/ ∼ F arehomeomorphic, where F = T i ∈ I F i . In that sense, note that if ( F i , r i ) ≤ ( F j , r j ) then π F j ,F i ◦ π F j = π F i thanks to (3.6), and by the universal property of lim ←− X/ ∼ F i there is a uniquecontinuous map e : X → lim ←− X/ ∼ F i such that π i ◦ e = π F i , for each i ∈ I . We claim that e is surjective. In fact, since π F i is surjective for each i ∈ I we know that e ( X ) is dense thanksto the first item of Lemma 3.9. Moreover lim ←− X/ ∼ F i is Hausdorff by Theorem 3.2 for each i .Then e ( X ) = lim ←− X/ ∼ F i .The prove that induced function e : X/ ∼ F → lim ←− X/ ∼ F i is homeomorphism is analogousto the prove given in [11, Proposition 2.9]. Now, let i, j ∈ I such that ( F i , r i ) ≤ ( F j , r j ).We have r i = π F j ,F i ◦ r j , then by the universal property of lim ←− X/ ∼ F i there is a uniquecontinuous function r : X/ ∼ G → lim ←− X/ ∼ F i such that π i ◦ r = r i for each i ∈ I . Consider q : X/ ∼ G → X/ ∼ F defined by q = e − ◦ r . Let us prove that ( F, q ) ∈ X , take x ∈ X and z := G x · x we need to check that π F,G ( q ( z )) = z . Let t ∈ X such that q ( z ) = F t · t, then r ( z ) = e ( F t · t ) = e ( t ), that is ( r i ( G x · x )) i ∈ I = ( π F i ( t )) i ∈ I . If i ∈ I then G t · t = π F i ,G ( π F i ( t )) = π F i ,G ( r i ( G x · x )) = G x · x, and π F,G ( q ( z )) = G t · t = G x · x = z . Finally we have to prove that ( F i , r i ) ≤ ( F, q ) , for each i ∈ I . Let i ∈ I , z = G x · x ∈ X/ ∼ G and t ∈ X such that q ( z ) = F t · t it follows by theprevious reasoning that r i ( z ) = π F i ( t ), which is π F,F i ( q ( z )) = π F,F i ( F t · t ) = π F i ( t ) = r i ( z ).This shows that ( F i , r i ) ≤ ( F, q ) and by Zorn’s Lemma we conclude that there is a maximalelement (
N, t ) ∈ X . To finish the proof we show that N = { } . Suppose that there exist a open subgroup M of N such that M is normal in G . Since N acts continuously on X we get by Lemma 3.7that N/M acts partially and continuously on T := X/ ∼ M and there is a homeomorphism ψ : T / ∼ N/M → X/ ∼ N . Moreover by [16, Lemma 2.1.2] the group N/M is finite thus there is λ : T / ∼ N/M → T a continuous section of π N/M : T → T / ∼ N/M , thanks to Proposition 3.4 andfollows by 3.8 that λ ◦ ψ − is a continuous section of π M,N , then α := λ ◦ ψ − ◦ t is a continuous11ection of π M,G and (
M, α ) ∈ X . Besides π M,N ◦ α = π M,N ◦ λ ◦ ψ − ◦ t = Id X/ ∼ N ◦ t = t, then ( N, t ) ≤ ( M, α ). Since (
N, t ) is maximal, we conclude that N = M and by the seconditem of Lemma 3.9 we have that N = { } , that is, t is a continuous section of π G . Example 3.11.
Let G be a profinite group with internal operation > , then G acts freelyand continuously on itself with action > . Let us take a open subgroup U of G and consider θ = { θ g : D g − −→ D g } g ∈ G the restriction of > on U , where for each g ∈ G , D g = U ∩ gU .It’s clear that θ is free and continuous. On the other hand note that G ∗ U = G × U andfrom [16, Lemma 2.1.2] it follows that G ∗ U is clopen. By the Theorem 3.10 we conclude that π G : U −→ U/ ∼ G admits a continuous section. Remark 3.12.
Notice that in general the assumption that η acts freely on X, cannot beomitted even when η acts globally, see for instance [16, Example 5.6.8]. Let µ be the globalization of η and X G the enveloping space of X according to Definition 2.9,we apply our results to the study of X G and the relation between continuous sections of themaps π G and Π G , being Π G the corresponding quotient map of the enveloping action µ. Proposition 3.13.
Let η be a continuous partial action of a profinite group G on a profinitespace X. Then the following statements hold.1. If G ∗ X is closed, then X G is profinite.2. If µ is free and G ∗ X is clopen, then Π G has a continuous section.3. If π G has a continuous section so does Π G .4. If G ∗ X is open and q is a continuous section of Π G such that im q ⊆ ι ( X ) , then π G hasa continuous section.5. If Π G and ˆΠ G have continuous sections, then π G has continuous section, where ˆΠ G is thequotient map G × X → X G . Proof.
