aa r X i v : . [ m a t h . R A ] J a n GROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS
VOLODYMYR NEKRASHEVYCH
Abstract.
We study growth and complexity of ´etale groupoids in relation togrowth of their convolution algebras. As an application, we construct simplefinitely generated algebras of arbitrary Gelfand-Kirillov dimension ≥ Contents
1. Introduction 12. ´Etale groupoids 53. Compactly generated groupoids 63.1. Cayley graphs and their growth 63.2. Complexity 83.3. Examples 84. Convolution algebras 114.1. Definitions 114.2. Growth of k [ G ] 134.3. Finite generation 144.4. Examples 144.5. Modules k G x Introduction
Topological groupoids are extensively used in dynamics, topology, non-commutativegeometry, and C ∗ -algebras, see [13, 29, 30]. With recent results on topological fullgroups (see [22, 15, 16]) new applications of groupoids to group theory were dis-covered.Our paper studies growth and complexity for ´etale groupoids with applications tothe theory of growth and Gelfand-Kirillov dimension of algebras. We give examplesof groupoids whose convolution algebras (over an arbitrary field) have prescribedgrowth. In particular, we give first examples of simple algebras of quadratic growthover finite fields and simple algebras of Gelfand-Kirillov dimension 2 that do nothave quadratic growth.A groupoid G is the set of isomorphisms of a small category, i.e., a set G withpartially defined multiplication and everywhere defined operation of taking inversesatisfying the following axioms:(1) If the products ab and bc are defined, then ( ab ) c and a ( bc ) are defined andare equal. (2) The products a − a and bb − are always defined and satisfy abb − = a and a − ab = b whenever the product ab is defined.It follows from the axioms that ( a − ) − = a and that a product ab is defined ifand only if bb − = a − a . The elements of the form aa − are called units of thegroupoid. We call o ( g ) = g − g and t ( g ) = gg − the origin and the target of theelement g ∈ G .A topological groupoid is a groupoid together with topology such that multipli-cation and taking inverse are continuous. It is called ´etale if every element has abasis of neighborhoods consisting of bisections , i.e., sets F such that o : F −→ o ( F )and t : F −→ t ( F ) are homeomorphisms.For example, if G is a discrete group acting (from the left) by homeomorphismson a topological space X , then the topological space G × X has a natural structureof an ´etale groupoid with respect to the multiplication( g , g ( x )) · ( g , x ) = ( g g , x ) . In some sense ´etale groupoids are generalization of actions of discrete groups ontopological spaces.We consider two growth functions for an ´etale groupoid G with compact totallydisconnected space of units. The first one is the most straightforward and classical:growth of fibers of the origin map. If S is an open compact generating set of G then,for a given unit x , we can consider the growth function γ S ( r, x ) equal to the numberof groupoid elements with origin in x that can be expressed as a product of at most n elements of S ∪ S − . This notion of growth of a groupoid has appeared in manysituations, especially in amenability theory for topological groupoids, see [17, 1].See also Theorem 3.1 of our paper, where for a class of groupoids we show howsub-exponential growth implies absence of free subgroups in the topological fullgroup of the groupoid.This notion of growth does not capture full complexity of a groupoid preciselybecause it is “fiberwise”. Therefore, we introduce the second growth function:complexity of the groupoid. Let S be a finite covering by open bisections of anopen compact generating set S of G . For a given natural number r and units x, y ∈ G (0) we write x ∼ r y if for any two products S S . . . S n and R R . . . R m of elements of S ∪ S − such that n, m ≤ r we have S S . . . S n x = R R . . . R m x ifand only if S S . . . S n y = R R . . . R m y . In other words, x ∼ r y if and only if ballsof radius r with centers in x and y in the natural S -labeled Cayley graphs of G areisomorphic. Then the complexity function δ ( r, S ) is the number of ∼ r -equivalenceclasses of points of G (0) .This notion of complexity (called in this case factor complexity , or subword com-plexity ) is well known and studied for groupoids of the action of shifts on closedshift-invariant subsets of X Z , where X is a finite alphabet. There is an extensiveliterature on it, see [8, 10] An interesting result from the group-theoretic point ofview is a theorem of N. Matte Bon [21] stating that if complexity of a subshift isstrictly sub-quadratic, then the topological full group of the corresponding groupoidis Liouville. Here the topological full group of an ´etale groupoid G is the group ofall G -bisections A such that o ( A ) = t ( A ) = G (0) . ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 3
It seems that complexity of groupoids in more general ´etale groupoids has notbeen well studied yet. It would be interesting to understand how complexity func-tion (together with the growth of fibers) is related with the properties of the topo-logical full group of an ´etale groupoid. For example, it would be interesting toknow if there exists a non-amenable (e.g., free) group acting faithfully on a compacttopological space so that the corresponding groupoid of germs has sub-exponentialgrowth and sub-exponential complexity functions.We relate growth and complexity of groupoids with growth of algebras naturallyassociated with them. Suppose that A is a finitely generated algebra with a unitover a field k . Let V be the k -linear span of a finite generating set containing theunit. Denote by V n the linear span of all products a a . . . a n for a i ∈ V . Then A = S ∞ n =1 V n . Growth of A is the function γ ( n ) = dim V n . It is easy to see that if γ , γ are growth functions defined using different finitegenerating sets, then there exists C > γ ( n ) ≤ γ ( Cn ) and γ ( n ) ≤ γ ( Cn ). Gelfand-Kirillov dimension of A is defined as lim sup n →∞ log dim V n log n , which in-formally is the degree of polynomial growth of the algebra. If A is not finitelygenerated, then its Gelfand-Kirillov dimension is defined as the supremum of theGelfand-Kirillov dimensions of all its sub-algebras. See the monograph [19] for asurvey of results on growth of algebras and their Gelfand-Kirillov dimension.It is known, see [34] and [19, Theorem 2.9], that Gelfand-Kirillov dimensioncan be any number in the set { , } ∪ [2 , ∞ ]. The values in the interval (1 , d ∈ [2 , ∞ ],see [33], but it seems that no examples of simple algebras of arbitrary Gelfand-Kirillov dimension over finite fields were known so far.A naturally defined convolution algebra k [ G ] over arbitrary field k is associatedwith every ´etale groupoid G with totally disconnected space of units. If the groupoid G is Hausdorff, then k [ G ] is the convolution algebra of all continuous functions f : G −→ k with compact support, where k is taken with the discrete topology.Here convolution f · f of two functions is the function given by the formula f ( g ) = X g g = g f ( g ) f ( g ) . In the non-Hausdorff case we follow A. Connes [9] and B. Steinberg [32], anddefine k [ G ] as the linear span of the functions that are continuous on open compactsubsets of G . Equivalently, k [ G ] is the linear span of the characteristic functions ofopen compact G -bisections.Note that the set B ( G ) of all open compact G -bisections (together with theempty one) is a semigroup. The algebra k [ G ] is isomorphic to the quotient of thesemigroup algebra of B ( G ) by the ideal generated by the relations F − ( F + F )for all triples F, F , F ∈ B ( G ) such that F = F ∪ F and F ∩ F = ∅ .We prove the following relation between growth of groupoids and growth of theirconvolution algebras. Theorem 1.1.
