aa r X i v : . [ m a t h . D S ] N ov TOTALLY NONFREE ACTIONSAND THE INFINITE SYMMETRICGROUP
A. M. Vershik ∗ ”October 13, 2018 To the memory of my beloved Lyotya
ABSTRACT
We define the notion of a totally nonfree (TNF) action of a group andstudy its properties. Such an action may be regarded as the adjoint actionof the group on the lattice of its subgroups with a special adjoint-invariantmeasure (TNF measure). The main result is the description of all adjoint-invariant and TNF measures on the lattice of subgroups of the infinite sym-metric group S N . The problem is closely related to the theory of charactersand factor representations of groups. The main result of this paper is a precise description of all adjoint-invariantergodic probability Borel measures on the lattice of all subgroups of theinfinite symmetric group S N . The reason why problems of this type areof importance can be briefly formulated as follows: the adjoint action ofa group on the lattice of subgroups with an adjoint-invariant probabilitymeasure produces, in a certain way, a nontrivial character of the group and,consequently, determines a special factor representation of the group. ∗ St. Petersburg Department of Steklov Institute of Mathematics and Max Plank In-stitute Bonn. E-mail: [email protected] . Partially supported by the RFBR grant11-01-00677-a.
1n the case of the infinite symmetric group, it turns out that this methodyields, up to a small deformation, all characters of S N . This phenomenon wasfirst observed in [13], as a particular fact related to a certain model of factorrepresentations of the infinite symmetric group. The list of characters itselfwas known earlier (Thoma’s theorem [8]), but the new proof of this theorem,which used the ergodic approach and approximation suggested in [12], aswell as the dynamical approach, made it possible to introduce the so-calledgroupoid model of factor representations. More precisely, the connection isas follows: the value of an indecomposable character at a given element ofthe group is equal, up to a certain factor, to the measure of the set of fixedpoints of this element for some special action. But what actions can appearin this construction? It turns out that these are so-called totally nonfree ( TNF ) actions, so it is important to describe such actions for a given group.In this paper, we go in the opposite direction: first we define and studythe class of TNF actions of a group. From the point of view of ergodic theory,this kind of actions is of great interest, and, as far as we know, it has notyet been systematically investigated. Due to the lack of space, we decidedto separate the discussion of the link between the questions considered hereand the theory of representations, characters, and factors; these topics willbe treated in another article.In the first section, we introduce the main notions and fix definitionsrelated to nonfree and totally nonfree actions. We develop a systematicapproach to nonfree actions. Although we are mainly interested in totallynonfree actions, we also consider intermediate cases and the reduction of ageneral action to a TNF action. The main open problem that arises in thiscontext concerns the existence of TNF and similar actions for a given group .We use the language of the lattice of subgroups and the adjoint action ofthe original group on this lattice; as far as we know, the dynamics on thislattice has never been considered from the point of view of ergodic theory.The previous question is equivalent to that of the existence or nonexistenceof continuous adjoint-invariant measures. The universal example of a TNFaction is the adjoint action of the group on the lattice of its subgroups witha TNF measure, i.e., a measure concentrated on the set of self-normalizingsubgroups (= subgroups coinciding with their normalizers). An importantresult asserts that this measure is a complete metric invariant of the action.In general, there are other adjoint-invariant measures that are not TNF; wedescribe a procedure (which may be infinite) that produces all TNF measures.All these facts heavily depend on the properties of the group. It is clear2hat for many Lie groups, as well as for groups close to Abelian ones, sucha measure does not exist. Of course, for the problem to be interesting, thegroup should have a continuum of subgroups. For some “large” groups, wehave many TNF measures (or many TNF actions), which, fortunately, canbe listed up to isomorphism (in contrast to the usual situation in ergodictheory). These measures on the lattice of subgroups can be regarded, ina natural way, as “random subgroups”; for different reasons, the notion ofa random subgroup was considered in [6]. The author believes that thisquestion is also of interest within ergodic theory itself.In the second section, we study the case of the infinite symmetric group.We use a fundamental classical fact about its subgroups; namely, the in-finite symmetric group has no primitive subgroups except the alternatingsubgroup and the whole group itself. This follows from a classical theoremdue to C. Jordan (H. Wielandt was perhaps the first to observe this, see [4]).We successively exclude from consideration all other subgroups that cannotlie in the support of an adjoint-invariant measure and reduce the questionto a de Finetti-like problem. The description of adjoint-invariant measureson the group S N relies on the new notion of a signed Young subgroup , whichis a generalization of the classical notion of a Young subgroup. It is naturalto understand a random signed Young subgroup exactly as a random sub-group whose distribution is an adjoint-invariant measure on L ( S N ). The listof parameters α for these measures is exactly the list of Thoma’s param-eters. We briefly compare our formula with that of Thoma at the end ofthe paper; in a sense, our list of adjoint-invariant measures can be regardedas an explanation for the list of characters. We will return to this questionelsewhere.After our short announcement [11] about the concept of a TNF action,it became clear that the papers by R. Grigorchuk and his colleagues [5, 1]contain examples of TNF actions of groups acting on trees. Also, L. Bowen[2] found such examples for the free group. However, an explicit descriptionof the list of all such actions is known only for S N . It turns out that theanswer is even closer to the group-theoretic structure; we will apply it tothe theory of characters and factor representations in a subsequent paper.Perhaps, our methods can also be applied to other similar groups, such asthe group od rational interval exchange, U ( ∞ ) (the infinite unitary group), GL ( F q , ∞ ), etc.I am grateful to Natalia Tsilevich for her help in editing of this article.3 MAIN DEFINITIONS. TOTALLY NON-FREE ACTIONS σ -FIELDS Let ( X, A , µ ) be a Lebesgue space with a probability measure µ defined on a σ -field A of classes of mod0 coinciding measurable sets, and let a countablegroup G act on this space by µ -preserving transformations. We will consideronly effective actions , which means that only the identity e ∈ G of the groupacts as the identity transformation mod0.For each element g ∈ G , we define a measurable set X g called the set offixed points of g : X g = { x ∈ X, gx = x } . Consider the map Φ : G → A ; Φ : g X g . Definition 1.
The fixed point σ -field corresponding to the action of G underconsideration is the sub- σ -field A G of the σ -field A generated by the family ofall sets X g , g ∈ G . The sets X g are well defined for arbitrary actions of countable groups and,more generally, for pointwise, or measurable actions of arbitrary groups. Itis worth mentioning that the above definition of the σ -field A G applies to continuous actions of arbitrary groups , since the set of fixed points for agiven automorphism is well defined with respect to mod0: if g = g mod 0,then X g = X g mod 0. Remark 1.
An action of a group G is called free if µX g = 0 for all g = Id, g ∈ G , or, in short, if the σ -field A G is trivial (the trivial σ -field will bedenoted by N ). Recall that an action of a group G is called pointwise (or measurable) if there is ameasurable set of full measure on which the action of G is defined; an action is calledcontinuous (in Rokhlin’s terminology; the other term is mod 0-action) if a homomorphism G → Aut(
X, µ ) is defined, where Aut(
X, µ ) is the group of all classes of measure-preservingtransformations of (
X, µ ). For countable groups, as well for locally compact groups, thesetwo notions are equivalent. stationary subgroup ,or the stabilizer , of a point x ∈ X : G x = { g ∈ G : gx = x } . It is clear that if y = hx with y, x ∈ X , h ∈ G , then G y = h − G x h . Ingeneral, this notion is not well defined for uncountable groups; more exactly,it can be defined only if one can introduce the notion of the orbit partition.Now we are going to define another sub- σ -field of the σ -field A in thespace X . We start with the following definition. Definition 2.
Consider the partition ξ G of the space X into the classes ofpoints having the same stabilizer. We call it the iso-stable partition of thetriple ( X, G, µ ) . The iso-stable partition ξ G is measurable, because it is the limit, over anincreasing sequence of finite subsets K n ⊂ G , S n K n = G , of measurablepartitions ξ K n G : ξ G = lim n ξ K n G , where two points x, y ∈ X belong to the sameblock of ξ K n G if and only if K ⊂ G x , K ⊂ G y . The partition ξ G is obviously G -invariant, because an element of ξ G consists of all points that have thesame stabilizers. Definition 3.
Let A G be the sub- σ -field of A that consists of all sets measur-able with respect to the iso-stable partition ξ G . In the quotient space X/ξ G ,we have a well-defined action of the group G ; it will be called the reducedaction. Proposition 1.
Assume that there is a pointwise action of a group G on aspace ( X, µ ) with an invariant measure µ . Both sub- σ -fields A G and A G are G -invariant, and the following inclusion holds: A G ⊂ A G . For a countable group G , both sub- σ -fields coincide: A G = A G ≡ A ( G ) . Proof.
The first claim is trivial: two points that cannot be separated bytheir fixed point sets have the same stabilizers. By definition, the σ -field5 G is generated by the family of sets X g , g ∈ G . But, since the group G iscountable, a basis of the σ -field A G consists of the sets Y K = ∩ g ∈ K X g , where K ⊂ Stab( x ) ⊂ G is an arbitrary finite set. Thus the family X g , g ∈ G , generates both σ -fields in question.For continuous groups, the sub- σ -fields in question do not coincide ingeneral. For instance, considering the action of the orthogonal group SO (3)on the projective space RP , we see that in this case A G $ A G . Indeed, eachset of fixed points has zero measure, whence A G = N (where N is the trivial σ -field), while A G = A since the set of all rotations separate the points of P R . Denote by L ( G ) the set of all subgroups of a locally compact group G andequip it with the natural weak topology and the corresponding Borel struc-ture. For a countable group, the space L ( G ) equipped with this topology isa compact (Cantor) space. The adjoint action of the group G on L ( G ) isdefined as follows. Let g ∈ G , H ∈ L ( G ); thenAd( g ) H = gHg − . We will study the dynamical system ( L ( G ) , Ad( G )) from the point of viewof ergodic theory; namely, we will consider Ad( G )-invariant Borel measures.The key problem is the existence of continuous (nonatomic) invariant mea-sures. PROBLEM 1.
For what groups do there exist continuous
Ad( G ) -invariantBorel probability measures? Describe all such measures for a given group. We will solve this problem for the infinite symmetric group. Of course,the theory we develop here is interesting for countable groups that haveuncountably many subgroups. A neighborhood of a subgroup in the weak topology is the set of subgroups that havethe same intersection with a given compact subset of G . For a discrete group, L ( G ) is asubspace of the compact space of all subsets of G .
6t is known (see [2]) that a non-Abelian free group has a lot of suchmeasures, but one has no general description of these measures. In [5], actionsof groups on trees and more general graphs were considered, and it wasverified that these actions are TNF.A natural point of view on Ad( G )-invariant measures is to regard themas random subgroups of G ; more precisely, each Ad( G )-invariant measuredetermines a statistics on the set of subgroups, or a random subgroup. Theinvariance under conjugations is a natural condition for applications. Onemay refine this condition and consider random subgroups with additionalproperties (e.g., TNF measures, or Ad( G )-invariant measures on the set ofself-normalizing subgroups, see below). In the recent paper [6], a problemrelated to random subgroups arises for a different reason.The lattice structure on the space of subgroups L ( G ) is a very popularobject of algebraic studies (see, e.g., [7]); we will not use it. It is worthmentioning that an important and completely open question concerns theexistence of σ -finite invariant continuous measures on L ( G ). As far as weknow, ergodic aspects of the natural dynamical system ( L ( G ) , Ad( G ) , ν ),where ν is an Ad( G )-invariant measure, has not been seriously studied.Let us now connect these dynamical systems ( L ( G ) , Ad( G ) , ν ) with non-free actions of the group G . Namely, we can identify the quotient space withrespect to the iso-stable partition ξ G with L ( G ). Definition 4.
Given an action of a group G on a Lebesgue space ( X, µ ) ,consider the map Ψ : X → L ( G ) , Ψ( x ) = G x . It is a measurable homomorphism of the triple ( X, G, µ ) to the triple ( L ( G ) , Ad( G ) , Ψ ∗ µ ) ,where Ψ ∗ µ is an Ad( G ) -invariant Borel measure on L ( G ) , the image of themeasure µ under Ψ : Ψ ∗ ( µ )( B ) = µ { x : G x ∈ B ⊂ L ( G ) } . We will call Ψ ∗ µ the characteristic measure of the action ( X, G, µ ) . Proposition 2.