1) Let ˆ η be given by Example 2.4, by Proposition 2.7 the map ˆ η is continuous, denote ∼ ˆ G the orbit equivalence relation on G × X determined by ˆ η then by [13, Theorem 3.3] we get X G = ( G × X ) / ∼ ˆ G . Thus using Theorem 3.2 it is enough to check that G ∗ ( G × X ) is closed in G × G × X. In that sense, let ( g, ( h, x )) / ∈ G ∗ ( G × X ) then ( h, x ) / ∈ G × X g − and ( g, x ) / ∈ G ∗ X, since G ∗ X is closed there are open sets T ⊆ G and U ⊆ X such that ( g, x ) ⊆ T × U ⊆ ( G ∗ X ) c .12ote that ( g, ( h, x )) ∈ T × ( G × U ) ⊆ ( G ∗ ( G × X )) c . This shows that G ∗ ( G × X ) is closedand we conclude that X G is profinite.2) We know that X G is profinite and that µ is continuous, we check that it is free. Let g ∈ G and [ h, x ] ∈ X G such that [ h, x ] = µ g ([ h, x ]) = [ gh, x ], then by (2.3) x ∈ X h − gh and η h − g − h ( x ) = x, thus g = 1 because η is free. This shows that µ is free and by Theorem (3.10)we conclude the proof.3) Suppose that q : X/ ∼ G → X is a continuous section of π G . Consider s : ( X G / ∼ G ) ∋ G · [1 , x ] ι ( q ( π G ( x ))) ∈ X G , where ι is given by (2.5). We claim that s is a continuous section of Π G . First note that s iswell defined. In fact, let x, y ∈ X such that [1 , x ] ∼ G [1 , y ], we have that π G ( x ) = π G ( y ). Let z x = q ( π G ( x )) and z y = q ( π G ( y )), then π G ( z x ) = π G ( x ) = π G ( y ) = π G ( z y ). Thus s ( G · [1 , x ]) = [1 , z x ] = [1 , z y ] = s ( G · [1 , y ]),and s is well defined. Since q and ι are continuous we get that s is continuous. To finish theproof, take x ∈ X and let y x ∈ X such that q ( π G ( x )) = y x . Since π G ( y x ) = π G ( x ) there is g ∈ G x such that η g ( x ) = y x . Thus µ ( g, [1 , x ]) = [1 , y x ] and we have(Π G ◦ s )( G · [1 , x ]) = Π G ([1 , y x ]) = G · [1 , y x ] = G · [1 , x ] . This shows that s is continuous section of Π G .4) Let r : ( X/ ∼ G ) ∋ G x · x ι − ( q ( G [1 , x ])) ∈ X. It is not difficult to check that r is well defined, moreover the fact that G ∗ X is open implies that ι is open and thus r iscontinuous. Finally take x, z x ∈ X such that q ( G [1 , x ]) = [1 , z x ] , then G [1 , x ] = G [1 , z x ] whichgives π G ( x ) = π G ( z x ) , this implies π G ( r ( G x · x )) = π G ( z x ) = π G ( x ) and r is a continuoussection of π G .5) Let q : X G / ∼ G → X G and t : X G → G × X be continuous sections of Π G and ˆΠ G , respectively. Define p : X/ ∼ G ∋ G x · x proj ( t ( q ( G · [1 , x ]))) ∈ X. We check that p is a continuous section of π G . The map p is well defined and continuous since X/ ∼ G ∋ G x · x G [1 , x ] ∈ X G / ∼ G is continuous. On the other hand, take x ∈ X . Let h, k ∈ G and y, z ∈ X such that q ( G [1 , x ]) = [ h, y ] and t ([ h, y ]) = ( k, z ), then G [1 , x ] = G [ h, y ]and [ k, z ] = [ h, y ]. Thus η h − g ( x ) = y and η k − h ( y ) = z for some g ∈ G , then G x · x = G z · z and we have that π G ( p ( G x · x )) = G z · z = G x · x , as desired.It follows by [14, Lemma 4.2] that ˆΠ G has a Borel section, provided that G and X are alsoa Polish group and a Polish space, respectively. Thus by 1) of Proposition 3.13 and the proof5) of the same Proposition we have the next. 13 roposition 3.14. Let η be a continuous partial action of a profinite polish group G on aprofinite Polish space X such that G ∗ X is closed, then π G has a Borel section. The Exel’s semigroup S ( G ) of G appeared for the first time in [7], where it was shown thatthe actions of S ( G ) are in one-to-one correspondence with the partial actions of G, both inthe case of actions on a set. Later in [10] the authors proved that S ( G ) is isomorphic to theBirget-Rhodes expansion ˜ G R := { ( A, g ) ∈ P ( G ) × G : { , g } ∈ A } of G, where P ( G ) denotesthe finite subsets of G, and thus S ( G ) is identified with ˜ G R . In