Notes on the Schreier graphs of the Grigorchuk group
aa r X i v : . [ m a t h . D S ] D ec Notes on the Schreier graphsof the Grigorchuk group
Yaroslav Vorobets
Abstract
The paper is concerned with the space of the marked Schreier graphs ofthe Grigorchuk group and the action of the group on this space. In partic-ular, we describe the invariant set of the Schreier graphs corresponding tothe action on the boundary of the binary rooted tree and dynamics of thegroup action restricted to this invariant set.
This paper is devoted to the study of two equivalent dynamical systems of theGrigorchuk group G , the action on the space of the marked Schreier graphs andthe action on the space of subgroups. The main object of study is going to bethe set of the marked Schreier graphs of the standard action of the group on theboundary of the binary rooted tree and their limit points in the space of all markedSchreier graphs of G .Given a finitely generated group G with a fixed generating set S , to eachaction of G we associate its Schreier graph, which is a combinatorial object thatencodes some information about orbits of the action. The marked Schreier graphsof various actions form a topological space Sch( G, S ) and there is a natural actionof G on this space. Any action of G corresponds to an invariant set in Sch( G, S )and any action with an invariant measure gives rise to an invariant measure onSch(
G, S ). The latter allows to define the notion of a random Schreier graph,which is closely related to the notion of a random subgroup of G .A principal problem is to determine how much information about the originalaction can be learned from the Schreier graphs. The worst case here is a free action,for which nothing beyond its freeness can be recovered. Vershik [5] introduced thenotion of a totally nonfree action. This is an action such that all points havedistinct stabilizers. In this case the information about the original action can berecovered almost completely. Further extensive development of these ideas wasdone by Grigorchuk [3]. 1 ac cbddd bc d Figure 1: The marked Schreier graph of 0 ∞ = 000 . . . The Grigorchuk group was introduced in [1] as a simple example of a finitelygenerated infinite torsion group. Later it was revealed that this group has inter-mediate growth and a number of other remarkable properties (see the survey [2]).In this paper we are going to use the branching property of the Grigorchuk group,which implies that its action on the boundary of the binary rooted tree is totallynonfree in a very strong sense.The main results of the paper are summarized in the following two theorems.The first theorem contains a detailed description of the invariant set of the Schreiergraphs. The second theorem is concerned with the dynamics of the group actionrestricted to that invariant set.
Theorem 1.1
Let F : ∂ T →
Sch( G , { a, b, c, d } ) be the mapping that assigns toany point on the boundary of the binary rooted tree the marked Schreier graph ofits orbit under the action of the Grigorchuk group. Then(i) F is injective;(ii) F is measurable; it is continuous everywhere except for a countable set, theorbit of the point ξ = 111 . . . ;(iii) the Schreier graph F ( ξ ) is an isolated point in the closure of F ( ∂ T ) ; theother isolated points are graphs obtained by varying the marked vertex of F ( ξ ) ;(iv) the closure of the set F ( ∂ T ) differs from F ( ∂ T ) in countably many points;these are obtained from three graphs ∆ , ∆ , ∆ choosing the marked vertexarbitrarily;(v) as an unmarked graph, F ( ξ ) is a double quotient of each ∆ i ( i = 0 , , );also, there exists a graph ∆ such that each ∆ i is a double quotient of ∆ . Theorem 1.2
Using notation of the previous theorem, let Ω be the set of non-isolated points of the closure of F ( ∂ T ) . Then(i) Ω is a minimal invariant set for the action of the Grigorchuk group G on Sch( G , { a, b, c, d } ) ; c d a ad dbc c Figure 2: The marked Schreier graph of 1 ∞ = 111 . . . (ii) the action of G on Ω is a continuous extension of the action on the boundaryof the binary rooted tree; the extension is one-to-one everywhere except fora countable set, where it is three-to-one;(iii) there exists a unique Borel probability measure ν on Sch( G , { a, b, c, d } ) in-variant under the action of G and supported on the set Ω ;(iv) the action of G on Ω with the invariant measure ν is isomorphic to the actionof G on ∂ T with the uniform measure. The paper is organized as follows. Section 2 contains a detailed constructionof the space of marked graphs. The construction is more general than that in [3].Section 3 contains notation and definitions concerning group actions. In Section4 we introduce the Schreier graphs of a finitely generated group, the space ofmarked Schreier graphs, and the action of the group on that space. In Section5 we study the space of subgroups of a countable group and establish a relationof this space with the space of marked Schreier graphs. Section 6 is devoted togeneral considerations concerning groups of automorphisms of a regular rootedtree and their actions on the boundary of the tree. In Section 7 we apply theresults of the previous sections to the study of the Grigorchuk group and proveTheorems 1.1 and 1.2. The exposition in Sections 2–6 is as general as possible, tomake their results applicable to the actions of groups other than the Grigorchukgroup.The author is grateful to Rostislav Grigorchuk, Anatoly Vershik, and OlegAgeev for useful discussions. A graph Γ is a combinatorial object that consists of vertices and edges relatedso that every edge joins two vertices or a vertex to itself (in the latter case theedge is called a loop ). The vertices joined by an edge are its endpoints . Let V be the vertex set of the graph Γ and E be the set of its edges. Traditionally E is regarded as a subset of V × V , i.e., any edge is identified with the pair of itsendpoints. In this paper, however, we are going to consider graphs with multipleedges joining the same vertices. Also, our graphs will carry additional structure.3o accomodate this, we regard E merely as a reference set whereas the actualinformation about the edges is contained in their attributes , which are functionson E . In a plain graph any edge has only one attribute: its endpoints, which arean unordered pair of vertices. Other types of graphs involve more attributes.A directed graph has directed edges. The endpoints of a directed edge e areordered, namely, there is a beginning α ( e ) ∈ V and an end ω ( e ) ∈ V . Clearly, anundirected loop is no different from a directed one. An undirected edge joiningtwo distinct vertices may be regarded as two directed edges e and e with thesame endpoints and opposite directions, i.e., α ( e ) = ω ( e ) and ω ( e ) = α ( e ).This way we can represent any graph with undirected edges as a directed graph.Conversely, some directed graphs can be regarded as graphs with undirected edges(we shall use this in Section 7).A graph with labeled edges is a graph in which each edge e is assigned a label l ( e ). The labels are elements of a prescribed finite set. A marked graph is a graphwith a distinguished vertex called the marked vertex .The vertices of a graph are pictured as dots or small circles. An undirectededge is pictured as an arc joining its endpoints. A directed edge is pictured as anarrow going from its beginning to its end. The label of an edge is written next tothe edge. Alternatively, one might think of labels as colors and picture a graphwith labeled edges as a colored graph.Let Γ be a graph and V be its vertex set. To any subset V ′ of V we associate agraph Γ ′ called a subgraph of Γ. By definition, the vertex set of the graph Γ ′ is V ′ and the edges are those edges of Γ that have both endpoints in V ′ (all attributesare retained). If Γ is a marked graph and the marked vertex is in V ′ , it will alsobe the marked vertex of the subgraph Γ ′ . Otherwise the subgraph is not marked.Suppose Γ and Γ are graphs of the same type. For any i ∈ { , } let V i bethe vertex set of Γ i and E i be the set of its edges. The graph Γ is said to be isomorphic to Γ if there exist bijections f : V → V and φ : E → E that respectthe structure of the graphs. First of all, this means that f sends the endpoints ofany edge e ∈ E to the endpoints of φ ( e ). If Γ and Γ are directed graphs, weadditionally require that α ( φ ( e )) = f ( α ( e )) and ω ( φ ( e )) = f ( ω ( e )) for all e ∈ E .If Γ and Γ have labeled edges, we also require that φ preserve labels. If Γ andΓ are marked graphs, we also require that f map the marked vertex of Γ to themarked vertex of Γ . Assuming the above requirements are met, the mapping f of the vertex set is called an isomorphism of the graphs Γ and Γ . If Γ = Γ then f is also called an automorphism of the graph Γ . We call the mapping φ a companion mapping of f . If neither of the graphs Γ and Γ admits multipleedges with identical attributes, the companion mapping is uniquely determinedby the isomorphism f . Further, we say that the graph Γ is a quotient of Γ if all of the above requirements are met except the mappings f and φ need notbe injective. Moreover, Γ is a k -fold quotient of Γ if f is k -to-1. Finally, wesay that the graphs Γ and Γ coincide up to renaming edges if they have thesame vertices and there is a one-to-one correspondence between their edges that4reserves all attributes. An equivalent condition is that the identity map on thecommon vertex set is an isomorphism of these graphs.A path in a graph Γ is a sequence of vertices v , v , . . . , v n together with asequence of edges e , . . . , e n such that for any 1 ≤ i ≤ n the endpoints of the edge e i are v i − and v i . We say that the vertex v is the beginning of the path and v n is the end. The path is closed if v n = v . The length of the path is the number ofedges in the sequence (counted with repetitions), which is a nonnegative integer.The path is a directed path if the edges are directed and, moreover, α ( e i ) = v i − and ω ( e i ) = v i for 1 ≤ i ≤ n . If the graph Γ has labeled edges then the pathis assigned a code word l ( e ) l ( e ) . . . l ( e n ), which is a string of labels read off theedges while traversing the path.We say that a vertex v of a graph Γ is connected to a vertex v ′ if there is a pathin Γ such that the beginning of the path is v and the end is v ′ . The length of theshortest path with this property is the distance from v to v ′ . The connectivity isan equivalence relation on the vertex set of Γ. The subgraphs of Γ correspondingto the equivalence classes are connected components of the graph Γ. A graphis connected if all vertices are connected to each other. Clearly, the connectedcomponents of any graph are its maximal connected subgraphs.Let v be a vertex of a graph Γ. For any integer n ≥
0, the closed ball of radius n centered at v , denoted B Γ ( v, n ), is the subgraph of Γ whose vertex set consistsof all vertices in Γ at distance at most n from the vertex v . A graph is locallyfinite if every vertex is the endpoint for only finitely many edges. If the graph Γ islocally finite then any closed ball of Γ is a finite graph , i.e., it has a finite numberof vertices and a finite number of edges.Let MG denote the set of isomorphism classes of all marked directed graphswith labeled edges. For convenience, we regard elements of MG as graphs (i.e.,we choose representatives of isomorphism classes). It is easy to observe thatconnectedness and local finiteness of graphs are preserved under isomorphisms.Let MG denote the subset of MG consisting of connected, locally finite graphs.We endow the set MG with a topology as follows. The topology is generatedby sets U (Γ , V ) ⊂ MG , where Γ runs over all finite graphs in MG and V can be any subset of the vertex set of Γ . By definition, U (Γ , V ) is the set of allisomorphism classes in MG containing any graph Γ such that Γ is a subgraph ofΓ and every edge of Γ with at least one endpoint in the set V is actually an edgeof Γ . In other words, there is no edge in Γ that joins a vertex from V to a vertexoutside the vertex set of Γ . For example, U (Γ , ∅ ) is the set of all graphs in MG that have a subgraph isomorphic to Γ . On the other hand, if V is the entirevertex set of Γ then U (Γ , V ) contains only the graph Γ . As a consequence,every finite graph in MG is an isolated point. The following lemma implies thatsets of the form U (Γ , V ) constitute a base of the topology. Lemma 2.1
Any nonempty intersection of two sets of the form U (Γ , V ) can berepresented as the union of some sets of the same form. roof. Let Γ , Γ ∈ MG be finite graphs and V , V be subsets of their vertexsets. Consider an arbitrary graph Γ ∈ U (Γ , V ) ∩ U (Γ , V ). For any i ∈ { , } let f i : W i → W ′ i be an isomorphism of the graph Γ i with a subgraph of Γ such thatno edge of Γ joins a vertex from the set f i ( V i ) to a vertex outside W ′ i . Denoteby Γ the finite subgraph of Γ with the vertex set W = W ′ ∪ W ′ . Since thesubgraphs of Γ with vertex sets W ′ and W ′ are both connected and both containthe marked vertex of Γ, the subgraph Γ is also marked and connected. Besides,no edge of Γ joins a vertex from the set V = f ( V ) ∪ f ( V ) to a vertex outside W . Hence Γ ∈ U (Γ , V ). It is easy to observe that the entire set U (Γ , V ) iscontained in the intersection U (Γ , V ) ∩ U (Γ , V ). The lemma follows.Next we introduce a distance function on MG . Consider arbitrary graphsΓ , Γ ∈ MG . Let v be the marked vertex of Γ and v be the marked vertexof Γ . We let δ (Γ , Γ ) = 0 if the graphs Γ and Γ are isomorphic (i.e., theyrepresent the same element of MG ). Otherwise we let δ (Γ , Γ ) = 2 − n , where n is the smallest nonnegative integer such that the closed balls B Γ ( v , n ) and B Γ ( v , n ) are not isomorphic. Lemma 2.2
The graphs Γ and Γ are isomorphic if and only if the closed balls B Γ ( v , n ) and B Γ ( v , n ) are isomorphic for any integer n ≥ . Proof.