The characteristic measure Ψ ∗ µ is a metric invariant ofmeasure-preserving actions in the following sense: if two measure-preservingactions of a countable group G on spaces ( X i , µ i ) , i = 1 , , are metricallyisomorphic, then the corresponding measures Ψ ∗ µ i , i = 1 , , on L ( G ) coin-cide. roof. It suffices to observe that every isomorphism between two actions of G must send the set of points with a given stabilizer for one action to thesame set for the other action.The map Ψ is nothing else than the factorization of the space ( X, µ ) withrespect to the iso-stable partition ξ G , which identifies the quotient space X/ξ G with the image L ( G ). The quotient measure µ ξ G tautologically coincides withthe characteristic measure Ψ ∗ µ .For a free action, Ψ is a constant map and the characteristic measure isthe δ -measure at the identity subgroup { e } ∈ L ( G ). Definition-Theorem 1.
A measure-preserving action of a countable group G on a space ( X, µ ) is called totally nonfree (TNF) if one of the followingequivalent conditions holds: The σ -field A G (= A G = A ( G )) generated by all sets of fixed points co-incides with the whole σ -field A of all measurable subsets of X . Equivalently,the iso-stable partition ξ G coincides mod0 with the partition into separatepoints. For almost all (with respect to the measure µ × µ ) pairs of differentpoints ( x, y ) , x = y , their stabilizers do not coincide: G x = G y . The map
Ψ : X → L ( G ) = X/ξ G is a mod 0 isomorphism (= separatespoints mod0 on ( X, µ ) ). Consequently, the canonical skew product of theaction is isomorphic to the action itself.If an action is TNF, then we say that its characteristic measure is a TNFmeasure on L ( G ) . The equivalence of the above three conditions directly follows from thedefinitions of the previous section. It is also clear that the definitions arecorrect with respect to changing the actions on sets of zero measure.TNF actions are an opposite extreme to free actions.The metric classification of TNF actions of a countable group G reducesto the calculation of the characteristic measures Ψ ∗ µ on the lattice L ( G ); sothe classification problem for TNF actions is, in a sense, smooth (tame), incontrast to the general isomorphism problem in ergodic theory.In this paper, we will describe the TNF actions for the infinite symmetricgroup and explain their connection to the representation theory of this group.8 efinition 5. The normalizer of a subgroup Λ ⊂ G is the subgroup N (Λ) = { g ∈ G : g Λ g − = Λ } . A subgroup H ⊂ G for which N ( H ) = H is calledself-normalizing. Denote the set of all self-normalizing subgroups of G by LN ( G ) . The following claim is obvious.
Proposition 3.
A transitive action of a group G (the left action of G on ahomogeneous space G/H ) is totally nonfree if and only if the stabilizer (i.e., H ) is a self-normalizing subgroup ( N ( H ) = H , or H ∈ LN ( G ) ). Indeed, any two points belong to the same orbit (for any x, y ∈ X , thereexists h ∈ G such that y = hx ); if they have the same stabilizer G x , then h ∈ N ( G x ), where N ( G x ) = { h ∈ G : h − G x h = G x } is the normalizer of G x .Consequently, either N ( G x ) = G x and the action is not TNF, or N ( G x ) = G x and the action is TNF.A similar situation holds for general actions. Proposition 4. If a measure-preserving action of a group G on a space ( X, µ ) is TNF, then for almost all x ∈ X the stabilizers G x are self-normalizing: µ { x : G x ∈ LN ( G ) } = 1 . The adjoint action of the group G on the lattice L ( G ) with an Ad( G ) -invariant TNF measure ν is a TNF action. The adjoint action of the group G on the measure space ( L ( G ) , ν ) isTNF if and only if ν -almost all subgroups H ∈ L ( G ) are self-normalizing: ν ( LN ( G )) = 1 .Proof.
1. Assume that the action is TNF, but at the same time there existsa measurable set A of positive µ -measure such that the stabilizer G x of everypoint x ∈ A is not self-normalizing. Then there exists a point hx ∈ A with h ∈ G , h / ∈ G x such that hx = x but hG x h − = G x ; consequently, x and hx have the same stabilizer, which contradicts the TNF property.2. If ν is a TNF measure on L ( G ), then, by definition, it is the image ν = Ψ ∗ µ , where ( X, G, ν ) is a space with a TNF action of G . But the actionAd( G ) on ( L ( G ) , ν ) is isomorphic to the action of G on the space ( X, µ )(see Section 1). At the same time, the TNF property is invariant underisomorphism. It is more natural to call such a subgroup abnormal , or anormal .
9. We have already proved that the condition ν ( LN ( G )) = 1 is necessaryfor the action Ad( G ) to be TNF. The converse is obvious since Stab H ≡ N ( H ) = H . Remark 2.
Although, as we have proved, for the action of G on ( L ( G ) , ν ),the condition ν ( LN ( G )) = 1 is equivalent to the TNF property, for a generalmeasure-preserving action of G on a space ( X, µ ), the condition µ { x : G x ∈ LN ( G ) } = 1 is only necessary but not sufficient for the action to be TNF,because the stabilizers of two points x, y that belong to different orbits canbe the same self-normalizing subgroup: G x = G y ∈ LN ( G ).Now we can specify Problem 1 formulated above and reduce the descrip-tion of TNF actions of a group to the following question. PROBLEM 2.
Given a group G , describe all ergodic continuous (nonatomic)probability TNF measures on the lattice L ( G ) of its subgroups. Or, equiva-lently, describe all ergodic continuous (nonatomic) Borel probability Ad( G ) -invariant measures on the subset LN ( G ) of L ( G ) . Perhaps, for some groups (similar to the infinite symmetric group) thisproblem coincides with Problem 1 concerning the description of all Ad( G )-invariant measures, but in general these problems are different. An important problem is to characterize the TNF actions of a given group G from the point of view of ergodic theory and that of representation theory:what ergodic properties can have TNF actions, and what kind of factor rep-resentations can arise for TNF actions? L ( G ) It makes sense to consider types of actions intermediate between free andTNF actions, because they also can be studied by the suggested methods.First of all, we consider the factorization with respect to the iso-stable par-tition ξ G in order to define the canonical skew product structure forgeneral actions of groups .Consider a measure-preserving action of a countable group G on a Le-besgue space ( X, µ ) and the G -invariant iso-stable partition ξ G (see Sec-tion 1.1). The reduced action of G on ( X/ξ G , µ ξ G ) (see Definition 3) is iso-morphic to the adjoint action of G on the space L ( G ) equipped with the10haracteristic measure Ψ ∗ µ . We regard the triple ( L ( G ) , Ad( G ) , Ψ ∗ µ ) as thebase of the canonical skew product structure for the action ( X, G, µ ).Recall that a skew product is defined if we have an action of the groupon the base and a 1-cocycle on the base with values in the group of automor-phisms of the typical fiber (
Y, α ). For a free action of G , this skew product istrivial, the base consists of a single point, and the fiber is ( Y, α ) ≡ ( X, µ ). Inthe case of a TNF action, Y is a one-point space and the base coincides withthe whole space ( X, µ ) ≡ ( L ( G ) , Ψ ∗ µ ). In the general case, the action on thebase is the adjoint action, and the 1-cocycle c is a measurable function onthe product G × L ( G ) with values in the group of measure-preserving trans-formations of the fiber ( Y, α ). The general condition on a cocycle c on thespace L ( G ) with the adjoint action of the group G and an arbitrary group ofcoefficients has the form c ( g g , H ) = c ( g , H ) c ( g , g Hg − ) , c ( e, H ) = Id , where g ∈ G , H ∈ L ( G ), and c ( · , · ) is a measurable function on G × L ( G )with values in some group. From this we can conclude that for a fixed H , therestriction of the map g c ( g, H ) to g ∈ H is a homomorphism of the group H . But the cocycle that defines the canonical skew product has a strongerproperty. Proposition 5.
If the action of G on ( X, µ ) is ergodic, then the above con-struction defines a decomposition of the space ( X, µ ) into the direct product ( X ≈ L ( G ) × Y ; µ ξ G × α ) , where ( Y, α ) is the typical fiber of the skew product;the action of G on the base is the adjoint action with the Ad( G ) -invariantmeasure Ψ ∗ ( µ ) ; a 1-cocycle c ( · , · ) is a function on the space G × L ( G ) withvalues in the group Aut(
Y, ν ) of measure-preserving automorphisms of thespace ( Y, ν ) .A necessary and sufficient condition for a cocycle c to define the canonicalskew product is as follows. If ν is a TNF measure, then c ( h, H ) = Id for ν -almost all H ∈ L ( G ) and h ∈ H , where Id is the mod 0 identity map onthe space Y ; in other words, the homomorphism mentioned above is identical. If ν is not a TNF measure, then, in addition to the previous condition,the following property holds: for ν -almost all H ∈ L ( G ) and h ∈ N ( H ) \ H , α (Fix c ( h,H ) ) = 0 , here Fix φ = { y ∈ Y : φ ( y ) = y } is the set of fixed points of an automorphism φ . The first condition means that for ν -almost all subgroups H , for all h ∈ H ,almost all points of Y are fixed points of the automorphism c ( h, H ); and thesecond condition means that for ν -almost all subgroups H , for all h / ∈ H ,the set of fixed points of c ( h, H ) is, on the contrary, of zero measure. Theseconditions on a cocycle follow from that fact that, by definition, the fibers ofthe partition ξ coincide with the sets of points with a given stabilizer. Wewill not discuss details and similar questions. Unfortunately, it is difficultto verify whether there exists a cocycle satisfying this condition for a givenAd(G)-invariant measure ν . At the same time, for the analysis of a givenaction it is important to use the canonical skew product and to study thecorresponding Ad( G )-invariant measure on L ( G ) and cocycle.Now let us consider the base of the canonical skew product. Definition-Theorem 2.
The action of a group G on a space ( X, µ ) iscalled reduced totally nonfree (RTNF) if the reduced action on the base of thecanonical skew product ( X ξ G , µ G ) is a TNF action, or, equivalently, µ { x : N ( G x ) ∈ LN ( G ) } = 1 ⇔ µ { x : N ( G x ) = N ( G x ) } = 1 . A RTNF measureon L ( G ) is, by definition, the characteristic measure ν of a RTNF action: ν { H ∈ L ( G ) : N ( H ) = N ( H ) } = 1 (the second normalizer of a subgroupcoincides with its first normalizer). Recall that the TNF property is equivalent to the condition ν { H ∈ L ( G ) : N ( H ) = H } = 1. This follows from the results of the previous section. Ingeneral, the quotient of the canonical skew product (i.e., its base) is not aTNF action, because the stabilizer of a point of the quotient space is thenormalizer of the stabilizer of the original point, Stab Ψ( x ) = N (Stab x ), butdifferent stabilizers may have the same normalizers, and, consequently, wecan obtain different points with the same stabilizers. Thus we can apply themap Ψ to the base and consider the second canonical skew product of thebase, the third one, etc. This gives an hierarchy of Ad-invariant measures on L ( G ).The following commutative diagram shows how to continue our classifi-cation: X Ψ −−−→ L ( G ) π y y N X/ξ G Ψ −−−→ L ( G ) . y G y associates with a point y its stabilizer, π : X → X/ξ G is the canonical projection, and the map N : H N ( H ) associateswith a subgroup its normalizer. Definition 6.
We will denote by AD ( G ) the space of all Ad( G ) -invariantmeasures on the lattice L ( G ) (which will be called in short “AD-measures”),and by CM (“characteristic measures”), the subset of AD ( G ) consisting ofall characteristic measures Ψ ∗ µ = µ ξ G , for an arbitrary triple ( X, µ, G ) .Denote by N = N ∗ the operation on the set of measures on L ( G ) cor-responding to the normalization of subgroups: [( N ) ν ]( F ) = ν ( N − F ) , F ⊂ L ( G ) . It is clear that N sends AD-measures to AD-measures. It follows from our definitions that if a measure ν on L ( G ) is RTNF,then the measure N ( ν ) is TNF; in particular, if ν is a TNF measure, then N ( ν ) = ν . Thus the operation N : { RTNF measures } → {
TNF measures } is a projection.We have a hierarchy of AD-measures on the lattice L ( G ): AD ⊃ CM ⊃ N ( CM ) ⊃ N ( CM ) ⊃ · · · ⊃ RT N F ⊃ T N F.