Let G be an ´etale groupoid with compact totally disconnected spaceof units. Let S be a finite set of open compact G -bisections such that S = S S is a VOLODYMYR NEKRASHEVYCH generating set of G . Let V ⊂ k [ G ] be the linear span of the characteristic functionsof elements of S . Then dim V n ≤ γ ( r, S ) δ ( r, S ) , where γ ( r, S ) = max x ∈ G (0) γ S ( r, x ) . We say that a groupoid G is minimal if every G -orbit is dense in G (0) . We saythat G is essentially principal if the set of points x with trivial isotropy group isdense in G (0) . Here the isotropy group of a point x is the set { g ∈ G : o ( g ) = t ( g ) = x } . It is known, see [7], that for a Hausdorff minimal essentially principal groupoid G with compact totally disconnected set of units the algebra k [ G ] is simple. Wegive a proof of this fact for completeness in Proposition 4.1.We give in Proposition 4.4 a condition (related to the classical notion of an expansive dynamical system ) ensuring that k [ G ] is finitely generated.Fibers of the origin map provide us with naturally defined k [ G ]-modules. Namely,for a given unit x ∈ G (0) consider the vector space k G x of functions φ : G x −→ k with finite support, where G x = o − ( x ) is the set of elements of the groupoid G with origin in x . Then convolution f · φ for any f ∈ k [ G ] and φ ∈ k G x is an elementof k G x , and hence k G x is a left k [ G ]-module.It is easy to prove that if the isotropy group of x is trivial, then k G x is simpleand that growth of k G x is bounded by γ S ( x, r ), see Proposition 4.8.As an example of applications of these results, we consider the following family ofalgebras. Let X be a finite alphabet, and let w : X −→ Z be a bi-infinite sequenceof elements of X . Denote by D x , for x ∈ X the diagonal matrix ( a i,j ) i,j ∈ Z given by a i,j = (cid:26) i = j and w ( i ) = x ,0 otherwise.Let T be the matrix ( t i,j ) i,j ∈ Z of the shift given by t i,j = (cid:26) i = j + 1,0 otherwise.Fix a field k , and let A w be the k -algebra generated by the matrices D x , for x ∈ X ,by T , and its transpose T ⊤ .We say that w is minimal if for every finite subword ( w ( n ) , w ( n +1) , . . . , w ( n + k ))there exists R > i ∈ Z there exists j ∈ Z such that | i − j | ≤ R and ( w ( j ) , w ( j + 1) , . . . , w ( j + k )) = ( w ( n ) , w ( n + 1) , . . . , w ( n + k )). We say that w is non-periodic if there does not exist p = 0 such that w ( n + p ) = w ( n ) forall n ∈ Z . Complexity function p w ( n ) of the sequence w ∈ X Z is the number ofdifferent subwords ( w ( i ) , w ( i + 1) , . . . , w ( i + n − n in w .The following theorem is a corollary of the results of our paper, see Subsec-tion 4.4.1 and Example 4.6. Theorem 1.2.
Suppose that w ∈ X Z is minimal and non-periodic. Then thealgebra A w is simple, and its growth γ ( n ) satisfies C − n · p w ( C − n ) ≤ Cn · p w ( Cn ) for some C > . We can apply now results on complexity of sequences to construct simple algebrasof various growths. For example, if w is Sturmian , then p w ( n ) = n + 1, and hence A w has quadratic growth. For different Toeplitz sequences we can obtain simple
ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 5 algebras of arbitrary Gelfand-Kirillov dimension d ≥
2, or simple algebras of growth n log n , etc., see Subsection 4.4.1.Another class of examples of groupoids considered in our paper are groupoidsassociated with groups acting on a rooted tree. If G acts by automorphisms ona locally finite rooted tree T , then it acts by homeomorphisms on the boundary ∂T . One can consider the groupoid of germs G of the action. Convolution algebras k [ G ] are related to the thinned algebras studied in [31, 2]. In the case when G is a contracting self-similar group , Theorem 1.1 implies a result of L. Bartholdifrom [2] giving an estimate of Gelfand-Kirillov dimension for the thinned algebrasof contracting self-similar groups.2. ´Etale groupoids A groupoid is a small category of isomorphisms (more precisely, the set of itsmorphisms). For a groupoid G , we denote by G (2) the set of composable pairs, i.e.,the set of pairs ( g , g ) ∈ G × G such that the product g g is defined. We denoteby G (0) the set of units of G , i.e., the set of identical isomorphisms. We also denoteby o , t : G −→ G (0) the origin and target maps given by o ( g ) = g − g, t ( g ) = gg − . We interpret then an element g ∈ G as an arrow from o ( g ) to t ( g ). The product g g is defined if and only if t ( g ) = o ( g ).For x ∈ G (0) , denote G x = { g ∈ G : o ( g ) = x } , G x = { g ∈ G : t ( g ) = x } . The set G x ∩ G x is called the isotropy group of x . A groupoid is said to be principal (or an equivalence relation) if the isotropy group of every point is trivial.Two units x, y ∈ G (0) belong to one orbit if there exists g ∈ G such that o ( g ) = x and t ( g ) = y . It is easy to see that belonging to one orbit is an equivalence relation.A topological groupoid is a groupoid G with a topology on it such that multi-plication G (2) −→ G and taking inverse G −→ G are continuous maps. We donot require that G is Hausdorff, though we assume that the space of units G (0) ismetrizable and locally compact.A G -bisection is a subset F ⊂ G such that the maps o : F −→ o ( F ) and t : F −→ t ( F ) are homeomorphisms. Definition 2.1.