For any i ∈ { , } let V i denote the vertex set of the graph Γ i and E i denote its set of edges. Further, for any integer n ≥ V i ( n ) and E i ( n ) denotethe vertex set and the set of edges of the closed ball B Γ i ( v i , n ). First assume thatthe graph Γ is isomorphic to Γ . Let f : V → V be an isomorphism of thesegraphs and φ : E → E be its companion mapping. Clearly, f ( v ) = v . It iseasy to see that any isomorphism of graphs preserves distances between vertices.It follows that f maps V ( n ) onto V ( n ) for any n ≥
0. Consequently, φ maps E ( n ) onto E ( n ). Hence the restriction of f to the set V ( n ) is an isomorphismof the graphs B Γ ( v , n ) and B Γ ( v , n ).Now assume that for every integer n ≥ B Γ ( v , n ) and B Γ ( v , n ) are isomorphic. Let f n : V ( n ) → V ( n ) be an isomorphism of thesegraphs and φ n : E ( n ) → E ( n ) be its companion mapping. Clearly, f n ( v ) = v .Note that the closed ball B Γ i ( v i , n ) is also the closed ball with the same centerand radius in any of the graphs B Γ i ( v i , m ), m > n . It follows that the restrictionof the mapping f m to the set V ( n ) is an isomorphism of the graphs B Γ ( v , n ) and B Γ ( v , n ) while the restriction of φ m to E ( n ) is its companion mapping. Since thegraphs Γ and Γ are locally finite, the sets V ( n ) , V ( n ) , E ( n ) , E ( n ) are finite.Hence there are only finitely many distinct restrictions f m | V ( n ) or φ m | E ( n ) for anyfixed n . Therefore one can find nested infinite sets of indices I ⊃ I ⊃ I ⊃ . . . such that the restriction f m | V ( n ) is the same for all m ∈ I n and the restriction φ m | E ( n ) is the same for all m ∈ I n . For any integer n ≥ f ′ n = f m | V ( n ) and φ ′ n = φ m | E ( n ) , where m ∈ I n . By construction, f ′ n is a restriction of f ′ k and φ ′ n is a restriction of φ ′ k whenever n < k . Hence there exist maps f : V → V and6 : E → E such that all f ′ n are restrictions of f and all φ ′ n are restrictions of φ . Since the graphs Γ and Γ are connected, any finite collection of vertices andedges in either graph is contained in a closed ball centered at the marked vertex.As for any n ≥ f ′ n is an isomorphism of B Γ ( v , n ) and B Γ ( v , n )and φ ′ n is its companion mapping, it follows that f is an isomorphism of Γ andΓ and φ is its companion mapping.Lemma 2.2 implies that δ is a well-defined function on MG × MG . This isa distance function, which makes MG into an ultrametric space. Lemma 2.3
The distance function δ is compatible with the topology on MG . Proof.
The base of the topology on MG consists of the sets U (Γ , V ). Thebase of the topology defined by the distance function δ is formed by open balls B (Γ , ǫ ) = { Γ ∈ MG | δ (Γ , Γ ) < ǫ } , where Γ can be any graph in MG and ǫ >
0. We have to show that any element of either base is the union of someelements of the other base.First consider an open ball B (Γ , ǫ ). If ǫ > B (Γ , ǫ ) = MG , whichis the union of all sets U (Γ , V ). Otherwise let n be the largest integer suchthat ǫ ≤ − n . Clearly, B (Γ , ǫ ) = B (Γ , − n ). Let Γ = B Γ ( v , n ), where v isthe marked vertex of the graph Γ , and let V be the set of all vertices of Γ atdistance at most n − v . Consider an arbitrary graph Γ ∈ MG such thatΓ is a subgraph of Γ. Clearly, Γ is also a subgraph of the closed ball B Γ ( v , n ).If v is a vertex of Γ at distance k from the marked vertex v , then any vertexjoined to v by an edge is at distance at most k + 1 and at least k − v .Moreover, if k > v is joined to a vertex at distance exactly k − v .It follows that Γ = B Γ ( v , n ) if and only if no vertex from the set V is joined inΓ to a vertex that is not a vertex of Γ . Thus B (Γ , − n ) = U (Γ , V ).Now consider the set U (Γ , V ), where Γ is a finite graph in MG and V isa subset of its vertex set. Denote by v the marked vertex of Γ . Let n be thesmallest integer such that every vertex of Γ is at distance at most n from v and every vertex from V is at distance at most n − v . Take any graphΓ ∈ MG such that Γ is a subgraph of Γ and there is no edge in Γ joining avertex from V to a vertex outside the vertex set of Γ . Let Γ = B Γ ( v , n ) and V be the set of all vertices of Γ at distance at most n − v . By the above, U (Γ , V ) = B (Γ , − n ). At the same time, U (Γ , V ) ⊂ U (Γ , V ) since Γ is asubgraph of Γ and V is a subset of V . Thus for any graph Γ ∈ U (Γ , V ) theentire open ball B (Γ , − n ) is contained in U (Γ , V ). In particular, U (Γ , V ) is theunion of those open balls.Given a positive integer N and a finite set L , let MG ( N, L ) denote the subsetof MG consisting of all graphs in which every vertex is the endpoint for at most N edges and every label belongs to L . Further, let MG ( N, L ) = MG ( N, L ) ∩ MG . Proposition 2.4 MG ( N, L ) is a compact subset of the metric space MG . roof. We have to show that any sequence of graphs Γ , Γ , . . . in MG ( N, L )has a subsequence converging to some graph in MG ( N, L ). For any positiveinteger n let V n denote the vertex set of the graph Γ n , E n denote its sets of edges,and v n denote the marked vertex of Γ n . First consider the special case when eachΓ n is a subgraph of Γ n +1 . Let Γ be the graph with the vertex set V = V ∪ V ∪ . . . and the set of edges E = E ∪ E ∪ . . . . We assume that any edge e ∈ E n retainsits attributes (beginning, end, and label) in the graph Γ. The common markedvertex of the graphs Γ n is set as the marked vertex of Γ. Note that any finitecollection of vertices and edges of the graph Γ is already contained in some Γ n .As the graphs Γ , Γ , . . . belong to MG ( N, L ), it follows that Γ ∈ MG ( N, L )as well. In particular, for any integer k ≥ B Γ ( v , k ) is a finitegraph. Then it is a subgraph of some Γ n . Clearly, B Γ ( v , k ) is also a subgraphof the graphs Γ n +1 , Γ n +2 , . . . . Moreover, it remains the closed ball of radius k centered at the marked vertex in all these graphs. It follows that δ (Γ m , Γ) < − k for m ≥ n . Since k can be arbitrarily large, the sequence Γ , Γ , . . . converges toΓ in the metric space MG .Next consider a more general case when each Γ n is isomorphic to a subgraph ofΓ n +1 . This case is reduced to the previous one by repeatedly using the followingobservation: if a graph P is isomorphic to a subgraph of a graph P then thereexists a graph P ′ isomorphic to P such that P is a subgraph of P ′ .Finally consider the general case. For any graph in MG ( N, L ), the closed ballof radius k with any center contains at most 1 + N + N + · · · + N k − vertices whilethe number of edges is at most N times the number of vertices. Hence for any fixed k the number of vertices and edges in the balls B Γ n ( v n , k ) is uniformly bounded,which implies that there are only finitely many non-isomorphic graphs amongthem. Therefore one can find nested infinite sets of indices I ⊃ I ⊃ I ⊃ . . . such that the closed balls B Γ n ( v n , k ) are isomorphic for all n ∈ I k . Choose anincreasing sequence of indices n , n , n , . . . such that n k ∈ I k for all k , and let Γ ′ k be the closed ball of radius k in the graph Γ n k centered at the marked point v n k .Clearly, Γ ′ k ∈ MG ( N, L ) and δ (Γ ′ k , Γ n k ) < − k . By construction, Γ ′ k is isomorphicto a subgraph of Γ ′ m whenever k < m . By the above the sequence Γ ′ , Γ ′ , Γ ′ , . . . converges to a graph Γ ∈ MG ( N, L ). Since δ (Γ ′ k , Γ n k ) < − k for all k ≥
0, thesubsequence Γ n , Γ n , Γ n , . . . converges to the graph Γ as well. Let M be an arbitrary nonempty set. Invertible transformations φ : M → M form a transformation group. An action A of an abstract group G on the set M is a homomorphism of G into that transformation group. The action can beregarded as a collection of invertible transformations A g : M → M , g ∈ G , where A g is the image of g under the homomorphism. The transformations are to satisfy A g A h = A gh for all g, h ∈ G . We say that A g is the action of an element g within8he action A . Alternatively, the action of the group G can be given as a mapping A : G × M → M such that A ( g, x ) = A g ( x ) for all g ∈ G and x ∈ M . Such amapping defines an action of G if and only if the following two conditions hold: • A ( gh, x ) = A ( g, A ( h, x )) for all g, h ∈ G and x ∈ M ; • A (1 G , x ) = x for all x ∈ M , where 1 G is the unity of the group G .A nonempty set S ⊂ G is called a generating set for the group G if any element g ∈ G can be represented as a product g g . . . g k where each factor g i is an elementof S or the inverse of an element of S . The elements of the generating set arecalled generators of the group G . The generating set S is symmetric if it is closedunder taking inverses, i.e., s − ∈ S whenever s ∈ S . If S is a generating set for G then any action A of the group G is uniquely determined by transformations A s , s ∈ S .Suppose G is a topological group. An action of G on a topological space M isa continuous action if it is continuous as a mapping of G × M to M . Similarly, anaction of G on a measured space M is a measurable action if it is measurable as amapping of G × M to M . A measurable action A of the group G on a measuredspace M with a measure µ is measure-preserving if the action of every element of G is measure-preserving, i.e., µ (cid:0) A − g ( W ) (cid:1) = µ ( W ) for all g ∈ G and measurablesets W ⊂ M . In what follows, the group G will be a discrete countable group.In that case, an action A of G is continuous if and only if all transformations A g , g ∈ G are continuous. Likewise, the action A is measurable if and only if every A g is measurable.Given an action A of a group G on a set M , the orbit O A ( x ) of a point x ∈ M under the action A is the set of all points A g ( x ), g ∈ G . A subset M ⊂ M is invariant under the action A if A g ( M ) ⊂ M for all g ∈ G . Clearly, the orbit O A ( x ) is invariant under the action. Moreover, this is the smallest invariant setcontaining x . The restriction of the action A to a nonempty invariant set M is anaction of G obtained by restricting every transformation A g to M . Equivalently,one might restrict the mapping A : G × M → M to the set G × M . The action A is transitive if the only invariant subsets of M are the empty set and M itself.Equivalently, the orbit of any point is the entire set M . Assuming the action A is continuous, it is topologically transitive if there is an orbit dense in M , and minimal if every orbit of A is dense. The action is minimal if and only if theempty set and M are the only closed invariant subsets of M . Assuming the action A is measure-preserving, it is ergodic if any measurable invariant subset of M has zero or full measure. A continuous action on a compact space M is uniquelyergodic if there exists a unique Borel probability measure on M invariant underthe action (the action is going to be ergodic with respect to that measure).Given an action A of a group G on a set M , the stabilizer St A ( x ) of a point x ∈ M under the action A is the set of all elements g ∈ G whose action fixes x ,i.e., A g ( x ) = x . The stabilizer St A ( x ) is a subgroup of G . The action is free if9ll stabilizers are trivial. In the case when the action A is continuous, we definethe neighborhood stabilizer St oA ( x ) of a point x ∈ M to be the set of all g ∈ G whose action fixes the point x along with its neighborhood (the neighborhoodmay depend on g ). The neighborhood stabilizer St oA ( x ) is a normal subgroup ofSt A ( x ).Let A : G × M → M and B : G × M → M be actions of a group G on sets M and M , respectively. The actions A and B are conjugated if thereexists a bijection f : M → M such that B g = f A g f − for all g ∈ G . Anequivalent condition is that A ( g, x ) = B ( g, f ( x )) for all g ∈ G and x ∈ M .The bijection f is called a conjugacy of the action A with B . Two continuousactions of the same group are continuously conjugated if they are conjugatedand, moreover, the conjugacy can be chosen to be a homeomorphism. Similarly,two measurable actions are measurably conjugated if they are conjugated and,moreover, the conjugacy f can be chosen so that both f and the inverse f − are measurable. Also, two measure-preserving actions are isomorphic if they areconjugated and, moreover, the conjugacy can be chosen to be an isomorphismof spaces with measure. The measure-preserving actions are isomorphic modulozero measure if each action admits an invariant set of full measure such that thecorresponding restrictions are isomorphic.Given two actions A : G × M → M and B : G × M → M of a group G , theaction A is an extension of B if there exists a mapping f of M onto M such that B g f = f A g for all g ∈ G . The extension is k -to- f is k -to-1. The extension is continuous if the actions A and B are continuous and f can be chosen continuous. Let G be a finitely generated group. Let us fix a finite symmetric generatingset S for G . Given an action A of the group G on a set M , the Schreier graph Γ Sch ( G, S ; A ) of the action relative to the generating set S is a directed graphwith labeled edges. The vertex set of the graph Γ Sch ( G, S ; A ) is M , the set ofedges is M × S , and the set of labels is S . For any x ∈ M and s ∈ S the edge( x, s ) has beginning x , end A s ( x ), and carries label s . Clearly, the action A canbe uniquely recovered from its Schreier graph. Given another action A ′ of G onsome set M ′ , the Schreier graph Γ Sch ( G, S ; A ′ ) is isomorphic to Γ Sch ( G, S ; A ) ifand only if the actions A and A ′ are conjugated. Indeed, a bijection f : M → M ′ is an isomorphism of the Schreier graphs if and only if A ′ s = f A s f − for all s ∈ S ,which is equivalent to f being a conjugacy of the action A with A ′ .Any graph of the form Γ Sch ( G, S ; A ) is called a Schreier graph of the group G (relative to the generating set S ). Notice that any graph isomorphic to a Schreiergraph is also a Schreier graph up to renaming edges. This follows from the nextproposition, which explains how to recognize a Schreier graph of G .10 roposition 4.1 A directed graph Γ with labeled edges is, up to renaming edges,a Schreier graph of the group G relative to the generating set S if and only if thefollowing conditions are satisfied:(i) all labels are in S ;(ii) for any vertex v and any generator s ∈ S there exists a unique edge withbeginning v and label s ;(iii) given a directed path with code word s s . . . s k , the path is closed wheneverthe reversed code word s k . . . s s equals G when regarded as a product in G . Proof.