It is not clear to the author whether the class CM coincides with AD ,or, whether an arbitrary AD-measure on L ( G ) is the characteristic measurefor some action; the problem consists in defining an appropriate cocycle fora given AD-measure. I think that for some groups G , the chain of normal-izations above can be infinite or even transfinite. But the most interestingclasses of measures are the first and the last two ones: TNF, RTNF, CM,and AD.
1. Fibre bundle over L ( G ) . Each subgroup H is normal subgroup inits normalizer N ( H ), so we have a fibre bundle over L ( G ) with a fibre over H ∈ L ( G )— the group N ( H ) /H . This bundle is invariant under the adjointaction of the group. We will use it for the theory of characters of the group.
2. The TNF limit of AD-measures.
It is natural to assume that forevery measure ν from the class of AD-measures on a given group G , the AD-measure ∩ ν does exist, which is the limit in n of the sequence of successivenormalizations: ν
7→ N ∗ n ( ν ), n = 1 , , . . . . This limit (for some groups, itmay be transfinite) must be a TNF measure.13 . TNF actions for continuous groups. As we know, the σ -fields A G and A G can be different. It is natural to define a TNF action of ageneral group as an action for which the σ -field A G is the complete σ -field,or, for which the stabilizers separate points. In this case, we again have anisomorphism between a TNF action and the adjoint action on the lattice ofsubgroups.
4. The continuous version of combinatorial multi-transitivity.
The continuous counterpart of the notion of transitivity (or topological tran-sitivity) of actions of discrete groups is that of ergodicity. What is the analogof double transitivity? A common explanation is that this is the ergodicityof the action on the Cartesian square. But I believe that this parallel is tooweak. The definition of double transitivity in combinatorics can be formu-lated as the transitivity of the action of the stabilizer of a point x on thespace X \ x . Thus we suggest the notion of multiple transitivity, which isrelated to our consideration as follows. Definition 7.
Assume that a countable group G acts on a standard space ( X, µ ) with a G -invariant continuous measure. We say that the action isdouble transitive if for µ -almost every point x ∈ X , the action of the stabilizer G x ⊂ G on ( X, µ ) is transitive. We say that the action is k -transitive iffor almost every (in the sense of the measure µ k on X k ) choice of points x , x , . . . , x k , the action of the intersection of subgroups T ki =1 G x i L ( G ) on ( X, µ ) is transitive. It is natural to consider this definition only for TNF actions of G . It willbe clear that all TNF actions of the infinite symmetric group are k -transitivefor an arbitrary positive integer k . It is of interest to find all countable groups for which TNF k -transitiveactions exist for any positive integer k .This problem is perhaps related to the class of oligomorphic groups , whichwas defined by P. Cameron [3] (a subgroup G of the group S N of all finitepermutations of N is called oligomorphic if for any positive integer k , thenumber of orbits of the diagonal action of G in the Cartesian product N k isfinite). 14 THE LIST OF RANDOM SUBGROUPSOF THE INFINITE SYMMETRIC GROUP
We consider the countable group S N , the infinite symmetric group of all finitepermutations of the set of positive integers N (or an arbitrary countable set).In this section, we will give the list of all AD-measures on the lattice L ( S N )of subgroups of this group and, in particular, the list of TNF measures. Wewill use some classical facts about permutation groups and the probabilisticapproach.The lattice L ( S N ) is very large and contains very different types of sub-groups. Nevertheless, the support of an AD-measure consists of subgroupsof a very special kind: so-called signed Young groups. The topology and theBorel structure on L ( S N ) are defined as usual; this is a compact (Cantor)space. Definition 8 (Signed partitions) . A signed partition η of the set N is afinite or countable partition N = ∪ B ∈B B of N together with a decomposition B = B + ∪ B − ∪ B of the set of its blocks, where B is the set of all single-point blocks; elements of B + are called positive blocks, and elements of B − are called negative blocks (thus each positive or negative block contains atleast two points), and we denote by B the union of all single-point blocks: B = ∪ { x }∈B { x } .Denote the set of all signed partitions of N by SPart( N ) . Recall that in the theory of finite symmetric groups, the Young subgroup Y η corresponding to an ordinary partition η = { B , B , . . . , B k } is Q ki =1 S B i ,where S B is the symmetric group acting on B . We will define the moregeneral notion of a signed Young subgroup , which makes sense both for finiteand infinite symmetric groups. We will use the following notation: S + ( B ) isthe symmetric group of all finite permutations of elements of a set B ⊂ N ,and S − ( B ) is the alternating group on B . Traditionally, the alternating group is denoted by A n ; V. I. Arnold was very enthusi-astic about the idea to denote it by S − n in order not to confuse it with the Lie algebra A n ;I agree with this idea. efinition 9 (Signed Young subgroups) . The signed Young subgroup Y η corresponding to a signed partition η of N is Y η = Y B ∈B + S + ( B ) × Y B ∈B − S − ( B ) . Note that on the set B ⊂ N , the subgroup Y η act identically, so that thepartition into the orbits of Y η coincides with η .It is not difficult to describe the conjugacy class of Young subgroups withrespect to the group of inner automorphisms: Y η ∼ Y η ′ if and only if η and η ′ are equivalent up to the action of S N . But it is more important toconsider the conjugacy with respect to the group of outer automorphisms.This is the group S N of all permutations of N . Denote by r ± the numberof infinite positive (respectively, negative) blocks, and by r ± s the number offinite positive (respectively, negative) blocks of length s >
1. Obviously, thelist of numbers { r ± , r ± , . . . } is a complete set of invariants of the group ofouter automorphisms. Consider a sequence of positive numbers α = { α i } i ∈ Z such that α i ≥ α i +1 ≥ i > α i +1 ≥ α i ≥ i < α ≥ X i ∈ Z α i = 1 . Consider a sequence of Z -valued independent random variables ξ n , n ∈ N ,with the distributionProb { ξ n = v } = α v for all n ∈ N , v ∈ Z . Thus we have defined a Bernoulli measure µ α on the space of integer se-quences Z N = { ξ = { ξ n } n ∈ N : ξ n ∈ Z } . Definition 10 (A random signed Young subgroup and the measures ν α ) . Fix a sequence α = { α i , i ∈ Z } ; for each realization of the random sequence { ξ n } , n ∈ N , with the distribution µ α , define a random signed partition η ( ξ ) of N as follows: η ( ξ ) = { B i ⊂ N , i ∈ Z } , B i := { n ∈ N : ξ n = i } , here a block is positive (negative) if it has a positive (negative) index and B is understood as the union of one-point blocks. The correspondence ξ η ( ξ ) defines a probability measure on the set SPart( N ) of signed partitions, theimage of the Bernoulli measure µ α . Analogously, the correspondence ξ Y η ( ξ ) defines a measure, which we denote by ν α , on the set of signed Youngsubgroups, i.e., a measure on the lattice L ( S N ) of subgroups of S N . It is convenient to denote positive and negative blocks B i of a signedpartition by B + i and B − i , respectively. Note that all nonempty blocks of therandom signed partition η ( ξ ) that consist of more than one point are infinitewith ν α -probability one. Theorem 1 (The list of all AD and TNF measures for the group S N ) . Everymeasure ν α is a Borel ergodic AD-measure on the lattice L ( S N ) ; every ergodicprobability Borel AD-measure on this lattice coincides with the measure ν α for some α . The measure ν α is RTNF, and is TNF if and only if α i = 0 forall i ≤ . Proof.
1. The easy part of the proof is to check that the measures ν α areindeed ergodic AD-measures on L ( S N ). The invariance follows from the con-struction, because µ α , being a Bernoulli measure, is invariant under all per-mutations of indices. The symmetric, alternating, and identity subgroups ofthe symmetric groups S ( B ) are normal, so they are Ad( G )-invariant. Con-sequently, the measure ν α , being the image of µ α , is Ad( G )-invariant. Theergodicity with respect to permutations also follows from the ergodicity ofthe Bernoulli measure.2. Now suppose that we have an ergodic AD-measure on L ( S N ). We willfilter out, step by step, classes of subgroups of S N that cannot support anyAD-measure, and will finally obtain the class of signed Young groups as theonly possible class. Then we will construct all AD-measures on this class.a) A classical result of the theory of permutation groups asserts that thegroup S N has no primitive subgroups except the whole group S N and thealternating group. This is a more or less direct corollary of the fundamentalestimates obtained by C. Jordan for finite symmetric groups, which weregeneralized by H. Wielandt [14] (see, e.g., [4, Chapter 8]). Namely, this is A primitive subgroup is a subgroup that has no nontrivial invariant partitions.
17 corollary of Jordan’s theorem asserting that if a primitive subgroup of S n has an element with support of size k , then n < β ( k ); a sharp bound on β ( k )is still unknown.b) Now consider an AD-measure ν on the set of imprimitive but transitivesubgroups of S N . Assume that such a subgroup H ∈ L ( S N ) has an invariantpartition θ . For the action of H to be transitive, all nontrivial minimal blocksof θ must have the same length l >
1, which must be finite (because finitepermutations cannot move one infinite block to another one). Denote by θ ( H ) the partition of N into the minimal blocks for H . The map H → θ ( H )associates with ν -almost every imprimitive subgroup a partition into blocks oflength l ( H ); because of the Ad( G )-ergodicity of ν , this length is the same for ν -almost all subgroups H . Thus the map H θ ( H ) sends ν to a probabilitymeasure on the set of partitions of N with countably many blocks of the samelength l >
1, and this measure is invariant with respect to the action of S N on the space of such partitions. Let us show that there are no such finitemeasures. Lemma 1.
There are no probability measures on the space
Part( l ) of allpartitions of N into (countably many) blocks of length l > that are invariantwith respect to the group S N . Remark.
The space Part( l ) equipped with the weak topology is locally com-pact but not compact; its natural compactification consists of all partitionswhose blocks have length at most l . Proof.