A topological groupoid G is ´etale if the set of all open G -bisectionsis a basis of the topology of G .Let G be an ´etale groupoid. It is easy to see that product of two open bisectionsis an open bisection. It follows that for every bisection F the sets o ( F ) = F − F and t ( F ) = F F − are open, which in turn implies that G (0) is an open subset of G .If G is not Hausdorff, then there exist g , g ∈ G that do not have disjointbisections. Since G (0) is Hausdorff, this implies that o ( g ) = o ( g ) and t ( g ) = t ( g ).It follows that the unit x = o ( g ) and the element g − g of the isotropy group of x do not have disjoint open neighborhoods. In particular, it means that principal´etale groupoids are always Hausdorff, and that an ´etale groupoid is Hausdorff ifand only if G (0) is a closed subset of G . Example 2.1.
Let G be a discrete group acting by homeomorphisms on a space X . Then the space G × X has a natural groupoid structure with given by the VOLODYMYR NEKRASHEVYCH multiplication ( g , g ( x ))( g , x ) = ( g g , x ) . This is an ´etale groupoid, since every set { g }×X is an open bisection. The groupoid G × X is called the groupoid of the action , and is denoted G ⋉ X .Our main class of groupoids will be naturally defined quotients of the groupoidsof actions, called groupoids of germs. Example 2.2.
Let G and X be as in the previous example. A germ is an equiva-lence class of a pair ( g, x ) ∈ G × X where ( g , x ) and ( g , x ) are equivalent if thereexists a neighborhood U of x such that the maps g : U −→ X and g : U −→ X coincide. The set of germs is also an ´etale groupoid with the same multiplicationrule as in the previous example. We call it groupoid of germs of the action .The spaces of units in both groupoids are naturally identified with the space X (namely, we identify the pair or the germ (1 , x ) with x ). The groupoid of the actionis Hausdorff if X is Hausdorff, since it is homeomorphic to G × X . The groupoid ofgerms, on the other hand, is frequently non-Hausdorff, even for a Hausdorff space X .If every germ of every non-trivial element of G is not a unit (i.e., not equal to agerm of the identical homeomorphism), then the groupoid of the action coincideswith the groupoid of germs.Many interesting examples of ´etale groupoids appear in dynamics and topology,see [13, 6, 27]. 3. Compactly generated groupoids
For the rest of the paper, G is an ´etale groupoid such that G (0) is a compacttotally disconnected metrizable space. Note that then there exists a basis of topol-ogy of G consisting of open compact G -bisections. Note that we allow compactnon-closed and compact non-Hausdorff sets, since G in general is not Hausdorff.However, if F is an open compact bisection, then o ( F ) and t ( F ) are clopen (i.e.,closed and open) and F is Hausdorff.3.1. Cayley graphs and their growth.Definition 3.1.
A groupoid G with compact totally disconnected unit space is compactly generated if there exists a open compact subset S ⊂ G such that G = S n ≥ ( S ∪ S − ) n . The set S is called the generating set of G .This definition is equivalent (for ´etale groupoids with compact totally discon-nected unit space) to the definition of [14]. Example 3.1.
Let G be a group acting on a Cantor set X . If S is a finite generatingset of G , then S × X is an open compact generating set of the groupoid G ⋉ X .The set all of germs of elements of S is an open compact generating set of thegroupoid of germs of the action. Thus, both groupoids are compactly generated if G is finitely generated.Let S be an open compact generating set of G . Let x ∈ G (0) . The Cayley graph G ( x, S ) is the directed graph with the set of vertices G x in which we have an arrowfrom g to g whenever there exists s ∈ S such that g = sg . ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 7
We will often consider the graph G ( x, S ) as a rooted graph with root x . Morphism φ : Γ −→ Γ of rooted graphs is a morphism of graphs that maps the root of Γ to the root of Γ .Note that since S can be covered by a finite set of bisections, the degrees ofvertices of the graphs G ( x, S ) are uniformly bounded. Example 3.2.
Let G be a finitely generated group acting on a totally disconnectedcompact space X . Let S be a finite generating set of G , and let S × X be thecorresponding generating set of the groupoid of action G ⋉ X . The Cayley graphs G ⋉ X ( x, S × X ) coincide then with the Cayley graphs of G (with respect to thegenerating set S ).The groupoid of germs G will have smaller Cayley graphs. Let S ′ ⊂ G be theset of all germs of elements of S . Denote, for x ∈ X , by G ( x ) the subgroup of G consisting of all elements g ∈ G such that there exists a neighborhood U of x suchthat g fixes every point of U . Then G ( x, S ′ ) is isomorphic to the Schreier graph of G modulo G ( x ) . Its vertices are the cosets hG ( x ) , and a coset h G ( x ) is connected byan arrow with h G ( x ) if there exists a generator s ∈ S such that sh G ( x ) = h G ( x ) .Cayley graphs G ( x, S ) are closely related to the orbital graphs , which are definedas graphs Γ( x, S ) with the set of vertices equal to the orbit of x , in which a vertex x is connected by an arrow to a vertex x if there exists g ∈ S such that o ( s ) = x and t ( s ) = x . Orbital graph Γ( x, S ) is the quotient on the Cayley graph G ( x, S )by the natural right action of the isotropy group of x . In particular, orbital graphand the Cayley graph coincide if the isotropy group of x is trivial.Denote by B S ( x, n ) the ball of radius n with center x in the graph G ( x, S ) seenas a rooted graph (with root x ). Let γ S ( x, n ) = | B S ( x, n ) | , γ ( n, S ) = max x ∈ G (0) γ S ( x, n ) . If S and S are two open compact generating sets of G , then there exists m suchthat S ⊂ S ≤ k ≤ m ( S ∪ S − ) k and S ⊂ S ≤ k ≤ m ( S ∪ S − ) k . Then γ S ( x, mn ) ≥ γ S ( x, n ) and γ S ( x, mn ) ≥ γ S ( x, n ) for all n . It also follows that γ ( mn, S ) ≥ γ ( n, S ) and γ ( mn, S ) ≥ γ ( n, S ) for all n . In other words, the growth rate of thefunctions γ S ( x, n ) and γ ( n, S ) do not depend on the choice of S , if S is a generatingset.Condition of polynomial growth of Cayley graphs of groupoids (or, in the measure-theoretic category, of connected components of graphings of equivalence relations)appear in the study of amenability of groupoids, see [17, 1].Here is another example of applications of the notion of growth of groupoids. Theorem 3.1.