the topological context this semigroup was considered in [4] as follows: Let G be a locallycompact group and K ( G ) be the semilattice compact subsets of G containing 1 , we endow withthe Vietoris topology, set ˜ G Rc := { ( A, g ) ∈ K ( G ) × G : { , g } ∈ A } , with product ( A, g )( B, h ) = ( A ∪ gB, gh ) . It is shown in [4] that ˜ G Rc is a topological inversemonoid.Now suppose that G is a profinite group, we shall check that ˜ G Rc is profinite. First ofall it is not difficult to check that K ( G ) is a compact Hausdorff space, moreover by [12,Proposition 8.6] the space K ( G ) is zero-dimensional, from this we get that K ( G ) × G isprofinite and thus it is enough to show that ˜ G Rc is closed in K ( G ) × G . In that sense, take( A, g ) ∈ ( K ( G ) × G ) \ ˜ G Rc . Then g ∈ A c and there is a clopen U ⊆ G such that g ∈ U ⊆ A c .Note that ( A, g ) ∈ h U c i × U ⊆ ( K ( G ) × G ) \ ˜ G Rc , and we conclude that ˜ G Rc is closed. Thus˜ G Rc is profinite. The same reasoning is used to prove that ˜ G R is profinite and thus the Exelsemigroup is profinite whenever G is. The shortest definition of groupoid is the following: a groupoid G is a small category closedunder inversions and in which every morphism is an isomorphism.According to [10, p. 100] or [8, p. 8] using a partial action η of a group G on a set X wehave the action groupoid G , where G = G ∗ X as sets, · − : G ∋ ( g, x ) ( g − , g · x ) ∈ G is aunary and partial product ∗ : G × G −→ G is a partial binary operation defined as follows: if( g, x ) , ( h, y ) ∈ G , then ( g, x ) ∗ ( h, y ) = ( gh, y ) provided that x = h · y and undefined otherwise.The groupoid G plays an important role in the connection of partial actions with star injective14unctor on G (see [10, p. 602]) and as being isomorphic (as groupoids) to the graph groupoid itis also relevant in the study of C ∗ -algebras (see [2]).If G and X are profinite and G = G ∗ X is closed, then it is profinite as a topological space.It remains to check that the operations are continuous. We shall do this in the case that G is finite (then discrete). We start with the inversion map. For this take ( g, x ) ∈ G and anopen set V ⊆ G such that ( g − , η g ( x )) ∈ V . Now there are open sets T ⊆ G and T ⊆ X such that V = G ∩ T × T . Consider W := { g } × η g − ( T ∩ X g ), then W ⊆ G is open and( g, x ) ∈ W . Let ( m, n ) ∈ · − ( W ), then there exists ( g, w ) ∈ W such that · − ( g, w ) = ( m, n )( g − , θ g ( w )) = ( m, n ) and we have ( m, n ) ∈ V which shows that the map · − is continuous.Let us now show the continuity of ∗ , take (( g, η h ( x )); ( h, x )) ∈ G × G and U ⊆ G is a openset such that ( gh, x ) ∈ U . As above there are open sets U ⊆ G and U ⊆ X such that U = G ∩ U × U . Consider the open sets W = { g } × X h , W = { h } × ( X h − ∩ U ) and W := W × W , then (( g, θ h ( x )); ( h, x )) ∈ W. Finally, if ( s, t ) ∈ ∗ ( W ) there is m ∈ X h − ∩ U such that ( gh, m ) = ∗ (( g, θ h ( m )); ( h, m )) = ( s, t ) and from this we get ( s, t ) ∈ U and ∗ iscontinuous. References [1] F. Abadie, Enveloping actions and Takai duality for partial actions.
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Proc. Am. Math. Soc , (2004) 1037-1047.[3] P. Ara and R. Exel, Dynamical systems associated to separated graphs, graph algebras,and paradoxical decompositions, Advances Math. , (2014), 748-804.[4] K. Choi, Birget-Rhodes expansions of topological groups, Advanced Studies in Contempo-rary Mathematics. (2013), (1), 203 - 211.[5] M. Dokuchaev, Recent developments around partial actions, S˜ao Paulo J. Math. Sci. (2019) (1) 195-247.[6] R. Exel, Circle actions on C ∗ -algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequences, J. Funct. Anal. (1994), (3), 361 - 401.[7] R. Exel, Partial actions of group and actions of inverse semigroups,