First suppose Γ is a Schreier graph Γ
Sch ( G, S ; A ). Consider an arbi-trary directed path in the graph Γ. Let v be the beginning of the path and s s . . . s k be its code word. Then the consecutive vertices of the path are v = v, v , . . . , v k , where v i = A s i ( v i − ) for 1 ≤ i ≤ k . Hence the end of the path is A s k . . . A s A s ( v ) = A g ( v ), where g denotes s k . . . s s regarded as a product in G .Clearly, the path is closed whenever g = 1 G . Thus any Schreier graph of the group G satisfies the condition (iii). The conditions (i) and (ii) are trivially satisfied aswell. It is easy to see that the conditions (i), (ii), and (iii) are preserved underisomorphisms of graphs. In particular, they hold for any graph that coincideswith a Schreier graph up to renaming edges.Now suppose Γ is a directed graph with labeled edges that satisfies the con-ditions (i), (ii), and (iii). Let M denote the vertex set of Γ. Given a word w = s s . . . s k over the alphabet S , we define a transformation B w : M → M asfollows. The condition (ii) implies that for any vertex v ∈ M there is a uniquedirected path in Γ with beginning v and code word s k . . . s s (the word w re-versed). We set B w ( v ) to be the end of that path. For any words w = s s . . . s k and w ′ = s ′ s ′ . . . s ′ m over the alphabet S let ww ′ denote the concatenated word s s . . . s k s ′ s ′ . . . s ′ m . Then B ww ′ ( v ) = B w ( B w ′ ( v )) for all v ∈ M . Any word overthe alphabet S can be regarded as a product in the group G thus representing anelement g ∈ G . Clearly, the concatenation of words corresponds to the multiplica-tion in the group. The condition (iii) means that B w is the identity transformationwhenever the word w represents the unity 1 G . This implies that transformations B w and B w ′ are the same if the words w and w ′ represent the same element g ∈ G . Indeed, let w = s s . . . s k , w ′ = s ′ s ′ . . . s ′ m and consider the third word z = s − k . . . s − s − . The word z represents the inverse g − . Therefore the words wz and zw ′ both represent the unity. Then B w = B w B zw ′ = B wzw ′ = B wz B w ′ = B w ′ .Now for any g ∈ G we let A g = B w , where w is an arbitrary word over the al-phabet S representing g . By the above A g is a well-defined transformation of M , A G is the identity transformation, and A gg ′ = A g A g ′ for all g, g ′ ∈ G . Hence thetransformations A g , g ∈ G constitute an action A of the group G on the vertexset M . By construction, for any v ∈ M and s ∈ S the vertex A s ( v ) is the end11f the edge with beginning v and label s . In view of the conditions (i) and (ii),this means that the graph Γ coincides with the Schreier graph Γ Sch ( G, S ; A ) upto renaming edges.For any x ∈ M let Γ Sch ( G, S ; A, x ) denote the Schreier graph of the restrictionof the action A to the orbit of x . We refer to Γ Sch ( G, S ; A, x ) as the Schreiergraph of the orbit of x . It is easy to observe that Γ Sch ( G, S ; A, x ) is the connectedcomponent of the graph Γ
Sch ( G, S ; A ) containing the vertex x . In particular, theSchreier graph of the action A is connected if and only if the action is transitive,in which case Γ Sch ( G, S ; A, x ) = Γ
Sch ( G, S ; A ) for all x ∈ M . Let Γ ∗ Sch ( G, S ; A, x )denote a marked graph obtained from Γ
Sch ( G, S ; A, x ) by marking the vertex x .We refer to it as the marked Schreier graph of the point x (under the action A ).Notice that the point x and the restriction of the action A to its orbit are uniquelyrecovered from the graph Γ ∗ Sch ( G, S ; A, x ). Any graph of the form Γ ∗ Sch ( G, S ; A, x )is called a marked Schreier graph of the group G (relative to the generating set S ). Let Sch( G, S ) denote the set of isomorphism classes of all marked Schreiergraphs of the group G relative to the generating set S . A graph Γ ∈ MG belongs toSch( G, S ) if it is a marked directed graph that is connected and satisfies conditions(i), (ii), (iii) of Proposition 4.1.The group G acts naturally on the set of the marked Schreier graphs of G bychanging the marked vertex. The action A is given by A g (cid:0) Γ ∗ Sch ( G, S ; A, x ) (cid:1) =Γ ∗ Sch ( G, S ; A, A g ( x )), g ∈ G . It turns out that A is well defined as an action onSch( G, S ). Indeed, let Γ ∗ Sch ( G, S ; B, y ) be a marked Schreier graph isomorphic toΓ ∗ Sch ( G, S ; A, x ). Then any isomorphism f of the latter graph with the former oneis simultaneously a conjugacy of the restriction of the action A to the orbit of x with the restriction of the action B to the orbit of y . Since f ( x ) = y , it followsthat f ( A g ( x )) = B g ( y ) for all g ∈ G . Hence for any g ∈ G the map f is also anisomorphism of the graph Γ ∗ Sch ( G, S ; A, A g ( x )) with Γ ∗ Sch ( G, S ; B, B g ( y )). Proposition 4.2
Sch(
G, S ) is a compact subset of the metric space MG . Theaction of the group G (regarded as a discrete group) on Sch(
G, S ) is continuous. Proof.
Let N be the number of elements in the generating set S . Then everyvertex v of a graph Γ ∈ Sch(
G, S ) is the beginning of exactly N edges. Further-more, v is the end of an edge with beginning v ′ and label s if and only if v ′ isthe end of the edge with beginning v and label s − . It follows that v is also theend of exactly N edges. Hence any vertex of Γ is an endpoint for at most 2 N edges. Therefore Sch( G, S ) ⊂ MG (2 N, S ). Since all marked Schreier graphs areconnected, we have Sch(
G, S ) ⊂ MG (2 N, S ) ⊂ MG .Now let us show that the set Sch( G, S ) is closed in the topological space MG .Take any graph Γ ∈ MG not in that set. Then Γ does not satisfy at least oneof the conditions (i), (ii), and (iii) in Proposition 4.1. First consider the casewhen the condition (i) or (iii) does not hold. Since the graph Γ is locally finite,12t has a finite subgraph Γ for which the same condition does not hold. Since Γis connected, we can choose the subgraph Γ to be marked and connected so thatΓ ∈ MG . Clearly, the same condition does not hold for any graph Γ ′ such thatΓ is a subgraph of Γ ′ . It follows that the neighborhood U (Γ , ∅ ) of the graph Γis disjoint from Sch( G, S ). Next consider the case when Γ does not satisfy thecondition (ii). Let v be the vertex of Γ such that for some generator s ∈ S thereare either several edges with beginning v and label s or no such edges at all.Since Γ ∈ MG , there exists a finite connected subgraph Γ of Γ that containsthe marked vertex, the vertex v , and all edges for which v is an endpoint. ThenΓ ∈ MG and the open set U (Γ , { v } ) is a neighborhood of Γ. By construction,the condition (ii) fails in the entire neighborhood so that U (Γ , { v } ) is disjointfrom Sch( G, S ). Thus the set MG \ Sch(
G, S ) is open in MG . Therefore theset Sch( G, S ) is closed.Since the closed set Sch(
G, S ) is contained in MG (2 N, S ), which is a compactset due to Proposition 2.4, the set Sch(
G, S ) is compact as well.An action of the group G is continuous whenever the generators act continu-ously. To prove that the transformations A s , s ∈ S are continuous, we are goingto show that δ ( A s (Γ) , A s (Γ ′ )) ≤ δ (Γ , Γ ′ ) for any graphs Γ , Γ ′ ∈ Sch(
G, S ) andany generator s ∈ S . If the graphs Γ and Γ ′ are isomorphic, then the graphs A s (Γ) and A s (Γ ′ ) are also isomorphic so that δ ( A s (Γ) , A s (Γ ′ )) = δ (Γ , Γ ′ ) = 0.Otherwise δ (Γ , Γ ′ ) = 2 − n for some nonnegative integer n . Since the distance be-tween any graphs in MG never exceeds 1, it is enough to consider the case n ≥ v denote the marked vertex of Γ and v ′ denote the marked vertex of Γ ′ . Bydefinition of the distance function, the closed balls B Γ ( v, n −
1) and B Γ ′ ( v ′ , n − f of these graphs. Clearly, f ( v ) = v ′ .Let v denote the marked vertex of the graph A s (Γ) and v ′ denote the markedvertex of A s (Γ ′ ). Then v is the end of the edge with beginning v and label s inthe graph Γ. Similarly, v ′ is the end of the edge with beginning v ′ and label s inΓ ′ . It follows that f ( v ) = v ′ . Since the vertex v is joined to v by an edge, theclosed ball B Γ ( v , n −
2) is a subgraph of B Γ ( v, n − B Γ ( v , n − B Γ ( v, n − B Γ ′ ( v ′ , n −
2) is a subgraph of B Γ ′ ( v, n −
1) and it is also the closed ballof radius n − v ′ in the graph B Γ ′ ( v, n − f ( v ) = v ′ and anyisomorphism of graphs preserves distance between vertices, the restriction f of f to the vertex set of B Γ ( v , n −
2) is an isomorphisms of the graphs B Γ ( v , n − B Γ ′ ( v ′ , n − B A s (Γ) ( v , n −
2) differsfrom B Γ ( v , n −
2) in that the marked vertex is v and, similarly, B A s (Γ ′ ) ( v ′ , n − B Γ ′ ( v ′ , n −
2) in that the marked vertex is v ′ . Therefore f is alsoan isomorphism of B A s (Γ) ( v , n −
2) and B A s (Γ ′ ) ( v ′ , n − δ ( A s (Γ) , A s (Γ ′ )) ≤ − ( n − = 2 δ (Γ , Γ ′ ).Let A be an action of the group G on a set M . To any point x ∈ M weassociate three subgroups of G : the stabilizer St A ( x ) of x , the stabilizer St A (Γ ∗ x ) of13he marked Schreier graph Γ ∗ x = Γ ∗ Sch ( G, S ; A, x ), and the neighborhood stabilizerSt o A (Γ ∗ x ) (if the action A is continuous then there is the fourth subgroup, theneighborhood stabilizer of x ). Clearly, the graph Γ ∗ A g ( x ) coincides with Γ ∗ x if andonly if A g ( x ) = x . However this does not imply that the stabilizer of the graphis the same as the stabilizer of x . Since A is an action on isomorphism classes ofgraphs, we have g ∈ St A (Γ ∗ x ) if and only if the graph Γ ∗ A g ( x ) is isomorphic to Γ ∗ x . Lemma 4.3 (i) St A ( x ) is a normal subgroup of St A (cid:0) Γ ∗ Sch ( G, S ; A, x ) (cid:1) .(ii) The quotient of St A (cid:0) Γ ∗ Sch ( G, S ; A, x ) (cid:1) by St A ( x ) is isomorphic to the groupof all automorphisms of the unmarked graph Γ Sch ( G, S ; A, x ) .(iii) St A ( x ) is a subgroup of St o A (cid:0) Γ ∗ Sch ( G, S ; A, x ) (cid:1) . Proof.