Let G be a finitely generated subgroup of the automorphism groupof a locally finite rooted tree T . Consider the groupoid of germs G of the action of G on the boundary ∂T of the tree. If γ S ( x, n ) has sub-exponential growth for every x ∈ ∂T , then G has no free subgroups.Proof. By [26, Theorem 3.3], if G has a free subgroup, then either there exists afree subgroup F and a point x ∈ ∂T such that the stabilizer of x in F is trivial, orthere exists a free subgroup F and a point x ∈ ∂T such that x is fixed by F andevery non-trivial element g of F the germ ( g, x ) is non-trivial. But both conditionsimply that the Cayley graph G ( x, S ) has exponential growth. (cid:3) VOLODYMYR NEKRASHEVYCH
Complexity.
Let S be a finite set of open compact G -bisections such that S = S S is a generating set. Note that every compact subset of G can be coveredby a finite number of open compact G -bisections.Denote by G ( x, S ) the oriented labeled graph with the set of vertices G x in whichwe have an arrow from g to g labeled by A ∈ S if there exists s ∈ A such that g = sg .The graph G ( x, S ) basically coincides with G ( x, S ) for S = S S . The onlydifference is the labeling and that some arrows of G ( x, S ) become multiple arrowsin G ( x, S ). In particular, the metrics induced on the sets of vertices of graphs G ( x, S ) and G ( x, S ) coincide.We denote by B S ( x, r ) or just by B ( x, r ) the ball of radius r with center in x ,seen as a rooted oriented labeled graph. We write x ∼ r y if B S ( x, r ) and B S ( y, r )are isomorphic. Definition 3.2.
Complexity of S is the function δ ( r, S ) equal to the number of ∼ r -equivalence classes.It is easy to see that δ ( r, S ) is finite for every r and S .3.3. Examples.
Shifts.
Let X be a finite alphabet containing more than one letter. Considerthe space X Z of all bi-infinite words over X , i.e., maps w : Z −→ X . Denote by s : X Z −→ X Z the shift map given by the rule s ( w )( i ) = w ( i + 1). The space X Z is homeomorphic to the Cantor set with respect to the direct product topology(where X is discrete).A sub-shift is a closed s -invariant subset X ⊂ X Z . We always assume that X has no isolated points. For a sub-shift X , consider the groupoid S of the germsof the action of Z on X generated by the shift. It is easy to see that all germs ofnon-zero powers of the shift are non-trivial, hence the groupoid S coincides withthe groupoid Z ⋉ X of the action. As usual, we will identify X with the space ofunits S (0) . The set S = { ( s, x ) : x ∈ X } is an open compact generating set of S .The Cayley graphs S ( w, S ) are isomorphic to the Cayley graph of Z with respectto the generating set { } .If X is aperiodic , i.e., if it does not contain periodic sequences, then S is principal.Note that S is always Hausdorff.For x ∈ X , denote by S x set of germs of the restriction of s onto the cylindricalset { w ∈ X : w (0) = x } . Then S = { S x } x ∈ X is a covering of S by disjoint clopensubsets of S . Then for every w ∈ X , the Cayley graph S ( w, S ) basically repeats w :its set of vertices is the set of germs ( s n , w ), n ∈ Z ; for every n we have an arrowfrom ( s n , w ) to ( s n +1 , w ) labeled by S w ( n ) .In particular, we have δ ( n, S ) = p X (2 n ) , where p X ( k ) denotes the number of words of length k that appear as subwords ofelements of X .Complexity p X ( n ) of subshifts is a well studied subject, see [20, 10, 8] and ref-erences therein.Two classes of subshifts are especially interesting for us: Sturmian and Toeplitzsubshifts. ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 9
Let θ ∈ (0 ,
1) be an irrational number, and consider the rotation R θ : x x + θ (mod 1)of the circle R / Z . For a number x ∈ R / Z not belonging to the R θ -orbit of 0,consider the θ -itinerary I θ,x ∈ { , } Z given by I θ,x ( n ) = (cid:26) x + nθ ∈ (0 , θ ) (mod 1) , x + nθ ∈ ( θ,
1) (mod 1) . In other words, I θ,x describes the itinerary of x ∈ R / Z under the rotation R θ withrespect to the partition [0 , θ ) , [ θ,
1) of the circle R / Z . If x belongs to the orbit of0, then we define two itineraries I θ,x +0 = lim t → x +0 I θ,t and I θ,x − = lim t → x − I θ,t ,where t in the limits belongs to the complement of the orbit of 0.The set X θ of all itineraries is a subshift of { , } Z called the Sturmian subshift associated with θ . Informally, the space X θ is obtained from the circle R / Z by“cutting” it along the R θ -orbit of 0, i.e., by replacing each point x = nθ by twocopies x + 0 and x −
0. A basis of topology of X θ is the set of arcs of the form[ nθ + 0 , mθ − R θ .Complexity p X θ ( n ) of the Sturmian subshift is equal to the number of all possible R θ -itineraries of length n . Consider the set { R − kθ ( θ ) } k =0 , ,...,n . It separates thecircle R / Z into n + 1 arcs such that two points x, y have equal length n segments { , . . . , n − } −→ { , } of their itineraries I θ,x , I θ,y if and only if they belong toone arc. It follows that p X θ ( n ) = n + 1. The subshifts of the form X θ and theirelements are called Sturmian subshifts and
Sturmian sequences.A sequence w : X −→ Z is a Toeplitz sequence if it is not periodic and for every n ∈ Z there exists p ∈ N such that w ( n + kp ) = w ( n ) for all k ∈ Z . Complexity ofToeplitz sequences is well studied.It is known, for example, (see [20, Proposition 4.79]) that for any 1 ≤ α ≤ β ≤ ∞ there exists a Toeplitz subshift X (i.e., closure of the shift orbit of a Toeplitzsequence) such thatlim inf n →∞ ln p X ( n )ln n = α, lim sup n →∞ ln p X ( n )ln n = β. The following theorem is proved by M. Koskas in [18].
Theorem 3.2.