Consider the case l = 2. Each partition from Part(2) determines asymmetric matrix { a i,j } , a i,j = a j,i , a i,i = 0, i, j = 1 , , . . . , with only oneentry in each row and each column equal to 1, all the other entries beingequal to 0. But because of the S N -invariance, the distribution of the entry a i,j does not depend on i, j , i = j , and must be a probability measure { p , p } on {
0; 1 } . By the ergodic theorem,lim n n ! X g ∈ S n a gi,gj = ¯ a i,j = 0for all i, j and all matrices { a i,j } of this type. Passing to the limit, we obtain p = p = 0, which means that a measure on Part(2) with desired properties Note that the lengths of all blocks for a given imprimitive group may be either bounded(so-called “almost primitive groups”) or unbounded (“totally imprimitive subgroups”), see[4], but this difference is not important for our purposes. The only difference in the case l > L ( S N ) takes thevalue 0 on the set of all transitive subgroups; so we have reduced the analysisto intransitive subgroups. Fix such a generic intransitive subgroup H ∈ L ( S N ) and consider the partition η ( H ) into its transitive components. Theaction of the group H on each such component must be primitive, becauseimprimitive cases can be discarded for the same reason as in the previouspart of the proof. For the same reason, it is obvious that all componentsof this partition must be infinite. Consequently, the action of H on eachcomponent is either the action of the whole symmetric group, or that of thealternating group (see part a) of the proof), or that of the identity group onthe single-point blocks. We denote the blocks by B i , i >
0, when the actionof H is the action of the symmetric group of B i , and by B i , i <
0, whenthe action of H is the action of the alternating group of B i . The action ofthe identity group on all single-point blocks can be regarded as the identityaction on the union of such blocks B . Thus we have a signed partition η ( H )such that the action of H on each block B i , i >
0, is the action of S + ( B ),the action of H on each block B i , i <
0, is the action of S − ( B ), and theaction on B is the identity action. This means that H ⊂ Y η = Q i S ± ( B i )and the restriction of the action of H to B i is the action of S ± ( B ). But themeasure ν is an AD-measure, so for ν -almost every group H , the orbit of H under conjugation must belong to a set A of full ν -measure. Applyingto H the conjugation gHg − , where g ∈ S ± ( B i ), we obtain a subgroup H ′ which has the same intersection with the product Q i = i S ± ( B i ) as H , whence H ′ = S ± ( B i ) Q i = i S ± ( B i ). Thus if we average the measure δ H over the orbitof H under conjugation (“ergodic method”), we obtain the same measure ν ,and conclude that the set A contains H ′ . Continuing this process, we seethat the set of all ergodic AD-measures on the lattice L ( S N ) coincides withthe set of measures that are the limits of the averages of the δ -measures atsigned Young subgroups Y η .d) Now we must identify the required measures with the ergodic lim-its with respect to conjugation of signed Young subgroups. Because of thecorrespondence between signed Young subgroups and signed partitions, thisquestion is equivalent to the description of S N -invariant measures on the setof signed partitions. The last question is similar to the classical de Finetti The limit measure is the δ -measure at the partition of N into separate points. S N -invariant measures on the space of all functions on N (see [9]). The only small difference lies in the fact that, in contrast to theclassical situation, here we have three types of blocks of signed partitionsinstead of one type in the ordinary de Finetti theorem. Lemma 2 (An analog of de Finetti’s theorem) . Consider the space
SPart( N ) of signed partitions of N ; every ergodic S N -invariant measure on SPart( N ) isdetermined by a sequence α = { α i } i ∈ Z such that α i ≥ α i +1 ≥ for i > , α i +1 ≥ α i ≥ for i < , α ≥ , and P i ∈ Z α i = 1 , as described above.Proof. The lemma can be proved by any of the methods people use to provede Finetti’s theorem. For completeness, we present a proof, applying ourold ergodic method from [9]. In order to find all ergodic measures ν on acompact S N -space X using the pointwise ergodic theorem for the group S N (which is in fact a theorem on the convergence of martingales), it suffices tofind the weak limits of measures (when they do exist)lim 1 n ! X g ∈ S n δ gx for all x ∈ X . More exactly, we need to calculate the limitslim n n ! X g ∈ S n f ( gx )for continuous functions f ∈ C ( X ). In our case, it suffices to consider cylinderfunctions on SPart( N ) which depend on finitely many blocks. Fix a signedpartition η and label its blocks with integers in an arbitrary way so thatpositive (negative) integers correspond to positive (negative) blocks and B is the union of one-point blocks. Consider the Z -valued sequence x n , n ∈ N ,defined as follows: x n = s if n ∈ B s . Now we may say that the signedpartition η is the partition corresponding to the sequence { x n } , and eachsuch sequence determines a signed partition. The action of S N on the set ofsigned partitions and its action by permutations of coordinates of sequencesobviously agree, so our problem reduces to the description of S N -invariantmeasures on the space of all elements of Z N . But this is exactly de Finetti’sproblem. Start with an arbitrary sequence { x n } ∈ Z N and calculate the limitlim n n ! { g ∈ S n : x gn = v } = α v x ∈ X ). Thuswe obtain the one-dimensional distribution of the random (with respect tothe measure ν ) sequence x n . In order to prove that this measure is a Bernoullimeasure on Z N , we must calculate the joint distribution of several coordinatesof x n . But because of the complete transitivity of the action of S n , for anychoice of v , . . . , v t and for n ≫ t we have { g ∈ S n : x gi = v i , i = 1 , , . . . , t, n ≫ t } n ! ≈ t Y i =1 α v i , which means that the random variables x n , n = 1 , , . . . , are independent.Thus all AD-measures arise from Bernoulli measures on the space of signedpartitions, i.e., ν = ν α for some α .e) Consider the random signed Young subgroup Y η constructed from asequence α with α i = 0 for all i ≤
0. Then all blocks B i , i ≤
0, areempty with probability one. Then, obviously, the normalizer N ( Y η ) coincideswith Y η , since each block of η gives rise to the self-normalizing subgroup S + ( B i ). Consequently, the measure ν α is TNF. If α i > i ≤ B i is not empty with probability one, whence N ( S − B i ) = S + B i = S − B i , so that Y η is not self-normalizing. But N ( Y η ) = + ∞ Y i = −∞ S + B i . Thus N ( Y η ) = N ( Y η ), so that ν α is a RTNF measure in the terminology ofSection 1. This completes the proof of Theorem 1. Corollary 1.
The action of the group S N on the measure space ( L ( S N ) , ν α ) is ergodic. Indeed, this is a corollary of the fact that the measure ν α is the image ofthe Bernoulli measure µ α and the correspondence µ α ν α between measurescommutes with the action of the group. The corresponding representation ofthe group S N in the space L ν α ( L ( S N )) will be considered elsewhere. Corollary 2.
There are three degenerate measures ν α , in the following cases(in the parentheses we indicate the corresponding character and representa-tion, see below): α = 1 , α i = 0 , i = 1 ; in this case, ν α = δ S N ( χ ( g ) ≡ , the identityrepresentation); α − = 1 , α i = 0 , i = − ; in this case, ν α = δ S − N ( χ ( g ) = ( − sgn( g ) ,the alternating representation); α = 1 , α i = 0 , i = 0 ; in this case, ν α = δ Id S N ( χ ( g ) = δ e ( g ) , theregular representation).An ergodic AD-measure ν α is atomic only in these three cases (in whichit is if fact a δ -measure); in all the other cases, ν α is a continuous measure. Make sense to compare the language which we use here (the action on L ( G ))with that which was used in [13] (the action on the Bernoulli sequences).More concretely, consider the action of S N on the space Z N (instead of L ( S N )) and ask for a description of TNF and RTNF measures. The answeris a little bit different than for the action on the space of Young subgroups.Namely, the following result holds. Proposition 6.
The measure µ α on the space X = Z N with the action ofthe group S N is a TNF measure if and only if all α i , i = 0 , are distinct.If α i = α j for some i = j , then the action of S N is RTNF but not TNF.The canonical projection X → X/ξ G is the factorization with respect to thefollowing equivalence relation on X = Z N : two elements { x n } n ∈ N , { x ′ n } n ∈ N ∈ X are equivalent for if for every v ∈ Z either { n ∈ N : x n = v } = { n ∈ N : x ′ n = v } , or there exists v ′ ∈ Z with α v = α v ′ such that { n ∈ N : x n = v } = { n ∈ N : x ′ n = v ′ } , and { n ∈ N : x n = v ′ } = { n ∈ N : x ′ n = v } . Thus, in this case the action is RTNF not TNF if we have multiplicity inthe values of α : α i = α j , i = j ; this is not the case for the action in L ( S N ).The supports of the measures ν α in the topological sense (i.e., the minimalclosed subsets of full measure) coincide for all α that have the same numberof infinite blocks. The support of ν α for α having infinitely many infiniteblocks coincides with the space of all signed Young subgroups.22n the case of the infinite symmetric group, all AD-measures are invariantunder the group S N of all permutations of N . The conjugation with respectto this group is an extension of the usual conjugation; but for a genericsubgroup H from a set A of full ν α -measure, its orbit under the action of S N is much larger than A . In other words, the frequencies α i are invariant underthe usual conjugation, but not under its extension. This fact is related tothe so-called Kolmogorov effect (see [10]).Theorem 1 gives more than just the list of AD-measures on the group S N ; it helps to give a new proof of Thoma’s formula for indecomposablecharacters of this group. This will be the subject of our next article, andnow we merely carry out the calculations and give a short commentary. Herewe present the formula for characters in the “positive” case. Theorem 2.
For an ergodic AD-measure ν α , ν α ( F g ) ≡ ν α { H : gHg − = H } = ν α { H : g ∈ N ( H ) } = Y n> [ p n ( α )] c n ( g ) , where p n ( α ) = X i =0 α ni (Newton’s power sum) and c n ( g ) is the number of cycles of length n > of apermutation g . In the case where α i = 0 for i <
0, this formula coincides with Thoma’sformula [12] for characters of the infinite symmetric group, because the mea-sure of the set of fixed points is equal to the value of the character: χ α ( g ) = ν α (Fix( g )) . In the general case, Thoma’s formula involves super-Newton instead of New-ton sums: p n ( α ) = X i> α ni + ( − n − X i< α ni = X i =0 (sgn i ) n − α ni . The measure of the set of fixed points does not depend on the types ofblocks, but for a general parameter α , the value of the character is not equaljust to the measure of this set, the formula involving a certain multiplier (see[13]). We will return to this question and give a model of representations inthe next article. For convenience, we have sightly changed the notation: usually, α i ≡ β i for i <
0, and α ≡ γ . eferences [1] D. Angeli, A. Dorno, M. Mathe, and T. Nagnibeda, Schreier graphs ofthe Basilica group, J. Modern Dynamics , No. 1, 167–205 (2010).[2] L. Bowen, Private communication.[3] P. Cameron, Oligomorphic Permutation Groups
Camb.Univ.Press(1990).[4] J. Dixon and B. Mortimer,
Permutation Groups , Springer-Verlag, NewYork, 1996.[5] R. Grigorchuk, Some problems of the dynamics of the group action onthe rooted trees,
Proc. Steklov Math. Inst. , 79–191 (2011).[6] Y. Glassner, A zero-one law for random subgroups of totally discon-nected groups; arXiv:0902.3792 (2009).[7] R. Schmidt,
Subgroup Lattices of Groups , de Gruyter, Berlin, 1994.[8] E. Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen derabzhlbar unendlichen, symmetrischen Gruppe,
Math. Z. , 40–61(1964).[9] A. M. Vershik, Description of invariant measures for the actions of someinfinite-dimensional groups, Sov. Math. Dokl. , 1396–1400 (1974).[10] A. M. Vershik, Kolmogorov’s example (a survey of actions of infinite-dimensional groups with invariant measure), Theory Probab. Appl. ,No. 2, 373–378 (2004).[11] A. M. Vershik, Nonfree actions of countable groups and their characters, J. Math. Sci. , No. 1, 1–6 (2011).[12] A. M. Vershik and S. V. Kerov, Asymptotic theory of characters of thesymmetric group,
Funct. Anal. Appl. , No. 4, 246–255 (1981).[13] A. M. Vershik and S. V. Kerov, Characters and factor representationsof the infinite symmetric group, Sov. Math. Dokl. , 1037–1040 (1981).[14] H. Wielandt, Finite Permutation Groups , Academic Press, New York–London, 1964. 24 r X i v : . [ m a t h . D S ] N ov TOTALLY NONFREE ACTIONSAND THE INFINITE SYMMETRICGROUP
A. M. Vershik ∗ October 13, 2018
To the memory of my beloved Lyotya
ABSTRACT
We consider the totally nonfree (TNF) action of a groups and the corre-sponding adjoint invariant (AD) measures on the lattices of the subgroupsof the given group. The main result is the description of all adjoint-invariantand TNF-measures on the lattice of subgroups of the infinite symmetric group S N . The problem is closely related to the theory of characters and factor rep-resentations of groups. The main result of this paper is a precise description of all adjoint-invariantergodic probability Borel measures on the lattice of all subgroups of theinfinite symmetric group S N . The reason why problems of this type areof importance can be briefly formulated as follows: the adjoint action ofa group on the lattice of subgroups with an adjoint-invariant probabilitymeasure produces, in a certain way, a nontrivial character of the group and,consequently, determines a special factor representation of the group. ∗ St. Petersburg Department of Steklov Institute of Mathematics and Max Plank In-stitute Bonn. E-mail: [email protected] . Partially supported by the RFBR-grant11-01-00677-a; RFBR-grant 11-01-12092 OFI-M