Without loss of generality we can assume that the action A is transi-tive. For brevity, let Γ ∗ denote the marked graph Γ ∗ Sch ( G, S ; A, x ), Γ denote theunmarked graph Γ
Sch ( G, S ; A, x ), and R denote the group of all automorphismsof Γ. Consider an arbitrary f ∈ R . For any vertex y ∈ O A ( x ) and any label s ∈ S the unique edge of Γ with beginning y and label s has end A s ( y ). It followsthat f ( A s ( y )) = A s ( f ( y )). Since the action A is transitive, the automorphism f commutes with transformations A s , s ∈ S . Then f commutes with A g for all g ∈ G . Notice that the automorphism f is uniquely determined by the vertex f ( x ). Indeed, any vertex y of Γ is represented as A g ( x ) for some g ∈ G , then f ( y ) = f ( A g ( x )) = A g ( f ( x )). In particular, f is the identity if f ( x ) = x .To prove the statements (i) and (ii), we are going to construct a homomor-phism Ψ of the stabilizer St A (Γ ∗ ) onto the group R with kernel St A ( x ). An element g ∈ G belongs to St A (Γ ∗ ) if the graph Γ ∗ is isomorphic to Γ ∗ Sch ( G, S ; A, A g ( x )).An isomorphism of these marked graphs is an automorphism of the unmarkedgraph Γ that sends x to A g ( x ). Hence g ∈ St A (Γ ∗ ) if and only if A g ( x ) = ψ g ( x )for some ψ g ∈ R . By the above the automorphism ψ g is uniquely determinedby A g ( x ). Now we define a mapping Ψ : St A (Γ ∗ ) → R by Ψ( g ) = ψ g − . Itis easy to observe that Ψ maps St A (Γ ∗ ) onto R and the preimage of the iden-tity under Ψ is St A ( x ). Further, for any g, h ∈ St A (Γ ∗ ) we have ψ ( gh ) − ( x ) = A − gh ( x ) = A − h ( A − g ( x )) = A − h ( ψ g − ( x )). Recall that the automorphism ψ g − com-mutes with the action A , in particular, A − h ψ g − = ψ g − A − h . Then ψ ( gh ) − ( x ) = ψ g − ( A − h ( x )) = ψ g − ( ψ h − ( x )), which implies that Ψ( gh ) = Ψ( g )Ψ( h ). Thus Ψ isa homomorphism.We proceed to the statement (iii). Take any element g ∈ St A ( x ). It can berepresented as a product s s . . . s k , where each s i is in S . Let γ denote the uniquedirected path in Γ ∗ with beginning x and code word s k . . . s s . By construction,the end of the path γ is A g ( x ) so that the path is closed. Let Γ ∗ denote thesubgraph of Γ ∗ whose vertex set consists of all vertices of the path γ . Clearly,Γ ∗ is a marked graph, finite and connected. Hence Γ ∗ ∈ MG . Any graphΓ ∗ ∈ U (Γ ∗ , ∅ ) admits a closed directed path with beginning at the marked pointand code word s k . . . s s . If Γ ∗ = Γ ∗ Sch ( G, S ; B, y ), this implies that B g ( y ) = y .14ence g ∈ St B ( y ) ⊂ St A (cid:0) Γ ∗ Sch ( G, S ; B, y ) (cid:1) . Thus the transformation A g fixesthe set U (Γ ∗ , ∅ ) ∩ Sch(
G, S ), which is an open neighborhood of the graph Γ ∗ inSch( G, S ).Any group G acts naturally on itself by left multiplication. The action adj G : G × G → G , called adjoint , is given by adj G ( g , g ) = g g . The Schreier graphof this action relative to any generating set S is the Cayley graph of the group G relative to S . Given a subgroup H of G , the adjoint action of the group G descends to an action on G/H . The action adj
G,H : G × G/H → G/H is givenby adj
G,H ( g , gH ) = ( g g ) H . The Schreier graph of the latter action relative to agenerating set S is denoted Γ coset ( G, S ; H ). It is called a Schreier coset graph . The marked Schreier coset graph Γ ∗ coset ( G, S ; H ) is the marked Schreier graph of thecoset H under the action adj G,H . It is obtained from Γ coset ( G, S ; H ) by markingthe vertex H . Proposition 4.4
A marked Schreier graph Γ ∗ Sch ( G, S ; A, x ) is isomorphic to amarked Schreier coset graph Γ ∗ coset ( G, S ; H ) if and only if H = St A ( x ) . Proof.
Let H denote the stabilizer St A ( x ). Suppose A g ( x ) = A g ( x ) for some g , g ∈ G . Then A g − g ( x ) = A − g ( A g ( x )) = x so that g − g ∈ H . Hence g − g H = H and g H = g H . Conversely, if g H = g H then g = g h forsome h ∈ H . It follows that A g ( x ) = A g ( A h ( x )) = A g ( x ).Let us define a mapping f : G/H → O A ( x ) by f ( gH ) = A g ( x ). By theabove f is well defined and one-to-one. Clearly, it maps G/H onto the entireorbit O A ( x ). For any g , g ∈ G we have f ( g gH ) = A g g ( x ) = A g ( A g ( x )) = A g ( f ( gH )). Therefore f is a conjugacy of the action adj G,H with the restric-tion of the action A to the orbit O A ( x ). It follows that f is also an isomorphismof the unmarked graphs Γ coset ( G, S ; H ) and Γ Sch ( G, S ; A, x ). As f ( H ) = x ,the mapping f is an isomorphism of the marked graphs Γ ∗ coset ( G, S ; H ) andΓ ∗ Sch ( G, S ; A, x ) as well.Since any isomorphism of Schreier graphs of the group G is also a conjugacyof the corresponding actions, it preserves stabilizers of vertices. In particular,marked Schreier graphs cannot be isomorphic if the stabilizers of their markedvertices do not coincide. For any subgroup H of G the stabilizer of the coset H under the action adj G,H is H itself. Therefore the graph Γ ∗ coset ( G, S ; H ) is notisomorphic to Γ ∗ Sch ( G, S ; A, x ) if H = St A ( x ). Let G be a discrete countable group. Denote by Sub( G ) the set of all subgroupsof G . We endow the set Sub( G ) with a topology as follows. First we consider theproduct topology on { , } G . The set { , } G is in a one-to-one correspondence15ith the set of all functions f : G → { , } . Also, any subset H ⊂ G (in particular,any subgroup) is assigned the indicator function χ H : G → { , } defined by χ H ( g ) = (cid:26) g ∈ H, g / ∈ H. This gives rise to a mapping j : Sub( G ) → { , } G , which is an embedding. Nowthe topology on Sub( G ) is the smallest topology such that the embedding j iscontinuous. By definition, the base of this topology consists of sets of the form U G ( S + , S − ) = { H ∈ Sub( G ) | S + ⊂ H and S − ∩ H = ∅} , where S + and S − run independently over all finite subsets of G . Notice that U G ( S +1 , S − ) ∩ U G ( S +2 , S − ) = U G ( S +1 ∪ S +2 , S − ∪ S − ).The topological space Sub( G ) is ultrametric and compact (since { , } G is ul-trametric and compact, and j (Sub( G )) is closed in { , } G ). Suppose g , g , g , . . . is a complete list of elements of the group G . For any subgroups H , H ⊂ G let d ( H , H ) = 0 if H = H ; otherwise let d ( H , H ) = 2 − n , where n is the smallestindex such that g n belongs to the symmetric difference of H and H . Then d isa distance function on Sub( G ) compatible with the topology.Note that the above construction also applies to a finite group G , in whichcase Sub( G ) is a finite set with the discrete topology.The following three lemmas explore properties of the topological space Sub( G ). Lemma 5.1
The intersection of subgroups is a continuous operation on the space
Sub( G ) . Proof.
We have to show that the mapping I : Sub( G ) × Sub( G ) → Sub( G )defined by I ( H , H ) = H ∩ H is continuous. Take any finite sets S + , S − ⊂ G .Given subgroups H , H ⊂ G , the intersection H ∩ H is an element of the set U G ( S + , S − ) if and only if H ∈ U G ( S + , S ) and H ∈ U G ( S + , S ) for some sets S and S such that S ∪ S = S − . Clearly, the sets S and S are finite. It followsthat I − (cid:0) U G ( S + , S − ) (cid:1) = [ S ,S : S ∪ S = S − U G ( S + , S ) × U G ( S + , S ) . It remains to notice that any open subset of Sub( G ) is a union of sets of theform U G ( S + , S − ) while any set of the form U G ( S + , S ) × U G ( S + , S ) is open inSub( G ) × Sub( G ). Lemma 5.2
For any subgroups H and H of the group G , let H ∨ H denote thesubgroup generated by all elements of H and H . Then ∨ is a Borel measurableoperation on Sub( G ) . roof. We have to show that the mapping J : Sub( G ) × Sub( G ) → Sub( G )defined by J ( H , H ) = H ∨ H is Borel measurable. Take any g ∈ G and considerarbitrary subgroups H , H ∈ Sub( G ) such that J ( H , H ) ∈ U G ( { g } , ∅ ), i.e., H ∨ H contains g . The element g can be represented as a product g = h h . . . h k ,where each h i belongs to H or H . Let S denote the set of all elements of H in the sequence h , h , . . . , h k and S denote the set of all elements of H inthe same sequence. Then the element g belongs to K ∨ K for any subgroups K ∈ U G ( S , ∅ ) and K ∈ U G ( S , ∅ ). Hence the pair ( H , H ) is contained in thepreimage of U G ( { g } , ∅ ) under the mapping J along with its open neighborhood U G ( S , ∅ ) × U G ( S , ∅ ). Thus the preimage J − (cid:0) U G ( { g } , ∅ ) (cid:1) is an open set. Sincethe set U G ( ∅ , { g } ) is the complement of U G ( { g } , ∅ ), its preimage under J is closed.Given finite sets S + , S − ⊂ G , the set U G ( S + , S − ) is the intersection of sets U G ( { g } , ∅ ), g ∈ S + and U G ( ∅ , { h } ), h ∈ S − . By the above J − (cid:0) U G ( S + , S − ) (cid:1) isa Borel set, the intersection of an open set with a closed one. Finally, any opensubset of Sub( G ) is the union of some sets U G ( S + , S − ). Moreover, it is a finite orcountable union since there are only countably many sets of the form U G ( S + , S − ).It follows that the preimage under J of any open set is a Borel set. Lemma 5.3
Suppose H is a subgroup of G . Then Sub( H ) is a closed subset of Sub( G ) . Moreover, the intrinsic topology on Sub( H ) coincides with the topologyinduced by Sub( G ) . Proof.