For every rational number p/q > and every positive increasingdifferentiable function f ( x ) satisfying f ( n ) = o ( n α ) for all α > , and nf ′ ( n ) = o ( n α ) for all α > , there exists a Toeplitz subshift X and two constants c , c > satisfying c f ( n ) n p/q ≤ p X ( n ) ≤ c f ( n ) n p/q for all n ∈ N . Groups acting on rooted trees.
Let X be a finite alphabet, | X | ≥
2. Denoteby X ∗ the set of all finite words (including the empty word ∅ ). We consider X ∗ asa rooted tree with root ∅ in which every word v ∈ X ∗ is connected to the words ofthe form vx for all x ∈ X . The boundary of the tree is naturally identified with thespace X N of all one-sided sequences x x x . . . . Every automorphism of the rootedtree X ∗ naturally induces a homeomorphism of X N .Let g be an automorphism of the tree X ∗ . For every v ∈ X ∗ there exists aunique automorphism g | v of the tree X ∗ such that g ( vw ) = g ( v ) g | v ( w ) for all w ∈ X ∗ . We say that a group G of automorphisms of X ∗ is self-similar if g | v ∈ G for every g ∈ G and v ∈ X ∗ . For every v ∈ X ∗ and w ∈ X N the germ( g, vw ) depends only on the quadruple ( v, g ( v ) , g | v , w ). Example 3.3.
Consider the automorphism a of the binary tree { , } ∗ defined bythe recursive rules a (0 w ) = 1 w, a (1 w ) = 0 a ( w ) . It is called the adding machine , or odometer . The cyclic group generated by a isself-similar. Example 3.4.
Consider the automorphisms of { , } ∗ defined by the recursiverules a (0 w ) = 1 w, a (1 w ) = 0 w and b (0 w ) = 0 a ( w ) , b (1 w ) = 1 c ( w ) ,c (0 w ) = 0 a ( w ) , c (1 w ) = 1 d ( w ) ,d (0 w ) = 0 w, d (1 w ) = 1 b ( w ) . The group generated by a, b, c, d is the
Grigorchuk group , see [12].For more examples of self-similar groups and their applications, see [24].Let G be a finitely generated self-similar group, and let l ( g ) denote the lengthof an element g ∈ G with respect to some fixed finite generating set of G . The contraction coefficient of the group G is the number λ = lim sup n →∞ lim sup g ∈ G,l ( g ) →∞ max v ∈ X n l ( g | v ) l ( g ) . The group is said to be contracting if λ < Z and the Grigorchuk group are bothcontracting with contraction coefficient λ = 1 / Proposition 3.3.
Let G be a contracting self-similar group acting on the tree X ∗ ,and let λ be the contraction coefficient. Consider the groupoid of germs G of theaction of G on X N , let S be a finite generating set of G , and let S be the set of G -bisets of the form { ( s, w ) : w ∈ X N } for s ∈ S . Then we have lim sup n →∞ log γ ( n, S )log n ≤ log | X |− log λ , lim sup n →∞ log δ ( n, S )log n ≤ log | X |− log λ . Proof.
Let ρ be any number in the interval ( λ, n , l such thatfor all elements g ∈ G such that l ( g ) > l we have l ( g | v ) ≤ ρ n l ( g ) for all v ∈ X n .It follows that there exists a finite set N such that g | v ∈ N for all v ∈ X ∗ and forevery g ∈ G \ N we have l ( g | v ) ≤ ρ n l ( g ) for all words v ∈ X ∗ of length at least n .Then for every g ∈ G and for every word v ∈ X ∗ of length at least j log l ( g ) − log l − log ρ k + n we have g | v ∈ N . Let w = x x . . . ∈ X N , and denote v = x x . . . x n , w ′ = x n +1 x n +2 . . . for n = j log r − log l − log ρ k + n . Then for fixed w and all g suchthat l ( g ) ≤ r , the germ ( g, w ) depends only on g ( v ) and g | v . There are not morethan | X | n possibilities for g ( v ), hence the number of germs ( g, w ) is not more than |N | · | X | n ≤ |N | exp (cid:18) log | X | (cid:18) log r − log l − log ρ + n (cid:19)(cid:19) ≤ C r log | X |− log ρ ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 11 for C = |N | · | X | log l ρ + n . Consequently, for every ρ ∈ ( λ,
1) there exists C > γ ( r, S ) ≤ C r log | X |− log ρ , hence lim sup r →∞ log γ ( r, S )log r ≤ log | X |− log λ .It is enough, in order to know the ball B S ( w, r ), to know for every word g ∈ G oflength at most 2 r whether the germ ( g, w ) is a unit. Let, as above, w = vw ′ , wherelength of v is n = j log 2 r − log l − log ρ k + n . For every g ∈ G of length at most 2 r the germ( g, w ) is a unit if and only if g ( v ) = v and ( g | v , w ′ ) is a unit. We have g | v ∈ N ,so B S ( w, r ) depends only on v and the set T w ′ = { h ∈ N : ( h, w ′ ) ∈ G (0) } .Consequently, δ ( r, S ) ≤ |N | · | X | n ≤ C r log | X |− log ρ , where C = 2 |N | | X | log l − log 2log ρ + n , which shows that lim sup r →∞ log δ ( r, S )log r ≤ log | X |− log λ . (cid:3) Both estimates in Proposition 3.3 are not sharp in general. For example, considera self-similar action of Z over the alphabet X of size 5 associated with the virtualendomorphism given by the matrix A = (cid:18) (cid:19) − = (cid:18) / − / − / / (cid:19) , see [24,2.9, 2.12] and [28] for details. Note that the eigenvalues of A are (cid:16) ±√ (cid:17) − ∈ (0 , λ = −√ = √ . On the other hand γ ( r, S )grows as a quadratic polynomial, while δ ( r, S ) is bounded.4. Convolution algebras
Definitions.
Let G be an ´etale groupoid, and let k be a field. Support of afunction f : G −→ k is closure of the set of points x ∈ G such that f ( x ) = 0. If f , f are functions with compact support, then their convolution is given by theformula f ∗ f ( g ) = X h ∈ G o ( g ) f ( gh − ) f ( h ) . Note that since f has compact support, the set of elements h ∈ G o ( g ) such that f ( h ) = 0 is finite.It is easy to see that if f , f are supported on the space of units, then theirconvolution coincides with their pointwise product. If F , F are bisections, thentheir characteristic functions satisfy 1 F ∗ F = 1 F F .The set of all functions f : G −→ k with compact support forms an algebra over k with respect to convolution. But this algebra is too big, and its definition doesnot use the topology of G much. On the other hand, the algebra of all continuousfunctions (with discrete topology on k ) is too small in the non-Hausdorff case.Therefore, we adopt the next definition, following Connes [9], see also [29] and [32]. Definition 4.1.