1n the case of the infinite symmetric group, it turns out that this methodyields, all characters of S N This phenomenon was first observed in [16], as aparticular fact related to a certain model of factor representations of the in-finite symmetric group. The list of characters itself is well-known; E.Thomaposed and gave the first solution of the problem ([9]), the new proof of thistheorem, which used the ergodic approach and approximation suggested in[15]. This proof was based on the ideas of the dynamical approach, and ap-proximation of the characters of infinite group with the characters of finitesymmetric group. The same dynamical approach, namely, so-called group-poid model, gives the realization of corresponding factor-representations oftype II . But in this paper for us more important, that the value of anindecomposable character at a given element of the group is equal, (up to acertain factor), to the measure of the set of fixed points of this element forsome special action. The most important thing is that precise link betweenThoma’s parameters of the characters and parameters of the measure bothare the same.But what kind of the actions can appear in this construction? This iswhat we define in this paper: it turns out that these are so-called totallynonfree ( TNF ) actions, so it is important to describe such actions for a givengroup.In this paper, we go in the opposite direction: we start with definitionand studying of the class of TNF actions of a group. From the point ofview of ergodic theory, this kind of actions is of great interest, and, as faras we know, it has not yet been systematically investigated. Due to thelack of space, we decided to separate the discussion of the link between thequestions considered here and the theory of representations, characters, andfactors; these topics will be treated in another article.In the first part of paper (the second section), we introduce the mainnotions and fix definitions related to nonfree and totally nonfree actions. Wedevelop a systematic approach to nonfree actions. Although we are mainlyinterested in totally nonfree actions, we also consider intermediate cases andthe reduction of a general action to a TNF action. The main open problemthat arises in this context concerns the existence and the list of TNF-actionsfor a given group . We use the language of the lattice of subgroups and theadjoint action of the original group on this lattice. The previous questionis equivalent to that of the existence or nonexistence of continuous adjoint-invariant measures. The universal example of a TNF action is the adjointaction of the group on the lattice of its subgroups with a TNF measure,2.e., a measure concentrated on the set of self-normalizing subgroups (= sub-groups coinciding with their normalizers). An important fact asserts thatthis measure is a complete metric invariant of the action. In general, thereare other adjoint-invariant measures that are not TNF; for example so calledRTNF-measures which also produced TNF action. We describe so calledcanonical skew-product of any action and sequence (which may be infinite ofeven transfinite) of reduced actions.All these facts heavily depend on the properties of the group. We considerhere only countable groups Of course, for the problem to be interesting, thegroup should have a continuum of subgroups. It is clear that for manygroups, f.e for groups close to abelean ones, such a measure does not exist.But for some “large” groups, we have many TNF measures (or many TNFactions), which, fortunately, can be listed up to isomorphism (in contrastto the usual situation in ergodic theory). It is natural to consider adjointmeasures on the lattice of subgroups as “random subgroups”; the notion ofa random subgroup was considered in [5, 7, 6] but with the different kind ofapplications. The author believes that this question is also of interest withinergodic theory itself.In the second part of the paper (the third section), we study the case ofthe infinite symmetric group. We use a fundamental classical fact about itssubgroups; namely, the infinite symmetric group has no primitive subgroupsexcept the alternating subgroup and the whole group itself. This follows froma classical theorem due to C. Jordan (H. Wielandt was perhaps the first toobserve this, see [4]). We successively exclude from consideration all othersubgroups that cannot lie in the support of an adjoint-invariant measure andreduce the question to a de Finetti-like problem and to Kingman’s theoremabout random partitions of the naturals. The description of adjoint-invariantmeasures on the group S N relies on the new important generalization of theclassical notion of Young subgroup, - namely, a signed Young subgroup; it isnatural to understand a random signed Young subgroup exactly as a randomsubgroup whose distribution is an adjoint-invariant measure on L ( S N ). Thelist of parameters α for these measures is exactly the list of Thoma’s param-eters. We briefly compare our formula with that of Thoma at the end ofthe paper; in a sense, our list of adjoint-invariant measures can be regardedas an explanation for the list of characters. We will return to this questionelsewhere.The direct proof of the TNF-measures for infinite symmetric group withergodic method perhaps gives us a new proof of the list of the characters of3his group. The conception presented here partially is based on the paper[16], but the general approach and link to the theory of characters is new one,it was proclaimed firstly in the authors’ talk in the Henri Poincare institute[12]. The short announcement of the approach can be found in [14].Some papers on the different topics are tightly related to out topic: thepapers by R. Grigorchuk and his colleagues [1, 2] contain examples of TNFactions of groups acting on trees. Also, papers by L. Bowen [6], found suchexamples of nontrivial AD-measures for the free group. The papers [5, 7]devoted to IRS = invariant random subgroups or AD-measures on the latticesin our terms but the goals are different.As it known for author the explicit description of the list of all AD-measures and TNF-actions for the group S N which we present here, is thefirst result of his type. Perhaps, this methods can also be applied to othergroups similar to S N , such as the group of rational interval exchange, U ( ∞ )(the infinite unitary group), GL ( F q , ∞ ), etc. It turns out that our answer iseven more tightly connected to group-theoretic structure and to the theoryof characters, than it can be assumed before; we will apply it to the theory ofcharacters and factor-representations of S N and other groups in a subsequentpaper.Professors M.Abert, L.Bowen, Y.Glasner, R.Grigorchuk, Y.Guivarch, N.Gordeev,W.Knapp, T.Nagnibeda-Smirnova, G.Olshansky, M.Zischang gave me theimportant references on the subject. I am grateful to Natalia Tsilevich forher help in editing of this article. σ -FIELDS Let ( X, A , µ ) be a Lebesgue space with a probability measure µ defined on a σ -field A of classes of mod0 coinciding measurable sets, and let a countablegroup G act on this space by µ -preserving transformations. We will consideronly effective actions , which means that only the identity e ∈ G of the groupacts as the identity transformation mod0. Because of that we denote bythe same letter an element of the group ( g ∈ G ) and the corresponding4utomorphisms ( g : x gx ) of the space ( X, µ ).For each element g ∈ G , we define a measurable set X g called the set offixed points of g : X g = { x ∈ X, gx = x } . Consider the map Φ : G → A ; Φ : g X g . Definition 1.
The fixed point σ -field corresponding to the action of G underconsideration is the sub- σ -field A G of the σ -field A generated by the family ofall sets X g , g ∈ G . The sets X g are well defined for arbitrary actions of countable groups and,more generally, for pointwise, or measurable actions of arbitrary groups. Itis worth mentioning that the above definition of the σ -field A G applies to continuous actions of arbitrary groups , since the set of fixed points for agiven automorphism is well defined with respect to mod0: if g = g mod 0(as the automorphisms of the space ( X, µ )), then X g = X g mod 0. Remark 1.
An action of a group G is called free if µX g = 0 for all g = Id, g ∈ G , or, in short, if the σ -field A G is trivial (the trivial σ -field will bedenoted by N ).For pointwise actions, we can define the notion of the stationary subgroup ,or the stabilizer , of a point x ∈ X : G x = { g ∈ G : gx = x } . It is clear that if y = hx with y, x ∈ X , h ∈ G , then G y = h − G x h . Ingeneral, this notion is not well defined for uncountable groups; more exactly,it can be defined only if one can introduce the notion of the orbit partition.Now we are going to define another sub- σ -field of the σ -field A in thespace X . We start with the following definition. Recall that an action of a group G is called pointwise (or measurable) if there is ameasurable set of full measure on which the action of G is defined; an action is calledcontinuous (in Rokhlin’s terminology; the other term is mod 0-action) if a homomorphism G → Aut(
X, µ ) is defined, where Aut(
X, µ ) is the group of all classes of measure-preservingtransformations of (
X, µ ). For countable groups, as well for locally compact groups, thesetwo notions are equivalent. efinition 2. Consider the partition ξ G of the space X into the classes ofpoints having the same stabilizer. We call it the iso-stable partition of thetriple ( X, G, µ ) . The iso-stable partition ξ G is measurable, because it is the limit, over anincreasing sequence of finite subsets K n ⊂ G , S n K n = G , of measurablepartitions ξ K n G : ξ G = lim n ξ K n G , where two points x, y ∈ X belong to the sameblock of ξ K n G if and only if K ⊂ G x , K ⊂ G y . The partition ξ G is obviously G -invariant, because an element of ξ G consists of all points that have thesame stabilizers. Definition 3.
Let A G be the sub- σ -field of A that consists of all sets measur-able with respect to the iso-stable partition ξ G . In the quotient space X/ξ G ,we have a well-defined action of the group G with invariant quotient (projec-tion) measure µ ξ G ; the action of G on ( X/ξ G , µ ξ G ) will be called the reducedaction. Proposition 1.
Assume that there is a pointwise action of a group G on aspace ( X, µ ) with an invariant measure µ . Both sub- σ -fields A G and A G are G -invariant, and the following inclusion holds: A G ⊂ A G . For a countable group G , both sub- σ -fields coincide: A G = A G ≡ A ( G ) . Proof.
The first claim is trivial: two points that cannot be separated bytheir fixed point sets have the same stabilizers. By definition, the σ -field A G is generated by the family of sets X g , g ∈ G . But, since the group G iscountable, a basis of the σ -field A G consists of the sets Y K = ∩ g ∈ K X g , where K ⊂ Stab( x ) ⊂ G is an arbitrary finite set. Thus the family X g , g ∈ G , generates both σ -fields in question.For continuous groups, the sub- σ -fields in question do not coincide ingeneral. For instance, considering the action of the orthogonal group SO (3)on the projective space RP , we see that in this case A G $ A G . Indeed, eachset of fixed points has zero measure, whence A G = N (where N is the trivial σ -field), while A G = A since the set of all rotations separate the points of P R . 6 .2 THE LATTICE OF SUBGROUPS AND THE AD-JOINT ACTION Denote by L ( G ) the set of all subgroups of a locally compact group G andequip it with the natural weak topology and the corresponding Borel struc-ture. For a countable group, the space L ( G ) equipped with this topology isa compact (Cantor) space. The adjoint action of the group G on L ( G ) isdefined as follows. Let g ∈ G , H ∈ L ( G ); thenAd( g ) H = gHg − . We will study the dynamical system ( L ( G ) , Ad( G )) from the point of viewof ergodic theory; namely, we will consider Ad( G )-invariant Borel measures.The key problem is the existence of continuous (nonatomic) invariant mea-sures. PROBLEM 1.
For what groups do there exist continuous
Ad( G ) -invariantBorel probability measures? Describe all such measures for a given group. We will solve this problem for the infinite symmetric group. Of course,the theory we develop here is interesting for countable groups that haveuncountably many subgroups.It is known (see [6]) that a non-Abelean free group has a lot of suchmeasures, but one has no general description of these measures. In [1], actionsof groups on trees and more general graphs were considered, and it wasverified that these actions are TNF.A natural point of view on Ad( G )-invariant measures is to regard themas random subgroups of G ; more precisely, each Ad( G )-invariant measuredetermines a statistics on the set of subgroups, or a random subgroup. Theinvariance under conjugations is a natural condition for applications. Onemay refine this condition and consider random subgroups with additionalproperties (e.g., TNF measures, or Ad( G )-invariant measures on the set ofself-normalizing subgroups, see below). In the recent paper [7], a problemrelated to random subgroups arises for a different reason.The lattice structure on the space of subgroups L ( G ) is a very popularobject of algebraic studies (see, e.g., [8]); we will not use it. It is worth A neighborhood of a subgroup in the weak topology is the set of subgroups that havethe same intersection with a given compact subset of G . For a discrete group, L ( G ) is asubspace of the compact space of all subsets of G . σ -finite invariant continuous measures on L ( G ). As far as weknow, ergodic aspects of the natural dynamical system ( L ( G ) , Ad( G ) , ν ),where ν is an Ad( G )-invariant measure, has not been seriously studied.Let us now connect these dynamical systems ( L ( G ) , Ad( G ) , ν ) with non-free actions of the group G . Namely, we can identify the quotient space withrespect to the iso-stable partition ξ G with L ( G ). Definition 4.
Given an action of a group G on a Lebesgue space ( X, µ ) ,consider the map Ψ : X → L ( G ) , Ψ( x ) = G x . It is a measurable homomorphism of the triple ( X, G, µ ) to the triple ( L ( G ) , Ad( G ) , Ψ ∗ µ ) ,where Ψ ∗ µ is an Ad( G ) -invariant Borel measure on L ( G ) , the image of themeasure µ under Ψ : Ψ ∗ ( µ )( B ) = µ { x : G x ∈ B ⊂ L ( G ) } . We will call Ψ ∗ µ the characteristic measure of the action ( X, G, µ ) . From definitions it is clear that Ψ is isomorphism between the reducedactions of the group G on ( X/ξ G , µ ξ G ) and adjoint action on ( L ( G ) , Ψ ∗ µ . Proposition 2.
The characteristic measure Ψ ∗ µ is a metric invariant ofmeasure-preserving actions in the following sense: if two measure-preservingactions of a countable group G on spaces ( X i , µ i ) , i = 1 , , are metricallyisomorphic, then the corresponding measures Ψ ∗ µ i , i = 1 , , on L ( G ) coin-cide.Proof. It suffices to observe that every isomorphism between two actions of G must send the set of points with a given stabilizer for one action to thesame set for the other action.The map Ψ is nothing else than the factorization of the space ( X, µ ) withrespect to the iso-stable partition ξ G , which identifies the quotient space X/ξ G with the image L ( G ). The quotient measure µ ξ G tautologically coincides withthe characteristic measure Ψ ∗ µ .For a free action, Ψ is a constant map and the characteristic measureis the δ -measure at the identity subgroup { e } ∈ L ( G ). If the action of thegroup is effective then T x G x = { e } . 8 .3 TOTALLY NONFREE (TNF) ACTIONS Definition-Theorem 1.