The intrinsic topology on Sub( H ) is generated by all sets of the form U H ( P + , P − ), where P + and P − are finite subsets of H . The topology inducedby Sub( G ) is generated by all sets of the form U G ( S + , S − ) ∩ Sub( H ), where S + and S − are finite subsets of G . Clearly, U G ( S + , S − ) ∩ Sub( H ) = U H ( S + , S − ∩ H )if S + ⊂ H and U G ( S + , S − ) ∩ Sub( H ) = ∅ otherwise. It follows that the twotopologies coincide.For any g ∈ G the open set U G ( ∅ , { g } ) is also closed in Sub( G ) as it is thecomplement of another open set U G ( { g } , ∅ ). Then the set Sub( H ) is closed inSub( G ) since it is the intersection of closed sets U G ( ∅ , { g } ) over all g ∈ G \ H .Let A be an action of the group G on a set M . Let us consider the stabilizerSt A ( x ) of a point x ∈ M under the action (see Section 3) as the value of a mappingSt A : M → Sub( G ). Lemma 5.4
Suppose A is a continuous action of the group G on a Hausdorfftopological space M . Then(i) the mapping St A is Borel measurable;(ii) St A is continuous at a point x ∈ M if and only if the stabilizer of x underthe action coincides with its neighborhood stabilizer: St oA ( x ) = St A ( x ) ; iii) if a sequence of points in M converges to the point x and the sequence oftheir stabilizers converges to a subgroup H , then St oA ( x ) ⊂ H ⊂ St A ( x ) . Proof.
For any g ∈ G let Fix A ( g ) denote the set of all points in M fixed bythe transformation A g . Let us show that Fix A ( g ) is a closed set. Take any point x ∈ M not in Fix A ( g ). Since the points x and A g ( x ) are distinct, they havedisjoint open neighborhoods X and Y , respectively. Since A g is continuous, thereexists an open neighborhood Z of x such that A g ( Z ) ⊂ Y . Then X ∩ Z is an openneighborhood of x and A g ( X ∩ Z ) is disjoint from X ∩ Z . In particular, X ∩ Z isdisjoint from Fix A ( g ).For any finite sets S + , S − ⊂ G the preimage of the open set U G ( S + , S − ) underthe mapping St A is \ g ∈ S + Fix A ( g ) \ [ h ∈ S − Fix A ( h ) . This is a Borel set as Fix A ( g ) is closed for any g ∈ G . Since sets of the form U g ( S + , S − ) constitute a base of the topology on Sub( G ), the mapping St A isBorel measurable.The mapping St A is continuous at a point x ∈ M if and only if x is an interiorpoint in the preimage under St A of any set U G ( S + , S − ) containing St A ( x ). Thelatter holds true if and only if x is an interior point in any set Fix A ( g ) containingthis point. Clearly, x is an interior point of Fix A ( g ) if and only if g belongs tothe neighborhood stabilizer St oA ( x ). Thus St A is continuous at x if and only if anyelement of St A ( x ) belongs to St oA ( x ) as well.Now suppose that a sequence x , x , . . . of points in M converges to the point x and, moreover, the stabilizers St A ( x ) , St A ( x ) , . . . converge to a subgroup H .Consider an arbitrary g ∈ G . In the case g ∈ H , the subgroup H belongs to theopen set U G ( { g } , ∅ ). Since St A ( x n ) → H as n → ∞ , we have St A ( x n ) ∈ U G ( { g } , ∅ )for large n . In other words, x n ∈ Fix A ( g ) for large n . Since the set Fix A ( g ) isclosed, it contains the limit point x as well. That is, g ∈ St A ( x ). In the case g / ∈ H ,the subgroup H belongs to the open set U G ( ∅ , { g } ). Then St A ( x n ) ∈ U G ( ∅ , { g } )for large n . In other words, x n / ∈ Fix A ( g ) for large n . Since x n → x as n → ∞ ,the action of g fixes no neighborhood of x . That is, g / ∈ St oA ( x ).The group G acts naturally on the set Sub( G ) by conjugation. The action C : G × Sub( G ) → Sub( G ) is given by C ( g, H ) = gHg − . This action is continuous.Indeed, one easily observes that C − g (cid:0) U G ( S , S ) (cid:1) = U G ( g − S g, g − S g ) for all g ∈ G and finite sets S , S ⊂ G . Proposition 5.5
The action C of the group G on Sub( G ) is continuously conju-gated to the action A on the space Sch(
G, S ) of the marked Schreier graphs of G relative to a generating set S . Moreover, the mapping f : Sub( G ) → Sch(
G, S ) given by f ( H ) = Γ ∗ coset ( G, S ; H ) is a continuous conjugacy. roof. Proposition 4.4 implies that the mapping f is bijective.Consider arbitrary element g and subgroup H of the group G . The stabilizerof the coset gH under the action adj G,H consists of those g ∈ G for which g gH = gH . The latter condition is equivalent to g − g g ∈ H . Therefore the stabilizer is gHg − = C g ( H ). As A g (cid:0) Γ ∗ coset ( G, S ; H ) (cid:1) = Γ ∗ Sch ( G, S ; adj
G,H , gH ), it follows fromProposition 4.4 that A g ( f ( H )) = f ( C g ( H )). Thus f conjugates the action C with A . Now we are going to show that for any finite sets S + , S − ⊂ G the imageof the open set U G ( S + , S − ) under the mapping f is open in Sch( G, S ). LetΓ ∗ Sch ( G, S ; A, x ) be an arbitrary graph in that image. Any element g ∈ G canbe represented as a product s s . . . s k , where s i ∈ S . Let us fix such a representa-tion and denote by γ g the unique directed path in Γ ∗ Sch ( G, S ; A, x ) with beginning x and code word s k . . . s s . Then the end of the path γ g is A g ( x ). In particular,the path γ g is closed if and only if g ∈ St A ( x ). By Proposition 4.4, the preimageof the graph Γ ∗ Sch ( G, S ; A, x ) under f is St A ( x ). Since St A ( x ) ∈ U G ( S + , S − ), thepath γ g is closed for g ∈ S + and not closed for g ∈ S − . Let Γ denote the smallestsubgraph of Γ ∗ Sch ( G, S ; A, x ) containing all paths γ g , g ∈ S + ∪ S − . Clearly, Γ is a marked graph, finite and connected. Hence Γ ∈ MG . For any markedSchreier graph Γ ∗ Sch ( G, S ; B, y ) in U (Γ , ∅ ), the directed path with beginning y and the same code word as in γ g is closed for all g ∈ S + and not closed for all g ∈ S − . It follows that St B ( y ) ∈ U G ( S + , S − ). Therefore the graph Γ ∗ Sch ( G, S ; A, x )is contained in f ( U G ( S + , S − )) along with its neighborhood U (Γ , ∅ ) ∩ Sch(
G, S ).Any open set in Sub( G ) is the union of some sets U G ( S + , S − ). Hence it followsfrom the above that the mapping f maps open sets onto open sets. In other words,the inverse mapping f − is continuous. Since the topological spaces Sub( G ) andSch( G, S ) are compact, f is continuous as well.Proposition 5.5 allows for a short (although not constructive) proof of thefollowing statement. Proposition 5.6
Any subgroup of finite index of a finitely generated group is alsofinitely generated.
Proof.
Suppose G is a finitely generated group and H is a subgroup of G offinite index. Let S be a finite symmetric generating set for G . By Proposition 5.5,the space Sub( G ) of subgroups of G is homeomorphic to the space Sch( G, S ) ofmarked Schreier graphs of G relative to the generating set S . Moreover, there isa homeomorphism that maps the subgroup H to the marked Schreier coset graphΓ ∗ coset ( G, S ; H ). The vertices of the graph are cosets of H in G . Since H has finiteindex in G , the graph Γ ∗ coset ( G, S ; H ) is finite. Notice that any finite graph in thetopological space MG , which contains Sch( G, S ), is an isolated point. It followsthat H is an isolated point in Sub( G ). Then there exist finite sets S + , S − ⊂ G such that H is the only element of the open set U G ( S + , S − ). Let H be thesubgroup of G generated by the finite set S + . Since S + ⊂ H and S − ∩ H = ∅ , thesubgroup H is disjoint from S − . Thus H ∈ U G ( S + , S − ) so that H = H .19 Automorphisms of regular rooted trees
Consider an arbitrary graph Γ. Let γ be a path in this graph, v , v , . . . , v m beconsecutive vertices of γ , and e , . . . , e m be consecutive edges. A backtracking inthe path γ occurs if e i +1 = e i for some i (then v i +1 = v i − ). The graph Γ is calleda tree if it is connected and admits no closed path of positive length withoutbacktracking. In particular, this means no loops and no multiple edges. A rootedtree is a tree with a distinguished vertex called the root . Clearly, the root is asynonym for the marked vertex. For any integer n ≥ level n (or the n thlevel) of the tree is defined as the set of vertices at distance n from the root. If n ≥ n th level is joined to exactly one vertex on thelevel n − n + 1. The rooted treeis called k -regular if every vertex on any level n is joined to exactly k vertices onlevel n + 1. The 2-regular rooted tree is also called binary .All k -regular rooted trees are isomorphic to each other. A standard modelof such a tree is built as follows. Let X be a set of cardinality k referred toas the alphabet (usually X = { , , . . . , k − } ). A word (or finite word ) in thealphabet X is a finite string of elements from X (referred to as letters ). The setof all words in the alphabet X is denoted X ∗ . X ∗ is a monoid with respect tothe concatenation (the unit element is the empty word, denoted ∅ ). Moreover, itis the free monoid generated by elements of X . Now we define a plain graph T with the vertex set X ∗ in which two vertices w and w are joined by an edge if w = w x or w = w x for some x ∈ X . Then T is a k -regular rooted tree withthe root ∅ . The n th level of the tree T consists of all words of length n .A bijection f : X ∗ → X ∗ is an automorphism of the rooted tree T if and onlyif it preserves the length of any word and the length of the common beginningof any two words. Given an automorphism f and a word u ∈ X ∗ , there exists aunique transformation h : X ∗ → X ∗ such that f ( uw ) = f ( u ) h ( w ) for all w ∈ X ∗ .It is easy to see that h is also an automorphism of the tree T . This automorphismis called the section of f at the word u and denoted f | u . A set of automorphismsof the tree T is called self-similar if it is closed under taking sections. For anyautomorphisms f and h and any word u ∈ X ∗ one has ( f h ) | u = f | h ( u ) h | u and f − | u = (cid:0) f | f − ( u ) (cid:1) − . It follows that any group of automorphisms generated by aself-similar set is itself self-similar.Suppose G is a group of automorphisms of the tree T . Let α denote thenatural action of G on the vertex set X ∗ . Given a word u ∈ X ∗ , the sectionmapping g g | u is a homomorphism when restricted to the stabilizer St α ( u ). If G is self-similar then this is a homomorphism to G . The self-similar group G iscalled self-replicating if for any u ∈ X ∗ the mapping g g | u maps the subgroupSt α ( u ) onto the entire group G .Suppose that letters of the alphabet X are canonically ordered: x , x , . . . , x k .For any permutation π on X and automorphisms h , h , . . . , h k of the tree T wedenote by π ( h , h , . . . , h k ) a transformation f : X ∗ → X ∗ given by f ( x i w ) =20 ( x i ) h i ( w ) for all w ∈ X ∗ and 1 ≤ i ≤ k . It is easy to observe that f is also anautomorphism of T and h i = f | x i for 1 ≤ i ≤ k . The expression π ( h , h , . . . , h k )is called the wreath recursion for f . Any self-similar set of automorphisms f j , j ∈ J satisfies a system of “self-similar” wreath recursions f j = π j ( f m ( j,x ) , f m ( j,x ) , . . . , f m ( j,x k ) ) , j ∈ J, where π j , j ∈ J are permutations on X and m maps J × X to J . Lemma 6.1
Any system of self-similar wreath recursions over the alphabet X issatisfied by a unique self-similar set of automorphisms of the regular rooted tree T . Proof.