The convolution algebra k [ G ] is the k -algebra generated by thecharacteristic functions 1 F of open compact G -bisections (with respect to convolu-tion).If G is Hausdorff, then k [ G ] is the algebra of all continuous (i.e., locally constant)functions f : G −→ k , where k has discrete topology. In the non-Hausdorff case the algebra k [ G ] contains discontinuous functions (e.g., characteristic functions ofnon-closed open compact bisections).From now on we will use the usual multiplication sign for convolution. The unitof the algebra k [ G ] is the characteristic function of G (0) , which we will often denotejust by 1.If G = G ⋉ X is the groupoid of an action, then k [ G ] is generated by thecommutative algebra of locally constant functions f : X −→ k (with pointwisemultiplication and addition) and the group ring k [ G ] subject to relations g − · f · g = f ◦ g, for all f : X −→ k and g ∈ G , where f ◦ g : X −→ k is given by ( f ◦ g )( x ) = f ( g ( x )).In other words, it is the cross-product of the algebra of functions and the groupring.Let T ⊂ G (0) be the set of units with trivial isotropy groups. The set T is G -invariant, i.e., is a union of G -orbits. Definition 4.2.
We say that G is essentially principal if the set T is dense in G (0) .It is principal if T = G (0) . The groupoid G is said to be minimal if every G -orbitis dense in G (0) . Example 4.1.
For every homeomorphism g of a metric space X , the set of points x ∈ X such that g ( x ) = x and the germ ( g, x ) is non-trivial is a closed nowheredense set. It follows that if G is a countable group of homeomorphisms of X , thengroupoid of germs of the action is essentially principal.Simplicity of essentially principal minimal groupoids is a well known fact, see [7]and a C ∗ -version in [30, Proposition 4.6]. We provide a proof of the following simpleproposition just for completeness. Proposition 4.1.
Suppose that G is essentially principal and minimal. Let I be the set of functions f ∈ k [ G ] such that f ( g ) = 0 for every g ∈ G such that o ( g ) , t ( g ) ∈ T . Then I is a two-sided ideal, and the algebra k [ G ] /I is simple. Inparticular, if G is Hausdorff, then k [ G ] is simple.Proof. The fact that I is a two-sided ideal follows directly from the fact that T is G -invariant.In order to prove simplicity of k [ G ] it is enough to show that if f ∈ k [ G ] \ I , thenthere exist elements a i , b i ∈ k [ G ] such that P ki =1 a i f b i = 1.If f ∈ k [ G ] \ I , then there exists g ∈ G such that o ( g ) , t ( g ) ∈ T and f ( g ) = 0.Let f = P mi =1 α i F i , where F i are open compact G -bisections. Let A = { ≤ i ≤ m : g ∈ F i } . Then f ( g ) = P i ∈ A α i . Since o ( g ) ∈ T , an equality oftargets t ( F i o ( g )) = t ( F j o ( g )) implies the equality F i o ( g ) = F j o ( g ) of groupoidelements. It follows that t ( F i o ( g )) = t ( g ) for every i / ∈ A . We can find therefore aclopen neighborhood U of o ( g ) such that U ⊂ o ( F i ), F i U = F j U , for all i, j ∈ A , U ∩ o ( F j ) = ∅ for all j / ∈ A , and t ( F i U ) ∩ t ( F j U ) = ∅ for all i ∈ A and j / ∈ A .Denote F i U = F for any i ∈ A . We have 1 F − f U = P i ∈ A α i U . It follows that1 U = α F − f U for some α ∈ k .The groupoid G is minimal, hence for every x ∈ G (0) there exists h ∈ G suchthat o ( h ) = x and t ( h ) ∈ U . There exists therefore an open compact G -bisection H such that x ∈ o ( H ) and t ( H ) ⊂ U . Then 1 o ( H ) = 1 H − U H = α H − F − f U H .It follows that G (0) can be covered by a finite collection of sets V i such that 1 V i ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 13 can be written in the form a i f b i for some a, b ∈ k [ G ]. Note that if V ′ i is a clopensubset of V i , then 1 V ′ i = 1 V ′ i V i , hence we may replace the covering { V i } by a finitecovering by disjoint clopen sets. But in that case we have 1 = P V i . (cid:3) Growth of k [ G ] .Theorem 4.2. Let G be an ´etale groupoid with compact totally disconnected unitspace. Let S be a finite set of open compact G -bisections. Let V ⊂ k [ G ] be the k -subspace generated by the characteristic functions of the elements of S . Then dim V n ≤ γ ( n, S ) δ ( n, S ) . Proof.
Fix n , and let S n be the set of all products S S . . . S n of length n of elementsof S . Then V n is the linear span of the characteristic functions of elements of S n .Denote, for x ∈ G (0) , A x = \ F ∈S n ,x ∈ o ( F ) o ( F ) \ [ F ∈S n ,x/ ∈ o ( F ) o ( F ) . Since o ( F ) is clopen for every F ∈ S n , the sets A x are also clopen. Note that forevery F ∈ S n and x ∈ G (0) , either A x ⊂ o ( F ), or A x ∩ o ( F ) = ∅ .If F , F are open G -bisections and F · x = F · x for a unit x , then the set ofpoints y such that F · y = F · y is equal to the intersection of F − F with G (0) .Since G is ´etale, this set is open. Denote by B x the set of all points y ∈ A x suchthat F · x = F · x implies F · y = F · y for all F , F ∈ S n . Then B x is open and x ∈ B x .Note that if x ∼ n y , then A x = A y , as belonging of a point y to the domainof a product S S . . . S n of elements of S is equivalent to the existence of a pathin G ( y, S ) of length n starting at y and labeled by the sequence S n , S n − , . . . , S .Similarly, if x ∼ n y , then B x = B y , since an equality F · x = F · x is equivalentto coincidence of endpoints of the paths corresponding to the products F and F starting at x .Let B = { B x : x ∈ G (0 } . Since B x = B y for x ∼ n y , the set B consists of atmost δ ( n, S ) elements. Lemma 4.3.