A measure-preserving action of a countable group G on a space ( X, µ ) is called totally nonfree (TNF) if one of the followingequivalent conditions holds: The σ -field A G (= A G = A ( G )) generated by all sets of fixed points co-incides with the whole σ -field A of all measurable subsets of X . Equivalently,the iso-stable partition ξ G coincides mod0 with the partition into separatepoints. The map
Ψ : X → X/ξ G ≃ L ( G ) is a mod0 isomorphism mod0 ofthe action of G on ( X, µ ) and adjoint action on ( L ( G ) , Ψ ∗ µ .If an action is TNF, then we say that its characteristic measure is a TNFmeasure on L ( G ) . The equivalence of the above two conditions directly follows from thedefinitions of the previous section. It is also clear that the definitions arecorrect with respect to changing the actions on sets of zero measure.TNF actions are an opposite extreme to free actions.
The characteristic measure of the ergodic TNF-action is complete metricinvariant therefore the metric classification of TNF actions of a countablegroup G reduces to the calculation of the characteristic measures Ψ ∗ µ on thelattice L ( G ). Thus the classification problem for TNF actions is, in a sense,smooth (tame), in contrast to the general isomorphism problem in ergodictheory. Definition 5.
The normalizer of a subgroup Λ ⊂ G is the subgroup N (Λ) = { g ∈ G : g Λ g − = Λ } . A subgroup H ⊂ G for which N ( H ) = H is calledself-normalizing. Denote the set of all self-normalizing subgroups of G by LN ( G ) . The following claim is obvious.
Proposition 3.
A transitive action of a group G (the left action of G on ahomogeneous space G/H ) is totally nonfree if and only if the stabilizer (i.e., H ) is a self-normalizing subgroup ( N ( H ) = H , or H ∈ LN ( G ) ). Indeed, any two points belong to the same orbit (for any x, y ∈ X , thereexists h ∈ G such that y = hx ); if they have the same stabilizer G x , then It is more natural to call such a subgroup abnormal , or anormal . ∈ N ( G x ), where N ( G x ) = { h ∈ G : h − G x h = G x } is the normalizer of G x .Consequently, either N ( G x ) = G x and the action is not TNF, or N ( G x ) = G x and the action is TNF.A similar situation holds for general actions. Proposition 4. If a measure-preserving action of a group G on a space ( X, µ ) is TNF, then for almost all x ∈ X the stabilizers G x are self-normalizing: N ( G x ) = G x ; or µ { x : G x ∈ LN ( G ) } = 1 . In other words, the char-acteristic measure of the TNF-action is concentrated on the set LN ( G ) ofself-normalizing subgroups. The adjoint action of the group G on the measure space ( L ( G ) , ν ) isTNF if and only if ν -almost all subgroups H ∈ L ( G ) have self-normalizingnormalizator: N ( N ( H )) ≡ N ( H ) = N ( H ) . In particular, the adjoint actionof the group on the lattice ( L ( G ) , ν ) with an Ad( G ) -invariant TNF-measure ν is a TNF action.Proof.
1. Assume that the action is TNF, but at the same time there existsa measurable set A of positive µ -measure such that the stabilizer G x of everypoint x ∈ A is not self-normalizing. Then there exists a point hx ∈ A with h ∈ G , h / ∈ G x such that hx = x but hG x h − = G x ; consequently, x and hx have the same stabilizer, which contradicts the TNF property.2. For adjoint action of the group G on L ( G ) the stabilizer G H = N ( H ),so condition N ( G H ) = G H is equivalent to the condition N ( H ) = N ( H ) for ν -almost all H ; by the item 1 we have TNF-action. Remark 2. L ( G ) , ν ) could be TNF-action not only for TNF-measures but for ν with property ν { H : N ( H ) = H } = 1 . We will call this measures RTNF-measures. In other words -TNF adjoint action takes place for measures ν which are concentrated on LN ( G ) : ν ( LN ( G )) = 1 . We will see that for infinite symmetric group wehave the examples of those measures.2.The condition µ { x : G x ∈ LN ( G ) } = 1 is only necessary but not suf-ficient for the action to be TNF, because the stabilizers of two points x, y that belong to different orbits can be the same self-normalizing subgroup: G x = G y ∈ LN ( G ) . Now we can specify Problem 1 formulated above and reduce the descrip-tion of TNF actions of a group to the following question.10
ROBLEM 2.
Given a group G , describe all ergodic continuous (nonatomic)probability TNF (correspondingly RTNF) measures on the lattice L ( G ) ofits subgroups. Or, equivalently, describe all ergodic continuous (nonatomic)Borel probability Ad( G ) -invariant measures on the subset LN ( G ) (corre-spondingly on LN ( G ) ) of whole space L ( G ) . We will see the different answer on the Problem 1 and Problem 2 forinfinite symmetric group.Remark that for ergodic TNF-measure ν the zero-one law with respectto LN ( G ) takes place: either ν ( LN ( G )) = 0 or ν ( LN ( G )) = 1. It isinteresting to characterize the TNF actions of a given group G from thepoint of view of ergodic theory and that of representation theory: whatkind of ergodic properties can have TNF actions, and what kind of factorrepresentations can arise for TNF actions? etc.It is interesting also to describe other classes of Ad( G )-invariant measuresdepending on the property of subgroup of full measure (or to describe randomsubgroup of the various algebraic types). Now we consider the general actions and describe the canonical reductionwhich leads to a TNF actions. First of all, we consider the factorizationwith respect to the iso-stable partition ξ G in order to define the the firstcanonical skew product structure for general actions of groups .Consider a measure-preserving action of a countable group G on a Le-besgue space ( X, µ ) and the G -invariant iso-stable partition ξ G (see Sec-tion 1.1). The reduced action (quotient action) of G on the space ( X/ξ G , µ ξ G )(see Definition 3) is isomorphic to the adjoint action of G on the space L ( G ) equipped with the characteristic measure Ψ ∗ µ . We regard the triple( L ( G ) , Ad( G ) , Ψ ∗ µ ) as the base of the canonical skew product structure forthe action ( X, G, µ ).Recall that a skew product is defined if we have an action of the groupon the base and a 1-cocycle on the base with values in the group of automor-phisms of the typical fiber (
Y, α ). For a free action of G , this skew productis trivial, the base consists of a single point, and the fiber is ( Y, α ) ≡ ( X, µ ).In the case of a TNF action, Y is a one-point space and the base coincideswith the whole space ( X, µ ) ≡ ( L ( G ) , Ψ ∗ µ ).11 efinition 6. The action of the group G on the base ( X/ξ G , µ ξ G ) ≃ ( L ( G ) , Ψ ∗ µ, Ad( G ) , ) we call canonical reduced action. The action on the space (
X, µ, G ) becomes a skew-product with fiber(
Y, α ), and we have the 1-cocycle c which is a measurable function on theproduct of the group and base with values in the group of measure-preservingtransformations of the fiber ( Y, α ): c : G × L ( G ) → AutY . Recall that thegeneral condition on a cocycle c on the space L ( G ) with the adjoint actionof the group G and an arbitrary group of coefficients has the form c ( g g , H ) = c ( g , H ) c ( g , g Hg − ) , c ( e, H ) = Id , where g ∈ G , H ∈ L ( G ), and c ( · , · ) is a measurable function on G × L ( G )with values in some group. From this we can conclude that for a fixed H , therestriction of the map g c ( g, H ) to g ∈ H is a homomorphism of the group H . But our cocycle that defines the canonical skew product has a strongerproperty. Proposition 5.
If the action of G on ( X, µ ) is ergodic, then the above con-struction defines a decomposition of the space ( X, µ ) into the direct product ( X ≈ L ( G ) × Y ; µ ξ G × α ) , where ( Y, α ) is the typical fiber of the skew product;the action of G on the base is the adjoint action with the Ad( G ) -invariantmeasure Ψ ∗ ( µ ) ; a 1-cocycle c ( · , · ) is a function on the space G × L ( G ) withvalues in the group Aut(
Y, ν ) of measure-preserving automorphisms of thespace ( Y, ν ) .A necessary and sufficient condition for a cocycle c to define the canonicalskew product is as follows. If ν is a TNF measure, which means that action on the base is TNF-action, then c ( h, H ) = Id for ν -almost all H ∈ L ( G ) and h ∈ H , where Id is the mod 0 identity map onthe space Y ; in other words, the homomorphism mentioned above is identical. If ν is not a TNF measure, then, in addition to the previous condition,the following property holds: for ν -almost all H ∈ L ( G ) and h ∈ N ( H ) \ H , α (Fix c ( h,H ) ) = 0 , where Fix φ = { y ∈ Y : φ ( y ) = y } is the set of fixed points of an automorphism φ . ν -almost all subgroups H , for all h ∈ H ,almost all points of Y are fixed points of the automorphism c ( h, H ); and thesecond condition means that for ν -almost all subgroups H , for all h / ∈ H ,the set of fixed points of c ( h, H ) is, on the contrary, of zero measure. Theseconditions on a cocycle follow from that fact that, by definition, the fibers ofthe partition ξ coincide with the sets of points with a given stabilizer. Wewill not discuss details and similar questions. Unfortunately, it is difficultto verify whether there exists a cocycle satisfying this condition for a givenAd(G)-invariant measure ν . At the same time, for the analysis of a givenaction it is important to use the canonical skew product and to study thecorresponding Ad( G )-invariant measure on L ( G ) and cocycle.Now let us consider the action of the group G on the base of the canonicalskew-product. We assume that this action is effective, in opposite case wemust apply al arguments to the quotient group of group G over kernal. Definition-Theorem 2.
Let π : X → ( X/ξ G ) -a canonical projection overiso-stable partition; the stabilizer of the projection of the point with respectto reduced action is normalizer of the stabilizer: Stab π ( x ) = N ( Stab x ) . The action of a group G on a space ( X, µ ) is called reduced totally nonfree(RTNF) if the reduced action is a TNF action, or, equivalently, (see sectionabove) µ { x : N ( G x ) ∈ LN ( G ) } = 1 ⇔ µ { x : N ( G x ) = N ( G x ) } = 1 . The adjoint action of the element of g ∈ Stab π ( x ) must preserve the stabi-lizer of Stab x by construction, this means that g ∈ N ( Stab x ) and and and viceversa. A RTNF-measure on L ( G ) is, by definition, the characteristic measure ν of a RTNF action and has property: ν { H ∈ L ( G ) : N ( H ) = N ( H ) } = 1(the second normalizer of a subgroup coincides with its first normalizer).The following commutative diagram shows the first step of our classifica-tion: X Ψ −−−→ L ( G ) π y y N X/ξ G Ψ −−−→ L ( G ) . Here the map Ψ : y G y associates with a point y its stabilizer, π : X → X/ξ G is the canonical projection, and the map N : H N ( H ) associateswith a subgroup its normalizer. 13n general, the quotient of the canonical skew product is not a TNF action,because the stabilizer of a point of the quotient space is the normalizer of thestabilizer of the original point, Stab Ψ( x ) = N (Stab x ), but different stabilizersmay have the same normalizers, and, consequently, we can obtain differentpoints with the same stabilizers. Thus we can apply again the map Ψ tothe base ( X/ξ G , ν ξ ) and consider the second canonical skew product of thebase, the third one, etc. This gives an hierarchy of Ad-invariant measures on L ( G ). Definition 7.
We will denote by AD ( G ) the space of all Ad( G ) -invariantmeasures on the lattice L ( G ) (which will be called in short “AD-measures”).Denote by N = N ∗ the operation on the set of measures on L ( G ) correspond-ing to the normalization of subgroups: [( N ) ν ]( F ) = ν ( N − F ) , F ⊂ L ( G ) . Itis clear that N sends AD-measures to AD-measures. It follows from our definitions that if a measure ν on L ( G ) is RTNF,then the measure N ( ν ) is TNF; in particular, if ν is a TNF measure, then N ( ν ) = ν . Thus the operation N : { RTNF measures } → {
TNF measures } is a projection.We have a hierarchy of AD-measures on the lattice L ( G ): AD ⊃ N ( AD ) ⊃ N ( AD ) ⊃ · · · ⊃ RT N F ⊃ T N F.