Consider a system of wreath recursions f j = π j ( f m ( j,x ) , . . . , f m ( j,x k ) ), j ∈ J , where π j , j ∈ J are permutations on X and m is a mapping of J × X to J . We define transformations F j , j ∈ J of the set X ∗ inductively as follows. First F j ( ∅ ) = ∅ for all j ∈ J . Then, once the transformations are defined on wordsof a particular length n ≥
0, we let F j ( x i w ) = π j ( x i ) F m ( j,x i ) ( w ) for all j ∈ J ,1 ≤ i ≤ k , and words w of length n . By definition, each F j preserves the lengthof words. Besides, it follows by induction on n that F j is bijective when restrictedto words of length n and that F j preserves having a common beginning of length n for any two words. Therefore each F j is an automorphism of the tree T . Byconstruction, the automorphisms F j , j ∈ J form a self-similar set satisfying theabove system of wreath recursions. Moreover, they provide the only solution tothat system.An infinite path in the tree T is an infinite sequence of vertices v , v , v , . . . together with a sequence of edges e , e . . . such that the endpoints of any e i are v i − and v i . The vertex v is the beginning of the path. Clearly, the path isuniquely determined by the sequence of vertices alone. The boundary of the rootedtree T , denoted ∂ T , is the set of all infinite paths without backtracking that beginat the root. There is a natural one-to-one correspondence between ∂ T and theset X N of infinite words over the alphabet X . Namely, an infinite word x x x . . . corresponds to the path going through the vertices ∅ , x , x x , x x x , . . . . Theset X N is equipped with the product topology and the uniform Bernoulli measure.This allows us to regard the tree boundary ∂ T as a compact topological spacewith a Borel probability measure (called uniform ).Suppose G is a group of automorphisms of the regular rooted tree T . Thenatural action of G on the vertex set X ∗ gives rise to an action on the boundary ∂ T . The latter is continuous and preserves the uniform measure on ∂ T . Proposition 6.2 ([3])
Let G be a countable group of automorphisms of a regularrooted tree T . Then the following conditions are equivalent:(i) the group G acts transitively on each level of the tree; ii) the action of G on the boundary ∂ T of the tree is topologically transitive;(iii) the action of G on ∂ T is minimal;(iv) the action of G on ∂ T is ergodic with respect to the uniform measure;(v) the action of G on ∂ T is uniquely ergodic. Let G be a countable group of automorphisms of a regular rooted tree T . Let α denote the natural action of G on the vertex set of the tree T and β denote theinduced action of G on the boundary ∂ T of the tree. Proposition 6.3
The mapping St β is continuous on a residual (dense G δ ) set. Proof.
For any g ∈ G let Fix β ( g ) denote the set of all points in ∂ T fixed bythe transformation β g . If g ∈ St β ( ξ ) but g / ∈ St oβ ( ξ ), then ξ is a boundary pointof the set Fix β ( g ), and vice versa. Since Fix β ( g ) is a closed set, its boundary isa closed, nowhere dense set. It follows that the set of points ξ ∈ ∂ T such thatSt oβ ( ξ ) = St β ( ξ ) is the intersection of countably many dense open sets (it is densesince ∂ T is a complete metric space). By Lemma 5.4, the latter set consists ofpoints at which the mapping St β is continuous.The mapping St β is Borel due to Lemma 5.4. If St β is injective then, accordingto the descriptive set theory, it also maps Borel sets onto Borel sets (see, e.g., [4]).The following two lemmas show the same can hold under a little weaker condition. Lemma 6.4
Assume that for any points ξ, η ∈ ∂ T either St β ( ξ ) = St β ( η ) or St β ( ξ ) is not contained in St β ( η ) . Then the mapping St β maps any open set, anyclosed set, and any intersection of an open set with a closed one onto Borel sets. Proof.
First let us show that St β maps any closed subset C of the boundary ∂ T onto a Borel subset of Sub( G ). For any positive integer n let C n denote the setof all words of length n in the alphabet X that are beginnings of infinite wordsin C . Further, let W n be the union of sets Sub(St α ( w )) over all words w ∈ C n .Finally, let W be the intersection of the sets W n over all n ≥
1. By Lemma 5.3,the set Sub( H ) is closed in Sub( G ) for any subgroup H ∈ Sub( G ). Hence each W n is closed as the union of finitely many closed sets. Then the intersection W isclosed as well.The stabilizer St β ( ξ ) of an infinite word ξ ∈ ∂ T is a subgroup of the stabilizerSt α ( w ) of a finite word w ∈ X ∗ whenever w is a beginning of ξ . It follows thatSt β ( ξ ) ∈ W for all ξ ∈ C . By construction of the set W , any subgroup of anelement of W is also an element of W . Hence W contains all subgroups of thegroups St β ( ξ ), ξ ∈ C .Conversely, for any subgroup H ∈ W there is a sequence of words w , w , . . . such that w n ∈ C n and H ⊂ St α ( w n ) for n = 1 , , . . . . Since the number of words22f a fixed length is finite, one can find nested infinite sets of indices I ⊃ I ⊃ . . . such that the beginning of length k of the word w n is the same for all n ∈ I k .Choose an increasing sequence of indices n , n , . . . such that n k ∈ I k for all k , andlet w ′ k be the beginning of length k of the word w n k . Then w ′ k ∈ C k as w n k ∈ C n k .Besides, St α ( w n k ) ⊂ St α ( w ′ k ), in particular, the group H is a subgroup of St α ( w ′ k ).By construction, the word w ′ k is a beginning of w ′ m whenever k < m . Thereforeall w ′ k are beginnings of the same infinite word ξ ′ ∈ ∂ T . Since every beginning of ξ ′ coincides with a beginning of some infinite word in C and the set C is closed, itfollows that ξ ′ ∈ C . The stabilizer St β ( ξ ′ ) is the intersection of stabilizers St α ( w ′ k )over all k ≥
1. Hence H is a subgroup of St β ( ξ ′ ).By the above a subgroup H of G belongs to the set W if and only if it isa subgroup of the stabilizer St β ( ξ ) for some ξ ∈ C . The assumption of thelemma implies that stabilizers St β ( ξ ), ξ ∈ C can be distinguished as the maximalsubgroups in the set W . That is, such a stabilizer is an element of W whichis not a proper subgroup of another element of W . For any g ∈ G we define atransformation ψ g of Sub( G ) by ψ g ( H ) = h g i ∨ H , where h g i is a cyclic subgroupof G generated by g . The group ψ g ( H ) is generated by g and all elements ofthe group H . Clearly, a subgroup H ∈ W is not maximal in W if and only if ψ g ( H ) ∈ W for some g / ∈ H . An equivalent condition is that H belongs to the set W ′ g = W ∩ ψ − g ( W ) ∩ U G ( ∅ , { g } ). It follows from Lemma 5.2 that the mapping ψ g is Borel measurable. Therefore W ′ g is a Borel set. Now the image of the set C under the mapping St β is the difference of the closed set W and the union ofBorel sets W ′ g , g ∈ G . Hence this image is a Borel set.Any open set D ⊂ ∂ T is the union of a finite or countable collection of cylinders Z , Z , . . . , which are both open and closed sets. By the above each cylinderis mapped by St β onto a Borel set in Sub( G ). Then the union D is mappedonto the union of images of the cylinders, which is a Borel set as well. Further,for any closed set C ⊂ ∂ T the intersection C ∩ D is the union of closed sets C ∩ Z , C ∩ Z , . . . . Hence it is also mapped by St β onto a Borel set. Lemma 6.5
Under the assumption of Lemma 6.4, if the mapping St β is finite-to-one, i.e., the preimage of any subgroup in Sub( G ) is finite, then it maps Borelsets onto Borel sets. Proof.
Recall that the class B of the Borel sets in ∂ T is the smallest collectionof subsets of ∂ T that contains all closed sets and is closed under taking countableintersections, countable unions, and complements. Let U denote the smallestcollection of subsets of ∂ T that contains all closed sets and is closed under takingcountable intersections of nested sets and countable unions of any sets. Note that U is well defined; it is the intersection of all collections satisfying these conditions.In particular, U ⊂ B . Further, let W denote the collection of all Borel sets in ∂ T mapped onto Borel sets in Sub( G ) by the mapping St β .23or any mapping f : ∂ T →
Sub( G ) and any sequence U , U , . . . of subsetsof ∂ T the image of the union U ∪ U ∪ . . . under f is the union of images f ( U ) , f ( U ) , . . . . On the other hand, the image of the intersection U ∩ U ∩ . . . under f is contained in f ( U ) ∩ f ( U ) ∩ . . . but need not coincide with the latterwhen the mapping f is not one-to-one. The two sets do coincide if f is finite-to-one and U ⊃ U ⊃ . . . . Since the mapping St β is assumed to be finite-to-one,it follows that the collection W is closed under taking countable intersections ofnested sets and countable unions of any sets. By Lemma 6.4, W contains all closedsets. Therefore U ⊂ W .To complete the proof, we are going to show that U = B , which will implythat W = B . Given a set Y ∈ U , let U Y denote the collection of all sets U ∈ U such that the intersection U ∩ Y also belongs to U . For any sequence U , U , . . . of elements of U Y we have (cid:16)[ n ≥ U n (cid:17) ∩ Y = [ n ≥ ( U n ∩ Y ) , (cid:16)\ n ≥ U n (cid:17) ∩ Y = \ n ≥ ( U n ∩ Y ) . Besides, the sets U ∩ Y, U ∩ Y, . . . are nested whenever the sets U , U , . . . arenested. It follows that the class U Y is closed under taking countable intersectionsof nested sets and countable unions of any sets. Consequently, U Y = U whenever U Y contains all closed sets. The latter condition obviously holds if the set Y isitself closed. Notice that for any sets Y, Z ∈ U we have Z ∈ U Y if and only if Y ∈ U Z . Since U Y = U for any closed set Y , it follows that U Z contains all closedsets for any Z ∈ U . Then U Z = U for any Z ∈ U . In other words, the class U is closed under taking finite intersections. Combining finite intersections withcountable intersections of nested sets, we can obtain any countable intersectionof sets from U . Namely, if U , U , . . . are arbitrary elements of U , then theirintersection coincides with the intersection of sets Y n = U ∩ U ∩ · · · ∩ U n , n =1 , , . . . , which are nested: Y ⊃ Y ⊃ . . . . Therefore U is closed under taking anycountable intersections.Let U ′ be the collection of complements in ∂ T of all sets from U . For anysubsets U , U , . . . of ∂ T the complement of their union is the intersection of theircomplements ∂ T \ U , ∂ T \ U , . . . while the complement of their intersection isthe union of their complements. Since the class U is closed under taking countableintersections and countable unions, so is U ′ . Further, any open subset of ∂ T is theunion of at most countably many cylinders, which are closed (as well as open) sets.Therefore U contains all open sets. Then U ′ contains all closed sets. Now it followsthat U ⊂ U ′ . In other words, the class U is closed under taking complements.Thus the collection U is closed under taking any countable intersections andcomplements. This implies that U = B .Let A be a continuous action of a countable group G on a compact metricspace M . Let Ω denote the image of M under the mapping St A .24 emma 6.6 Assume that for any distinct points x, y ∈ M the neighborhood sta-bilizer St oA ( x ) is not contained in the stabilizer St A ( y ) . Then the inverse of St A ,defined on the set Ω , can be extended to a continuous mapping of the closure of Ω onto M . Proof.