There exists a covering e B = { e B } B ∈B of G (0) by disjoint clopen setssuch that e B ⊂ B for every B ∈ B . We allow some of the sets e B to be empty. Proof.
By the Shrinking Lemma, we can find for every B ∈ B an open set B ′ ⊂ B such that { B ′ } B ∈B is a covering of G (0) , and closure of B ′ is contained in B . Thenclosure of B ′ is compact, and can be covered by a finite collection of clopen subsetsof B . Hence, after replacing B ′ by the union of these clopen subsets, we mayassume that B ′ are clopen. Order the set B into a sequence B , B , . . . , B m , define e B = B ′ , and inductively, e B i = B ′ i \ ( B ′ ∪ B ′ ∪ · · · ∪ B ′ i − ). Then { e B } B ∈B satisfiesthe conditions of the lemma. (cid:3) Let x , x , . . . , x m be a transversal of the ∼ n equivalence relation, where m = δ ( n, S ). For every F ∈ S n and x i ∈ o ( F ), consider the restriction F · e B x i of F onto e B x i . Since { e B x i } i =1 ,...,m is a covering of G (0) by disjoint subsets, the sets F · e B x i form a covering of F by disjoint subsets, and 1 F = P mi =1 F · e B xi . If F , F ∈ S n and x i are such that x i ∈ o ( F ) ∩ o ( F ), and F · x i = F · x i ,then for every y ∈ e B x i we have y ∈ o ( F ) ∩ o ( F ) and F · y = F · y , hence F · e B x i = F · e B x i . It follows that F · e B x i depends only on F · x i , and we have notmore than γ ( n, x i , S ) ≤ γ ( n, S ) non-empty sets of the form F · e B x i , for every given x i . Hence we have at most γ ( n, S ) δ ( n, S ) functions of the form 1 F · x i in total, andevery function 1 F , for F ∈ S n is equal to the sum of a subset of these functions,which finishes the proof of the theorem. (cid:3) Finite generation.
For a given finite set S of open compact G -bisections,generating G , denote A x,n = \ F ∈S n ,x ∈ o ( F ) o ( F ) \ [ F ∈S n ,x/ ∈ o ( F ) o ( F ) , see the proof of Theorem 4.2. Recall that the sets A x,n are clopen. It is also easyto see that two sets A x,n and A y,n are either disjoint or coincide. Note also that A x,n ⊂ A x,m if n > m . It follows that for any x, y ∈ G (0) and n > m , either A x,n ⊂ A y,m , or A x,n ∩ A y,m = ∅ . Definition 4.3.
We say that S is expansive if for any two different points x, y ∈ G (0) there exists n such that A x,n and A y,n are disjoint. Proposition 4.4. If S is expansive, then the set { S : S ∈ S ∪ S − } generates k [ G ] .Proof. Let A be the algebra generated by the functions 1 S for S ∈ S ∪ S − . Notethat o ( F ) = F − F , hence 1 F ∈ A for every F ∈ ( S ∪ S − ) n . Note also that1 A ∩ B = 1 A · B , 1 A \ B = 1 A · (1 A − B ), and 1 A ∪ B = 1 A + 1 B − A B for every A, B ⊂ G (0) . It follows that 1 A x,n ∈ A for all x ∈ G (0) and n .Let us show that for every open set A ⊂ G (0) and every x ∈ A there exists n suchthat A x,n ⊂ A . For every y / ∈ A there exists n y such that A x,n y ∩ A y,n y = ∅ . Since G (0) \ A is compact, there exists a finite covering A y ,n y , A y ,n y , . . . , A y m ,n ym of G (0) \ A . Let n = max n y i . Then A x,n ⊂ A .Let F be an arbitrary open compact G -bisection. For every g ∈ F there exists n and F ′ ∈ ( S ∪ S − ) n such that g ∈ F ′ . There also exists n g such that A o ( g ) ,n g ⊂ o ( F ) and F · A o ( g ) ,n g = F ′ · A o ( g ) ,n g . We get a covering of F by sets of the form F ′ · A x,m , where F ′ ∈ ( S ∪ S − ) n . Since any two sets of the form A x,n are eitherdisjoint or one is a subset of the other, we can find a covering of F by disjoint setsof the form F ′ · A x,m for F ′ ∈ ( S ∪ S − ) n . This implies that 1 F ∈ A , which finishesthe proof. (cid:3) Examples.
Subshifts.
Let
X ⊂ X Z be a subshift, and let S be the groupoid of germsgenerated by the shift s : X −→ X . Let, as in 3.3.1, S x = { ( s, w ) : w (0) = x } , S = { S x } x ∈ X . Note that for every word x x . . . x n domain of the product S x S x · · · S x n is the set of words w ∈ X such that w (0) = x n , w (1) = x n − , . . . , w ( n −
1) = x . It follows that the set S ∪ S − is expansive, and by Proposition 4.4, { S } S ∈S∪S − is a generating set of k [ S ].Since S coincides with the groupoid of the Z -action on X defined by the shift,the algebra k [ S ] is the corresponding cross-product of the algebra of continuous k -valued functions with the group algebra of Z . Every its element is uniquely written ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 15 as a Laurent polynomial P a n · t n , where t ∈ k [ G ] is the characteristic function ofthe set of germs of the shift s : X −→ X , and a n are continuous k -valued functions.Multiplication rule for such polynomials follows from the relations t · a = b · t , where a, b : X −→ k satisfy b ( w ) = a ( s − ( w )) for every w ∈ X . Proposition 4.5.
Let V be the linear span of { } ∪ { S } S ∈S∪S − . Then j n k p X (cid:16)j n k(cid:17) ≤ dim V n ≤ (2 n + 1) p X (2 n ) . Proof.