It is natural to assume that for some groups G , the chain of these normal-izations as well as the chain of the steps of reductions above can be infinite oreven transfinite. The most interesting classes of AD -measures TNF, RTNF,and AD itself.Remark that for RTNF measure ν the measure N ( ν ) is TNF measure,and, although ν is not TNF-measure, the adjoint action of the group G onthe ( L ( G ) , ν ) is TNF-action. Indeed, by definition of RTNF for ν -almost allsubgroup H , N ( H ) = N ( H ), but N ( H ) is the stabilizer of H , so ν -almostall stabilizers a self-normal. Moreover, adjoint action of G on the space( L ( G ) , ν ) for RTNF-measure ν is metrically isomorphic the adjoint action of G on the space ( L ( G ) , N ( ν )) and normalization N : L ( G ) → L ( G ) is thatisomorphism of the spaces and actions. AD -measures The natural question -is it true that each ergodic AD -measures is characteristic measure for some ergodic action of the group G .14e formulate the necessary and sufficient condition on AD measure to becharacteristic. Proposition 6.
Suppose ν is ergodic AD -measure on L ( G ) ; ν is charac-teristic measure for an ergodic action of G on a space ( X, µ ) iff there existprobability AD -measure ¯ ν with properties:1) adjoint action of G on ( L ( G ) , ¯ ν ) is ergodic;2) N (¯ ν ) = ν In this case we can define X = L ( G ) , µ = ¯ ν . It is not clear if such a measure ¯ ν exists for all AD -measures ν .
2. Fibre bundle over L ( G ) . Each subgroup H is normal subgroup inits normalizer N ( H ), so we have a fibre bundle over L ( G ) with a fibre over H ∈ L ( G )— the group N ( H ) /H . This bundle is invariant under the adjointaction of the group. We will use it for the theory of characters of the group.
3. The TNF limit of the normalizations of AD-measures.
It isnatural to assume that for every measure ν from the class of AD-measureson a given group G , the AD-measure ∩ ν does exist, which is the limit in n of the sequence of successive normalizations: ν
7→ N ∗ n ( ν ), n = 1 , , . . . . Thislimit (for some groups, it may be transfinite) must be a TNF measure.
4. TNF actions for continuous groups.
As we know, the σ -fields A G and A G can be different. It is natural to define a TNF action of ageneral group as an action for which the σ -field A G is the complete σ -field,or, for which the stabilizers separate points. In this case, we again have anisomorphism between a TNF action and the adjoint action on the lattice ofsubgroups.
5. The continuous version of combinatorial multi-transitivity.
The continuous counterpart of the notion of transitivity (or topological tran-sitivity) of actions of discrete groups is that of ergodicity. What is the analogof double transitivity? A common explanation is that this is the ergodicityof the action on the Cartesian square. But I believe that this parallel is tooweak. The definition of double transitivity in combinatorics can be formu-lated as the transitivity of the action of the stabilizer of a point x on thespace X \ x . Thus we suggest the notion of multiple transitivity, which isrelated to our consideration as follows. Definition 8.
Assume that a countable group G acts on a standard space ( X, µ ) with a G -invariant continuous measure. We say that the action is etrically double transitive if for µ -almost every point x ∈ X , the actionof the stabilizer G x ⊂ G on ( X, µ ) is ergodic. We say that the action ismetrically k -transitive if for almost every (in the sense of the measure µ k on X k ) choice of points x , x , . . . , x k − , the action of the intersection ofsubgroups T k − i =1 G x i on ( X, µ ) is ergodic. It is natural to consider this definition only for TNF actions of G . It willbe clear that all TNF actions of the infinite symmetric group are k -transitivefor an arbitrary positive integer k . It is of interest to find all countable groups for which TNF k -transitiveactions exist for any positive integer k .This problem is perhaps related to the class of oligomorphic groups , whichwas defined by P. Cameron [3] (a subgroup G of the group S N of all finitepermutations of N is called oligomorphic if for any positive integer k , thenumber of orbits of the diagonal action of G in the Cartesian product N k isfinite). We consider the countable group S N , the infinite symmetric group of all finitepermutations of the set of positive integers N (or an arbitrary countable set).In this section, we will give the list of all AD-measures on the lattice L ( S N )of subgroups of this group and, in particular, the list of TNF measures. Wewill use some classical facts about permutation groups and the probabilisticapproach.The lattice L ( S N ) is very large and contains very different types of sub-groups. Nevertheless, the support of an AD-measure consists of subgroupsof a very special kind: so-called signed Young groups. The topology and theBorel structure on L ( S N ) are defined as usual; this is a compact (Cantor)space. Definition 9 (Signed partitions) . A signed partition η of the set N is afinite or countable partition N = ∪ B ∈B B of N together with a decomposition = B + ∪ B − ∪ B of the set of its blocks, where B is the set of all single-point blocks; elements of B + are called positive blocks, and elements of B − are called negative blocks (thus each positive or negative block contains atleast two points), and we denote by B the union of all single-point blocks: B = ∪ { x }∈B { x } .Denote the set of all signed partitions of N by SPart( N ) . Recall that in the theory of finite symmetric groups, the Young subgroup Y η corresponding to an ordinary partition η = { B , B , . . . , B k } is Q ki =1 S B i ,where S B is the symmetric group acting on B . We will define the moregeneral notion of a signed Young subgroup , which makes sense both for finiteand infinite symmetric groups. We will use the following notation: S + ( B ) isthe symmetric group of all finite permutations of elements of a set B ⊂ N ,and S − ( B ) is the alternating group on B . Definition 10 (Signed Young subgroups) . The signed Young subgroup Y η corresponding to a signed partition η of N is Y η = Y B ∈B + S + ( B ) × Y B ∈B − S − ( B ) . Note that on the set B ⊂ N , the subgroup Y η act identically, so that thepartition into the orbits of Y η coincides with η .It is not difficult to describe the conjugacy class of Young subgroups withrespect to the group of inner automorphisms: Y η ∼ Y η ′ if and only if η and η ′ are equivalent up to the action of S N . But it is more important toconsider the conjugacy with respect to the group of outer automorphisms.This is the group S N of all permutations of N . Denote by r ± the numberof infinite positive (respectively, negative) blocks, and by r ± s the number offinite positive (respectively, negative) blocks of length s >
1. Obviously, thelist of numbers { r ± , r ± , . . . } is a complete set of invariants of the group ofouter automorphisms. Traditionally, the alternating group is denoted by A n ; V. I. Arnold was very enthusi-astic about the idea to denote it by S − n in order not to confuse it with the Lie algebra A n ;I agree with this idea. .2 STATEMENT OF THE MAIN RESULT Consider a sequence of positive numbers α = { α i } i ∈ Z such that α i ≥ α i +1 ≥ i > α i +1 ≥ α i ≥ i < α ≥ X i ∈ Z α i = 1 . Consider a sequence of Z -valued independent random variables ξ n , n ∈ N ,with the distributionProb { ξ n = v } = α v for all n ∈ N , v ∈ Z . Thus we have defined a Bernoulli measure µ α on the space of integer se-quences Z N = { ξ = { ξ n } n ∈ N : ξ n ∈ Z } . Definition 11 (A random signed Young subgroup and the measures ν α ) . Fix a sequence α = { α i , i ∈ Z } , and corresponding Bernoulli measure µ α ; foreach realization of the random sequence { ξ n } , n ∈ N , with the distribution µ α , define a random signed partition η ( ξ ) of N as follows: η ( ξ ) = { B i ⊂ N , i ∈ Z } , B i := { n ∈ N : ξ n = i } , here B + = { B i , i > } ; B − = { B i , i < } , and B is understood as theunion of one-point blocks. The correspondence ξ η ( ξ ) defines a probabilitymeasure on the set SPart( N ) of signed partitions, or random signed partition;the image of the Bernoulli measure µ α . The correspondence ξ Y η ( ξ ) definesa measure, which we denote by ν α , on the set of signed Young subgroups, i.e.,a measure on the lattice L ( S N ) of subgroups of S N . Note that all nonempty blocks of the random signed partition η ( ξ ) thatconsist of more than one point are infinite with ν α -probability one.Now we describe the list of all AD and TNF measures for the group S N . Theorem 1. ν α is a Borel ergodic AD-measure on thelattice L ( S N ); every ergodic probability Borel AD-measure on this latticecoincides with the measure ν α for some α .2. The measure ν α is RTNF-measure for all alpha, and is TNF-measureif and only if α i = 0 for all i ≤
0. So adjoint action of the group S N on thelattice L ( S N ) with any AD-measure is TNF-action.18 .3 PROOFS Proof.
1. The easy part of the proof is to check that the measures ν α areindeed ergodic AD-measures on L ( S N ). The invariance follows from the con-struction, because µ α , being a Bernoulli measure, is invariant under all per-mutations of indices. The symmetric, alternating, and identity subgroups ofthe symmetric groups S ( B ) are normal, so they are Ad( G )-invariant. Con-sequently, the measure ν α , being the image of µ α , is Ad( G )-invariant. Theergodicity with respect to permutations also follows from the ergodicity ofthe Bernoulli measure.2. Now suppose that we have an ergodic AD-measure on L ( S N ). We willfilter out, step by step, classes of subgroups of S N that cannot support anyAD-measure, and will finally obtain the class of signed Young groups as theonly possible class. Then we will construct all AD-measures on this class.a) A classical result of the theory of permutation groups asserts that thegroup S N has no primitive subgroups except the whole group S N and thealternating group. This is a more or less direct corollary of the fundamentalestimates obtained by C. Jordan for finite symmetric groups, which weregeneralized by H. Wielandt [17] (see, e.g., [4, Chapter 8]). Namely, this isa corollary of Jordan’s theorem asserting that if a primitive subgroup of S n has an element with support of size k , then n < β ( k ); a sharp bound on β ( k )is still unknown.b) Now consider an AD-measure ν on the set of imprimitive but transitivesubgroups of S N . Assume that such a subgroup H ∈ L ( S N ) has an invariantpartition θ . For the action of H to be transitive, all nontrivial minimal blocksof θ must have the same length l >
1, which must be finite (because finitepermutations cannot move one infinite block to another one). Denote by θ ( H ) the partition of N into the minimal blocks for H . The map H → θ ( H )associates with ν -almost every imprimitive subgroup a partition into blocks oflength l ( H ); because of the Ad( G )-ergodicity of ν , this length is the same for ν -almost all subgroups H . Thus the map H θ ( H ) sends ν to a probabilitymeasure on the set of partitions of N with countably many blocks of the samelength l >
1, and this measure is invariant with respect to the action of S N on the space of such partitions. Let us show that there are no such finite A primitive subgroup is a subgroup that has no nontrivial invariant partitions. Note that the lengths of all blocks for a given imprimitive group may be either bounded(so-called “almost primitive groups”) or unbounded (“totally imprimitive subgroups”), see[4], but this difference is not important for our purposes.
Lemma 1.
There are no probability measures on the space
Part( l ) of allpartitions of N into (countably many) blocks of length l > that are invariantwith respect to the group S N . Remark.
The space Part( l ) equipped with the weak topology is locally com-pact but not compact; its natural compactification consists of all partitionswhose blocks have length at most l . Proof.