Since St oA ( x ) is a subgroup of St A ( x ) for any x ∈ M , the assumption ofthe lemma implies that the mapping St A is one-to-one so that the inverse is welldefined on Ω. To prove that the inverse can be extended to a continuous mappingof the closure of Ω onto M , it is enough to show that any sequence x , x , . . . ofpoints in M is convergent whenever the sequence of stabilizers St A ( x ) , St A ( x ) , . . . converges in Sub( G ). Suppose that St A ( x n ) → H as n → ∞ . Since M is acompact metric space, the sequence x , x , . . . has at least one limit point. ByLemma 5.4(iii), any limit point x satisfies St oA ( x ) ⊂ H ⊂ St A ( x ). In particular,St oA ( x ) ⊂ St A ( y ) for any limit points x and y . Then x = y due to the assumptionof the lemma. It follows that the sequence x , x , . . . is convergent. Let X = { , } be the binary alphabet, X ∗ be the set of finite words over X regarded as the vertex set of a binary rooted tree T , and X N be the set of infinitewords over X regarded as the boundary ∂ T of the tree T .We define the Grigorchuk group G as a self-similar group of automorphismsof the tree T (for alternative definitions, see [2]). The group is generated byfour automorphisms a, b, c, d that, together with the trivial automorphism, forma self-similar set. Consider the following system of wreath recursions: a = (0 1)( e, e ) ,b = ( a, c ) ,c = ( a, d ) ,d = ( e, b ) ,e = ( e, e ) . By Lemma 6.1, this system uniquely defines a self-similar set of automorphismsof the tree T . The automorphism e is clearly the identity (e.g., by Lemma 6.1).It is the unity of the group G . We shall denote the unity by 1 G to avoid confusionwith a letter of the alphabet X . The set S = { a, b, c, d } shall be considered thestandard set of generators for the group G .All 4 generators of the Grigorchuk group are involutions. Indeed, the transfor-mations a , b , c , d , G form a self-similar set satisfying wreath recursions a =(1 G , G ), b = ( a , c ), c = ( a , d ), d = (1 G , b ), and 1 G = (1 G , G ). Then Lemma6.1 implies that a = b = c = d = 1 G . This fact allows us to regard the Schreiergraphs of the group G relative to the generating set S as graphs with undirectededges (as explained in Section 2). 25ince a = 1 G , the automorphisms bcd , cdb , dbc , and 1 G form a self-similar setsatisfying wreath recursions bcd = (1 G , cdb ), cdb = (1 G , dbc ), dbc = (1 G , bcd ), and1 G = (1 G , G ). Lemma 6.1 implies that bcd = cdb = dbc = 1 G . Then bc = bcd = d = d bc = bc . It follows that { G , b, c, d } is a subgroup of G isomorphic to theKlein 4-group.We denote by α the generic action of the group G on vertices of the binaryrooted tree T . The induced action on the boundary ∂ T of the tree is denoted β .For brevity, we write g ( ξ ) instead of β g ( ξ ). The action of the generator a is verysimple: it changes the first letter in every finite or infinite word while keeping theother letters intact. In particular, the empty word is the only word fixed by a .To describe the action of the other generators, we need three observations. Firstof all, b , c , and d fix one-letter words. Secondly, any word beginning with 0 isfixed by d while b and c change only the second letter in such a word. Thirdly,the section mapping g g | induces a cyclic permutation on the set { b, c, d } . Itfollows that a finite or infinite word w is simultaneouly fixed by b , c , and d if itcontains no zeros or the only zero is the last letter. Otherwise two of the threegenerators change the letter following the first zero in w (keeping the other lettersintact) while the third generator fixes w . In the latter case, it is the position k ofthe first zero in w that determines the generator fixing w . Namely, b ( w ) = w if k ≡ c ( w ) = w if k ≡ d ( w ) = w if k ≡ Lemma 7.1
The group G is self-replicating. Proof.
We have to show that for any word w ∈ X ∗ the section mapping g g | w maps the stabilizer St α ( w ) onto the entire group G . Let W be the set of allwords with this property. Clearly, ∅ ∈ W as St α ( ∅ ) = G and g | ∅ = g for all g ∈ G . Suppose w , w ∈ W . Given an arbitrary g ∈ G , there exists g ′ ∈ G such that g ′ ( w ) = w and g ′ | w = g . Further, there exists g ′′ ∈ G such that g ′′ ( w ) = w and g ′′ | w = g ′ . Then g ′′ ( w w ) = g ′′ ( w ) g ′′ | w ( w ) = w w and g ′′ | w w = ( g ′′ | w ) | w = g . Since g is arbitrary, w w ∈ W . That is, the set W isclosed under concatenation.Any automorphism of the tree T either interchanges the vertices 0 and 1 or fixesthem both. Hence the stabilizer St α (0) coincides with St α (1). This stabilizer con-tains the elements b, c, d, aba, aca, ada . The wreath recursions for these elementsare b = ( a, c ), c = ( a, d ), d = (1 G , b ), aba = ( c, a ), aca = ( d, a ), ada = ( b, G ). Itfollows that the images of the group St α (0) under the section mappings g g | and g g | contain the generating set S . As the restrictions of these mappingsto St α (0) are homomorphisms, both images coincide with G . Therefore the words0 and 1 are in the set W . By the above W is closed under concatenation andcontains the empty word. This implies W = X ∗ .The orbits of the actions α and β are very easy to describe.26 emma 7.2 The group G acts transitively on each level of the binary rooted tree T . Any two infinite words in ∂ T are in the same orbit of the action β if and onlyif they differ in only finitely many letters. Proof.
For any infinite word ξ ∈ ∂ T and any generator h ∈ { a, b, c, d } the infiniteword h ( ξ ) differs from ξ in at most one letter. Any g ∈ G can be represented asa product g = h h . . . h k , where each h i is in { a, b, c, d } . It follows that for any ξ ∈ ∂ T the infinite words g ( ξ ) and ξ differ in at most k letters. Thus any twoinfinite words in the same orbit of the action β differ in only finitely many letters.Now we are going to show that for any finite words w , w ∈ X ∗ of the samelength there exists g ∈ G such that g ( w ) = w and g | w = 1 G . Equivalently, g ( w ξ ) = w ξ for all ξ ∈ ∂ T . This will complete the proof of the lemma. Indeed,the claim contains the statement that the group G acts transitively on each levelof the tree T . Moreover, it implies that two infinite words in ∂ T are in the sameorbit of the action β whenever they differ in a finite number of letters.We prove the claim by induction on the length n of the words w and w . Thecase n = 0 is trivial. Here w and w are the empty words so that we take g = 1 G .Now assume that the claim is true for all pairs of words of specific length n ≥ w and w of length n + 1. Let x be the first letter of w and x be the first letter of w . Then w = x u and w = x u , where u and u are words of length n . By the inductive assumption, there exists h ∈ G such that h ( u ξ ) = u ξ for all ξ ∈ ∂ T . Since the group G is self-replicating, there exists g ∈ G such that g ( x η ) = x h ( η ) for all η ∈ ∂ T . In particular, g ( x u ξ ) = x u ξ for all ξ ∈ ∂ T . It remains to take g = g if x = x and g = ag otherwise. Then g ( x u ξ ) = x u ξ for all ξ ∈ ∂ T . Lemma 7.3
Suppose w and w are words in the alphabet { , } such that w is not a beginning of w while w , even with the last two letters deleted, is not abeginning of w . Then there exists g ∈ G that does not fix w while fixing all wordswith beginning w . Proof.
First we consider a special case when w = 100. To satisfy the assumptionof the lemma, the word w has to begin with 0. Then we can take g = d . Indeed,the transformation d fixes all words that begin with 0, which includes all wordswith beginning w . At the same time, d (100) = 1 b (00) = 10 a (0) = 101 = 100.Next we consider a slightly more general case when w is an arbitrary wordof length 3. By Lemma 7.2, the group G acts transitively on the third level ofthe tree T . Therefore h ( w ) = 100 for some h ∈ G . The words h ( w ) and h ( w )satisfy the assumption of the lemma since the words w and w do. By the above, dh ( w ) = h ( w ) while d ( h ( w ) u ) = h ( w ) u for all u ∈ X ∗ . Let g = h − dh . Then g ( w ) = w while g ( w w ) = w w for all w ∈ X ∗ .Finally, consider the general case. Let w be the longest common beginningof the words w and w . Then w = w u and w = w u , where the words u u also satisfy the assumption of the lemma. In particular, u is nonemptyand the length of u is at least 3. We have u = u ′ u ′′ , where u ′ , u ′′ ∈ X ∗ and thelength of u ′ is 3. Since the first letters of the words u and u ′ are distinct, thesewords satisfy the assumption of the lemma. By the above there exists g ∈ G such that g ( u ′ ) = u ′ and g ( u u ) = u u for all u ∈ X ∗ . Since the group G isself-replicating, there exists g ∈ G such that g ( w w ) = w g ( w ) for all w ∈ X ∗ .Then g does not fix the word w u ′ while fixing all words with beginning w . Since w u ′ is a beginning of w , the transformation g does not fix w as well. Lemma 7.4
For any distinct points ξ, η ∈ ∂ T the neighborhood stabilizer St oβ ( ξ ) is not contained in St β ( η ) . Proof.
Let n denote the length of the longest common beginning of the distinctinfinite words ξ and η . Let w be the beginning of ξ of length n + 1 and w be thebeginning of η of length n + 3. It is easy to see that the words w and w satisfythe assumption of Lemma 7.3. Therefore there exists a transformation g ∈ G thatdoes not fix w while fixing all finite words with beginning w . Clearly, the actionof g on ∂ T fixes all infinite words with beginning w . As such infinite words forman open neighborhood of the point ξ , we have g ∈ St oβ ( ξ ). At the same time, g does not fix the infinite word η since it does not fix its beginning w . Hence g / ∈ St β ( η ) so that St oβ ( ξ ) St β ( η ). Lemma 7.5 St oβ ( ξ ) = St β ( ξ ) for any infinite word ξ ∈ ∂ T containing infinitelymany zeros. Proof.
We are going to show that, given an automorphism g ∈ G and an infiniteword ξ ∈ ∂ T with infinitely many zeros, one has g | w = 1 G for a sufficiently longbeginning w of ξ . This claim implies the lemma. Indeed, in the case g ( ξ ) = ξ the action of g fixes all infinite words with beginning w , which form an openneighborhood of ξ .Let R be the set of all g ∈ G such that the claim holds true for g and any ξ ∈ ∂ T with infinitely many zeros. The set R contains the generating set S .Indeed, a | w = 1 G for any nonempty word w ∈ X ∗ and b | w = c | w = d | w = 1 G for anyword w that contains a zero which is not the last letter of w . Now suppose g, h ∈ R and consider an arbitrary ξ ∈ ∂ T with infinitely many zeros. Then h | w = 1 G for asufficiently long beginning w of ξ . Lemma 7.2 implies that the infinite word h ( ξ )also has infinitely many zeros. Since h ( w ) is a beginning of h ( ξ ) and g ∈ R , wehave g | h ( w ) = 1 G provided w is long enough. Since ( gh ) | w = g | h ( w ) h | w , we have( gh ) | w = 1 G provided w is long enough. Thus gh ∈ R . That is, the set R is closedunder multiplication. Since S ⊂ R and all generators are involutions, it followsthat R = G . 28he infinite word ξ = 111 . . . (also denoted 1 ∞ ) is an exceptional point forthe action β . Lemma 7.6
The quotient of St β ( ξ ) by St oβ ( ξ ) is the Klein -group. The cosetrepresentatives are G , b, c, d . Proof.