Consider the case l = 2, the same proof is true for an arbitrary l . Eachpartition from Part(2) determines a symmetric matrix (for l > { a i,j } , a i,j = a j,i , a i,i = 0, i, j = 1 , , . . . , with only one entry ineach row and each column equal to 1, all the other entries being equal to0. But because of the S N -invariance, we have a random symmetric matrix { a i,j } , unique element in each row which is equal to 1 must be uniformlydistributed along its row. It is impossible for infinite matrix.c) We have proved that an AD-measure on the lattice L ( S N ) takes thevalue 0 on the set of all transitive subgroups; so we have reduced the anal-ysis to intransitive subgroups. Fix such a generic intransitive subgroup H ∈ L ( S N ) and consider the maximal partition η ( H ) into its transitive com-ponents. The action of the group H on each such component must be prim-itive, because imprimitive cases can be discarded for the same reason as inthe previous part of the proof. For the same reason, it is obvious that allcomponents of this partition must be infinite. Consequently, the action of H on each component is either the action of the whole symmetric group, orthat of the alternating group (see part a) of the proof), or that of the identitygroup on the single-point blocks. We denote the blocks by B i , i >
0, whenthe action of H is the action of the symmetric group of B i , and by B i , i < H is the action of the alternating group of B i . The actionof the identity group on all single-point blocks can be regarded as the identityaction on the union of such blocks B . Thus we have a signed partition η ( H )such that the action of H on each block B i , i >
0, is the action of S + ( B ), theaction of H on each block B i , i <
0, is the action of S − ( B ), and the actionon B is the identity action. This means that H ⊂ Y η = Q i S ± ( B i ) and therestriction of the action of H to B i is the action of S ± ( B ).For each i = 0 denote the group K i = { g ∈ S ± ( B i ) : ∃ ¯ g ∈ H, ¯ g | B j = id, ∀ j = i, ¯ g | B i = g }
20r a subgroup of the all elements in H which acts as identity on all B j , j = i .It is clear that K i is a normal subgroup in S ± ( B i ) (because it is the kernalof homomorphism), so K i is either S ± ( B i ) or K i = { id } , and Y i =0 K i ⊂ H Thus we need to prove that K i = H | B i = S ± ( B i ) for all i = 0 (and inparticular K i = { id } if i = 0). There are no problem with i if K i = S + ( B i ) = H | B i . We must consider two cases: the first case when K i = { id } but H | B i = S ± ( B i ) (in this case it does not matter H | B i = S + or S − ,so i = 0), and the second case when K i = S − ( B i ) = H | B i = S + ( B i ).Let us consider the first case. Suppose for some i = 0 K i = { id } but H | B i = S ± ( B i ). Then there exist at least one j = i for which K i = { g ∈ S ± ( B i ) : ∃ ¯ g ∈ H, ¯ g | B j = id, ¯ g | B i = g } , indeed the intersection could be either { id } or S ± ( B i ) for all j and if theintersection in the definition of K i over all j = i is { id } = K i , then such j exists. It means that for this j and for h = id, h ∈ H | B i there exists h ′ ∈ H | B j and g ∈ H such that g | B i = h, g | B j = h ′ . So we have a map from H | B i to H | B j which is homomorphism, and consequently isomorphism whichis simply bijection - T - between B i and B j . This bijection could be arbitrarybecause of invariance under conjugation of the group. Thus the action onof the group H on B i ∪ B j is as follow: if n ∈ B i and T n = m ∈ B i , then gm = T gn . or gT = T g on B i ∪ B j . If we restrict the action of H on B i ∪ B j only, we obtain that the group H acts periodically (or ”simultaneously”) on B i and B j . Lemma 2.
There are no AD-invariant measures which are concentrated onthe intransitive subgroups H ⊂ S N of the following type: If N = N ′ × K , ( N ′ is infinite), then H = S N ′ × { id K } ⊂ S N H = { g : g = ( g ′ , id K ); g ′ ∈ S N ′ } , or periodic action on N ′ × K .Proof. The random group H of this type must define a S N -invariant randompartition of N onto | K | parts and S N ′ - invariant random bijections betweenall parts. The invariant random partitions do exist -see the next item butinvariant bijection do not because the absence of probability measure on thegroup S N ′ . 21o we don’t need to consider the subgroups H for which the first casetakes place and consequently we already proved that K i = S ± ( B i ) i = 0(we write S ± when it is not important either S + or S − ).Suppose now that for some i , K i ⊃ S − ( B i ), and H | B i = S + ( B i ). Againfind j = 0 for which K i = { g ∈ S + ( B i ) : ∃ ¯ g ∈ H, ¯ g | B j = id, ¯ g | B i = g } .Because of definition of K i it is clear that H | B i ∪ B j ⊃ S − ( B i ) × S ± ( B j ),and the last subgroup has index in H | B i ∪ B j at most two, but also we have H | B i = S + ( B i ), so H | B i ∪ B j = S + ( B i ) × S ± ( B j ). But this means that K i = H | B i = S + ( B i ).So we prove that H = Q i H | B i and each H | B i = S ± ( B i ) for i = 0, or inanother words we have proved that H must be a signed Young subgroups:only signed Young subgroups can carry AD-invariant measures on the lattice L ( S N ).The measures να which was defined above are concentrated on the signedYoung subgroup by definition.d)Now we will prove that indeed this case is realized: the random sub-groups in the infinite symmetric group or AD-invariant ergodic measure on L ( S N is one of the measure ν α and indeed each measure ν α are AD-invariantergodic measure on L ( S N .We must identify the required measures with the ergodic limits with re-spect to conjugation of signed Young subgroups. Because of the correspon-dence between signed Young subgroups and signed partitions, this questionis equivalent to the description of S N -invariant measures on the set of signedpartitions. The last question is similar to the classical de Finetti problemconcerning S N -invariant measures on the space of all functions on N (see[10]). The only small difference lies in the fact that, in contrast to the classi-cal situation, here we have three types of blocks of signed partitions insteadof one type in the ordinary de Finetti theorem. Lemma 3 (An analog of classical de Finetti’s theorem; Kingman’s theorem[13]) . Consider the space
SPart( N ) of signed partitions of N ; every ergodic S N -invariant measure on SPart( N ) is determined by a sequence α = { α i } i ∈ Z such that α i ≥ α i +1 ≥ for i > , α i +1 ≥ α i ≥ for i < , α ≥ , and P i ∈ Z α i = 1 , as described above.Proof. The lemma can be proved by any of the methods people use to provede Finetti’s theorem. For completeness, we present a proof, applying ourold ergodic method from [10]. In order to find all ergodic measures ν on acompact S N -space X using the pointwise ergodic theorem for the group S N n ! X g ∈ S n δ gx for all x ∈ X . More exactly, we need to calculate the limitslim n n ! X g ∈ S n f ( gx )for continuous functions f ∈ C ( X ). In our case, it suffices to consider cylinderfunctions on SPart( N ) which depend on finitely many blocks. Fix a signedpartition η and label its blocks with integers in an arbitrary way so thatpositive (negative) integers correspond to positive (negative) blocks and B is the union of one-point blocks. Consider the Z -valued sequence x n , n ∈ N ,defined as follows: x n = s if n ∈ B s . Now we may say that the signedpartition η is the partition corresponding to the sequence { x n } , and eachsuch sequence determines a signed partition. The action of S N on the set ofsigned partitions and its action by permutations of coordinates of sequencesobviously agree, so our problem reduces to the description of S N -invariantmeasures on the space of all elements of Z N . But this is exactly de Finetti’sproblem. Start with an arbitrary sequence { x n } ∈ Z N and calculate the limitlim n n ! { g ∈ S n : x gn = v } = α v under the assumption that it does exist (it exists for almost all x ∈ X ). Thuswe obtain the one-dimensional distribution of the random (with respect tothe measure ν ) sequence x n . In order to prove that this measure is a Bernoullimeasure on Z N , we must calculate the joint distribution of several coordinatesof x n . But because of the complete transitivity of the action of S n , for anychoice of v , . . . , v t and for n ≫ t we have { g ∈ S n : x gi = v i , i = 1 , , . . . , t, n ≫ t } n ! ≈ t Y i =1 α v i , which means that the random variables x n , n = 1 , , . . . , are independent.Thus all AD-measures arise from Bernoulli measures on the space of signedpartitions, i.e., ν = ν α for some α . The assertion of the theorem for the un-signed partitions is Kingman’s theorem ([13]), but our proof is different.23) Consider the random signed Young subgroup Y η constructed from asequence α with α i = 0 for all i ≤
0. Then all blocks B i , i ≤
0, areempty with probability one. Then, obviously, the normalizer N ( Y η ) coincideswith Y η , since each block of η gives rise to the self-normalizing subgroup S + ( B i ). Consequently, the measure ν α is TNF. If α i > i ≤ B i is not empty with probability one, whence N ( S − B i ) = S + B i = S − B i , so that Y η is not self-normalizing. But N ( Y η ) = + ∞ Y i = −∞ S + B i . Thus N ( Y η ) = N ( Y η ), so that ν α is a RTNF measure in the terminology ofSection 1. This completes the proof of Theorem 1. Corollary 1.
The action of the group S N on the measure space ( L ( S N ) , ν α ) is ergodic. Indeed, this is a corollary of the fact that the measure ν α is the image ofthe Bernoulli measure µ α and the correspondence µ α ν α between measurescommutes with the action of the group. The corresponding representation ofthe group S N in the space L ν α ( L ( S N )) will be considered elsewhere. Corollary 2.
There are three degenerate measures ν α , in the following cases(in the parentheses we indicate the corresponding character and representa-tion, see below): α = 1 , α i = 0 , i = 1 ; in this case, ν α = δ S N ( χ ( g ) ≡ , the identityrepresentation); α − = 1 , α i = 0 , i = − ; in this case, ν α = δ S − N ( χ ( g ) = ( − sgn( g ) ,the alternating representation); α = 1 , α i = 0 , i = 0 ; in this case, ν α = δ Id S N ( χ ( g ) = δ e ( g ) , theregular representation).An ergodic AD-measure ν α is atomic only in these three cases (in whichit is if fact a δ -measure); in all the other cases, ν α is a continuous measure. Make sense to compare the language which we use here (the action on L ( G ))with that which was used in [16] (the action on the Bernoulli sequences).24ore concretely, consider the action of S N on the space Z N (instead of L ( S N )) and ask for a description of TNF and RTNF measures. The answeris a little bit different than for the action on the space of Young subgroups.Namely, the following result holds. Proposition 7.
The measure µ α on the space X = Z N with the action ofthe group S N is a TNF measure if and only if all α i , i = 0 , are distinct.If α i = α j for some i = j , then the action of S N is RTNF but not TNF.The canonical projection X → X/ξ G is the factorization with respect to thefollowing equivalence relation on X = Z N : two elements { x n } n ∈ N , { x ′ n } n ∈ N ∈ X are equivalent for if for every v ∈ Z either { n ∈ N : x n = v } = { n ∈ N : x ′ n = v } , or there exists v ′ ∈ Z with α v = α v ′ such that { n ∈ N : x n = v } = { n ∈ N : x ′ n = v ′ } , and { n ∈ N : x n = v ′ } = { n ∈ N : x ′ n = v } . Thus, in this case the action is RTNF not TNF if we have multiplicity inthe values of α : α i = α j , i = j ; this is not the case for the action in L ( S N ).The supports of the measures ν α in the topological sense (i.e., the minimalclosed subsets of full measure) coincide for all α that have the same numberof infinite blocks. The support of ν α for α having infinitely many infiniteblocks coincides with the space of all signed Young subgroups.In the case of the infinite symmetric group, all AD-measures are invariantunder the group S N of all permutations of N . The conjugation with respectto this group is an extension of the usual conjugation; but for a genericsubgroup H from a set A of full ν α -measure, its orbit under the action of S N is much larger than A . In other words, the frequencies α i are invariant underthe usual conjugation, but not under its extension. This fact is related tothe so-called Kolmogorov effect (see [11]).Theorem 1 gives more than just the list of AD-measures on the group S N ; it helps to give a new proof of Thoma’s formula for indecomposablecharacters of this group. This will be the subject of our next article, andnow we merely carry out the calculations and give a short commentary. Herewe present the formula for characters in the “positive” case. For convenience, we have sightly changed the notation: usually, α i ≡ β i for i <
0, and α ≡ γ . heorem 2. For an ergodic AD-measure ν α , ν α ( F g ) ≡ ν α { H : gHg − = H } = ν α { H : g ∈ N ( H ) } = Y n> [ p n ( α )] c n ( g ) , where p n ( α ) = X i =0 α ni (Newton’s power sum) and c n ( g ) is the number of cycles of length n > of apermutation g . In the case where α i = 0 for i <
0, this formula coincides with Thoma’sformula [15] for characters of the infinite symmetric group, because the mea-sure of the set of fixed points is equal to the value of the character: χ α ( g ) = ν α (Fix( g )) . In the general case, Thoma’s formula involves super-Newton instead of New-ton sums: p n ( α ) = X i> α ni + ( − n − X i< α ni = X i =0 (sgn i ) n − α ni . The measure of the set of fixed points does not depend on the types ofblocks, but for a general parameter α , the value of the character is not equaljust to the measure of this set, the formula involving a certain multiplier (see[16]). We will return to this question and give a model of representations inthe next article. References [1] R. Grigorchuk. Some problems on the dynamics of the group actionsonthe rooted trees. Proc. Steklov Math.Inst. v 273, 72-191,(2011).[2] D’Angeli, A. Donno, M. Matter, T. Nagnibeda. Schreier graphs of theBasilica group,
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