Recall that H = { G , b, c, d } is a subgroup of G isomorphic to the Klein4-group. Clearly, H ⊂ St β ( ξ ). We are going to show that H ∩ St oβ ( ξ ) = { G } and St β ( ξ ) = St oβ ( ξ ) H , which implies the lemma.For any positive integer n let η n denote the infinite word over the alphabet X that has a single zero in the position n . The sequence η , η , . . . converges to ξ . One observes that any of the generators b , c , and d fixes η n only if n leavesa specific remainder under division by 3 (0 for b , 2 for c , and 1 for d ). It followsthat H ∩ St oβ ( ξ ) = { G } .Now let us show that any g ∈ St β ( ξ ) is contained in the set St oβ ( ξ ) H . Theproof is by strong induction on the length n of g , which is the smallest possiblenumber of factors in an expansion g = s m . . . s s such that each s i ∈ S . The case n = 0 is trivial as 1 G is the only element of length 0. Assume that the claim is truefor all elements of length less than some n > g ∈ St β ( ξ ) of length n . We have g = s n . . . s s , where each s i is a generator from S . Let ξ k = ( s k . . . s s )( ξ ), k = 1 , , . . . , n . If ξ k = ξ for some 0 < k < n , then g = s n . . . s k +1 and g = s k . . . s s both fix ξ . Since the length of g and g isless than n , they belong to St oβ ( ξ ) H by the inductive assumption. As St oβ ( ξ ) H is a group, so does g = g g . If ξ k = ξ for all 0 < k < n , then s i +1 | w i = 1 G forany 0 ≤ i < n and sufficiently long beginning w i of the infinite word ξ i . It followsthat g | w = 1 G for a sufficiently long beginning w of ξ . Thus g ∈ St oβ ( ξ ).Recall that we consider the Schreier graphs of the group G relative to thegenerating set S = { a, b, c, d } as graphs with undirected edges. The Schreiergraphs of all orbits of the action β except O β ( ξ ) are similar. Any vertex is joinedto two other vertices. Moreover, it is joined to one of the neighbors by a singleedge labeled a and to the other neighbor by two edges. Also, there is one loop ateach vertex. Hence the Schreier graph has a linear structure (see Figure 1) and allsuch graphs are isomorphic as graphs with unlabeled edges. The Schreier graphof the orbit of ξ = 1 ∞ is different in that there are three loops labeled b , c , and d at the vertex ξ (see Figure 2).Let F : ∂ T →
Sch( G , S ) be the mapping that assigns to any point on theboundary of the binary rooted tree T its marked Schreier graph under the action β . Using notation of Section 4, F ( ξ ) = Γ ∗ Sch ( G , S ; β, ξ ) for all ξ ∈ ∂ T . Lemma 7.7
The graph F ( ξ ) is an isolated point in the image F ( ∂ T ) . Proof.
Let Γ denote the marked graph with a single vertex and three loopslabeled b , c , and d . Recall that U (Γ , ∅ ) is an open subset of MG consisting of all29 ′ a ad dbc cb ′ a ad dbc cd ′ c ′ Figure 3: Limit graphs ∆ ∗ , ∆ ∗ , ∆ ∗ .graphs in MG that have a subgraph isomorphic to Γ . Hence U (Γ , ∅ ) ∩ Sch( G , S )is an open subset of Sch( G , S ). Given ξ ∈ ∂ T , the graph F ( ξ ) belongs to thatopen subset if and only if a ( ξ ) = ξ and b ( ξ ) = c ( ξ ) = d ( ξ ) = ξ . The latterconditions are satisfied only for ξ = ξ . The lemma follows.It turns out that the image F ( ∂ T ) is not closed in Sch( G , S ). The followingconstruction will help to describe the closure of F ( ∂ T ). Let us take two copiesof the Schreier graph Γ Sch ( G , S ; β, ξ ). We remove two out of three loops at thevertex ξ (loops with the same labels in both copies) and replace them with twoedges joining the two copies. Let c ′ and d ′ denote labels of the removed loopsand b ′ denote the label of the retained loop. Then b ′ , c ′ , d ′ is a permutation of b, c, d . To be rigorous, the new graph has the vertex set O β ( ξ ) × { , } , the setof edges O β ( ξ ) × { , } × S , and the set of labels S . An arbitrary edge ( ξ, i, s )has beginning ( ξ, i ) and label s . The end of this edge is ( s ( ξ ) , i ) unless ξ = ξ and s = c ′ or s = d ′ , in which case the end is ( s ( ξ ) , − i ) = ( ξ , − i ). Thereare three ways to perform the above construction depending on the choice of b ′ .We denote by ∆ , ∆ , and ∆ the graphs obtained when b ′ = b , b ′ = d , and b ′ = c , respectively. Further, for any i ∈ { , , } we denote by ∆ ∗ i a markedgraph obtained from ∆ i by marking the vertex ( ξ ,
0) (see Figure 3).Consider an arbitrary sequence of points η , η , . . . in ∂ T such that η n → ξ as n → ∞ , but η n = ξ . Let z n denote the position of the first zero in the infiniteword η n . Lemma 7.8
The marked Schreier graphs F ( η n ) converge to ∆ ∗ i , ≤ i ≤ , as n → ∞ if z n ≡ i mod 3 for large n . Proof.
For any n ≥ f n : O β ( ξ ) × { , } → O β ( η n ) as follows.Given ξ ∈ O β ( ξ ) and x ∈ { , } , let f n ( ξ, x ) be an infinite word obtained from η n after replacing the first z n − z n − ξ and adding x mod 2 to the ( z n + 1)-th letter. Clearly, f n ( ξ ,
0) = η n . Let i n be the remainder30 ad dbc ca ad dbc caa dd bcc aa dd bcc dd bbc c Figure 4: The Schreier coset graph of St oβ ( ξ ).of z n under division by 3. One can check that the restriction of f n to the vertexset of the closed ball B ∆ ∗ in (( ξ , , N ) is an isomorphism of this ball with theclosed ball B F ( η n ) ( η n , N ) whenever N ≤ z n − . Therefore δ ( F ( η n ) , ∆ ∗ i n ) → n → ∞ .One consequence of Lemma 7.8 is that the graphs ∆ , ∆ , and ∆ are Schreiergraphs of the group G . By construction, each of these graphs admits a nontrivialautomorphism, which interchanges vertices corresponding to the same vertex ofΓ Sch ( G , S ; β, ξ ). This property distinguishes ∆ , ∆ , and ∆ from the Schreiergraphs of orbits of the action β . Lemma 7.9
The Schreier graphs Γ Sch ( G , S ; β, ξ ) , ξ ∈ ∂ T do not admit nontrivialautomorphisms. The graphs ∆ , ∆ , and ∆ admit only one nontrivial automor-phism. Proof.
It follows from Proposition 4.4 and Lemma 7.4 that marked Schreiergraphs F ( ξ ) and F ( η ) are isomorphic only if ξ = η . Therefore the Schreier graphsΓ Sch ( G , S ; β, ξ ), ξ ∈ ∂ T admit no nontrivial automorphisms.The graphs ∆ , ∆ , and ∆ have linear structure. Namely, one can label theirvertices by v j , j ∈ Z so that each v j is adjacent only to v j − and v j +1 . If f isan automorphism of such a graph, then either f ( v j ) = v n − j for some n ∈ Z andall j ∈ Z or f ( v j ) = v n + j for some n ∈ Z and all j ∈ Z . Assume that some∆ i has more than one nontrivial automorphism. Then we can choose f above sothat the latter option holds with n = 0. Take any path in ∆ i that begins at v and ends at v n and let w be the code word of that path. Since f m ( v ) = v mn and f m ( v n ) = v ( m +1) n for any integer m , the path in ∆ i with beginning v mn andcode word w ends at v ( m +1) n . It follows that for any integer m > v and code word w m ends at v mn . In particular, this path is not closed.However every element of the Grigorchuk group G is of finite order (see [2]) sothat for some m > w m equals 1 G when regarded as a productin G . This conradicts with Proposition 4.1. Thus the graph ∆ i admits only onenontrivial automorphism. 31 emma 7.10 The Schreier graph Γ Sch ( G , S ; β, ξ ) is a double quotient of each ofthe graphs ∆ , ∆ , and ∆ . On the other hand, each of the graphs ∆ , ∆ , and ∆ is a double quotient of the Schreier coset graph Γ coset ( G , S ; St oβ ( ξ )) . Proof.
The Schreier coset graph of the subgroup St oβ ( ξ ) is shown in Figure4. In view of Lemmas 7.6 and 7.9, the automorphism group of this graph is theKlein 4-group. The quotient of the graph by the entire automorphism group isthe Schreier graph of the orbit of ξ . The quotients by subgroups of order 2 arethe graphs ∆ , ∆ , and ∆ .Now it remains to collect all parts in Theorems 1.1 and 1.2. Proof of Theorem 1.1.
We are concerned with the mapping F : ∂ T →
Sch( G , S ) given by F ( ξ ) = Γ ∗ Sch ( G , S ; β, ξ ). Let us also consider a mapping ψ : ∂ T →
Sub( G ) given by ψ ( ξ ) = St β ( ξ ) and a mapping f : Sub( G ) → Sch( G , S )given by f ( H ) = Γ ∗ coset ( G , S ; H ). By Proposition 4.4, F ( ξ ) = f ( ψ ( ξ )) for all ξ ∈ ∂ T . By Proposition 5.5, f is a homeomorphism. Lemma 7.4 implies thatthe mapping ψ is injective. It is Borel measurable due to Lemma 5.4. Also, ψ is continuous at a point ξ ∈ ∂ T if and only if St oβ ( ξ ) = St β ( ξ ). Lemmas 7.5and 7.6 imply that the latter condition fails only if the infinite word ξ containsonly finitely many zeros. According to Lemma 7.2, an equivalent condition isthat ξ is in the orbit of ξ = 1 ∞ under the action β . Since the mapping F is f postcomposed with a homeomorphism, it is also injective, Borel measurable, andcontinuous everywhere except the orbit of ξ .By Lemma 7.7, the graph F ( ξ ) is an isolated point of the image F ( ∂ T ).Since F ( g ( ξ )) = A g ( F ( ξ )) for any ξ ∈ ∂ T and g ∈ G and since the action A iscontinuous (see Proposition 4.2), the graph F ( g ( ξ )) is an isolated point of F ( ∂ T )for all g ∈ G . On the other hand, if ξ ∈ ∂ T is not in the orbit of ξ , then thegraph F ( ξ ) is not an isolated point of F ( ∂ T ) as the mapping F is injective andcontinuous at ξ .It follows from Lemma 7.9 that the image F ( ∂ T ) and the orbits O A (∆ ∗ i ), i ∈ { , , } are disjoint sets. Note that the orbit O A (∆ ∗ i ) consists of markedgraphs obtained from the graph ∆ i by marking an arbitrary vertex. Lemma 7.8implies the union of those 4 sets is the closure of F ( ∂ T ).Finally, the statement (v) of Theorem 1.1 follows from Lemma 7.10. Proof of Theorem 1.2.
Lemma 6.6 combined with Lemma 7.4 implies thatthe action of G on the closure of F ( ∂ T ) is a continuous extension of the action β . The extension is one-to-one everywhere except for the orbit O β ( ξ ) where it isfour-to-one. Namely, for any g ∈ G the point g ( ξ ) is covered by 4 graphs F ( g ( ξ )), A g (∆ ∗ ), A g (∆ ∗ ), and A g (∆ ∗ ). According to Theorem 1.1, the graph F ( g ( ξ )) isan isolated point of the closure of F ( ∂ T ). When we restrict our attention to theset Ω of non-isolated points of the closure, we still have a continuous extension ofthe action β , but it is three-to-one on the orbit O β ( ξ ).32y Lemma 7.2, the group G acts transitively on each level of the binary rootedtree T . Then Proposition 6.2 implies that the action β is minimal and uniquelyergodic, the only invariant Borel probability measure being the uniform measureon ∂ T . Since the action of G on the set Ω is a continuous extension of the action β that is one-to-one except for a countable set and since this action has no finiteorbits, it follows that the action is minimal, uniquely ergodic, and isomorphic to β as the action with an invariant measure. References [1] R. Grigorchuk,
On Burnside’s problem on periodic groups . Funct. Anal. Appl. (1980), 41–43.[2] R. Grigorchuk, Solved and unsolved problems around one group . L. Bartholdi(ed.) et al., Infinite groups: geometric, combinatorial and dynamical aspects.Basel, Birkh¨auser.
Progress in Mathematics , 117–218 (2005).[3] R. Grigorchuk,
Some topics in the dynamics of group actions on rooted trees .Proc. Steklov Inst. Math. (2011), no. 1, 64–175.[4] A. S. Kechris,
Classical descriptive set theory . Berlin, Springer.
GraduateTexts in Mathematics
156 (1995).[5] A. Vershik,
Nonfree actions of countable groups and their characters . J. Math.Sci., NY (2011), no. 1, 1–6.
Department of MathematicsMailstop 3368Texas A&M UniversityCollege Station, TX 77843-3368
E-mail: [email protected]@math.tamu.edu