The upper bound follows from Theorem 4.2. For the lower bound note that S x S x . . . S x n and S y S y . . . S y m are disjoint if x x . . . x n = y y . . . y m , hence theset of characteristic functions of all non-zero products of elements of S is linearlyindependent, so that P nk =0 p X ( n ) ≤ dim V n . Since p X ( n ) is non-decreasing, wehave (cid:4) n (cid:5) p X (cid:0)(cid:4) n (cid:5)(cid:1) ≤ P nk =0 p X ( n ). (cid:3) Note that since the characteristic functions of the products S x S x . . . S x n arelinearly independent, their linear span is a sub-algebra of k [ S ] isomorphic to thesemigroup algebra M X of the semigroup generated by the set { S x : x ∈ X } . Itis easy to see that M X is isomorphic to the quotient of the free associative algebragenerated by X modulo the ideal generated by all words w ∈ X ∗ such that w isnot a subword of any element of the subshift X . It follows from Proposition 4.5that growths of k [ S ] and M X are equivalent. Note that the algebras M X arethe original examples of algebras of arbitrary Gelfand-Kirillov dimension, see [34]and [19, Theorem 2.9]. Example 4.2.
Let X be a Sturmian subshift. It is minimal and p X ( n ) = n + 1,hence ( n + 1)( n + 2)2 ≤ dim V n ≤ n (2 n + 1) , so that k [ S ] is a quadratically growing finitely generated algebra. Note that it issimple by Proposition 4.1. This disproves Conjecture 3.1 in [4]. Example 4.3.
It is easy to see that every Toeplitz subshift is minimal. Conse-quently, known examples of Toeplitz subshifts (see Subsection 3.3.1) provide us withsimple finitely generated algebras of arbitrary Gelfand-Kirillov dimension α ≥ Self-similar groups.
Let G be a self-similar group of automorphisms of thetree X ∗ . Let G be the groupoid of germs of its action on the boundary X N of thetree. Suppose that G is self-replicating , i.e., for all x, y ∈ X and g ∈ G there exists h ∈ G such that g ( x ) = y and h | x = g . Then for all pairs of words v, u ∈ X ∗ ofequal length and every g ∈ G there exists h ∈ G such that h ( v ) = u and h | v = g . Inother words, the transformation vw ug ( w ) is an open compact G -bisection (morepedantically, the set of its germs is a bisection, but we will identify a G bisection F with the map o ( g ) t ( g ), g ∈ F ).Fix n ≥
0, and consider the set of all G -bisections of the form R u,g,v : vw ug ( w ) for v, u ∈ X n and g ∈ G . Note that these bisections are multiplied by therule(1) R u ,g ,v R u ,g ,v = (cid:26) v = u ; R u ,g g ,v if v = u . Let A n be the formal linear span of the elements R u,g,v for u, v ∈ X n and g ∈ G .Extend multiplication rule (1) to A n . It is easy to see then that A n is isomorphicto the algebra M d n × d n ( k [ G ]) of matrices of size d n × d n over the group ring k [ G ].The map R u,g,v P x ∈ X R ug ( x ) ,g | x ,vx induces a homomorphism A n A n +1 called the matrix recursion . More on matrix recursions for self-similar groups see [3,2, 23, 25, 11]. Example 4.4.
For the adding machine action (see Example 3.3) the matrix recur-sions replace every entry a n by (cid:18) a (cid:19) n , i.e., are induced by the map a (cid:18) a (cid:19) . For example, the image of a in A is a . For the Grigorchuk group the matrix recursions are induced by the map a (cid:18) (cid:19) , b (cid:18) a c (cid:19) ,c (cid:18) a d (cid:19) , d (cid:18) b (cid:19) . Proposition 4.6.
The convolution algebra k [ G ] of the groupoid of germs of theaction of G on X N is isomorphic to the direct limit of the matrix algebras A n ∼ = M d n × d n ( k [ G ]) with respect to the matrix recursions.Proof. Denote by A ∞ the direct limit of the algebras A n with respect to the matrixrecursions. Let φ : A ∞ −→ k [ G ] be the natural map given by φ ( R u,g,v ) = 1 R u,g,v .Note that 1 R u,g,v = P x ∈ X R ug ( x ) ,g | x,vx , hence the map φ is well defined. It alsofollows from equation (1) that φ is a homomorphism of algebras. It remains to showthat φ is injective. Let f be a non-zero element of k [ G ], and let ( g, w ) ∈ G be suchthat f ( g, w ) = 0. Suppose that φ ( f ) = P u,v ∈ X n α u,v R u,g u,v ,v for some α u,v ∈ k and g u,v ∈ G . Denote the set of all pairs ( u, v ) such that ( g, w ) ∈ R u,g u,v ,v and α u,v = 0by P . The set T ( u,v ) ∈ P R u,g u,v ,v is an open neighborhood of ( g, w ), hence thereexists a G -bisection R w ,h,w contained in T ( u,v ) ∈ P R u,g u,v ,v . Applying the matrixrecursion, we get a representation of f as an element P u,v ∈ X | w | β u,v R u,h u,v ,v ∈ A | w | such that ( g, w ) does not belong to any set R u,h u,v ,v , u, v ∈ X | w | , ( u, v ) =( w , w ). Then f ( g, w ) = β u,v = 0, hence φ ( f ) = 0. (cid:3) As a corollary of Proposition 3.3 and Theorem 4.2 we get the following result ofL. Bartholdi [2].
Proposition 4.7.
Let G be a contracting self-replicating group, and let G be thegroupoid of germs of its action on X N . Every finitely generated sub-algebra of k [ G ] has Gelfand-Kirillov dimension at most | X |− log λ , where λ is the contractioncoefficient of G . ROWTH OF ´ETALE GROUPOIDS AND SIMPLE ALGEBRAS 17
The image of the group ring k [ G ] in k [ G ] is called the thinned algebra . It wasdefined in [31], see also [2].Let us come back to the case of the Grigorchuk group. Since its contractioncoefficient is equal to 1 /
2, every finitely generated sub-algebra of k [ G ] has Gelfand-Kirillov dimension at most 2. It is easy to prove that it is actually equal to 2 inthis case. Moreover, it has quadratic growth, see [2].This example is also an illustration of the non-Hausdorffness phenomenon. Thegroupoid of germs of the Grigorchuk group is not Hausdorff: the germs ( b, . . . ),( c, . . . ), ( d, . . . ), and (1 , . . . ) do not have disjoint neighborhoods. Example 4.5.
Consider the convolution algebra F [ G ] for the groupoid of germsof the Grigorchuk group over the field with two elements. The matrix recursion forthe element b + c + d + 1 is b + c + d + 1 (cid:18) b + c + d (cid:19) . It follows that b + c + d is a non-trivial element of F [ G ] but, as a function on G iszero everywhere except for the germs of b, c, d, . . . , where it is equal to 1.This shows that the ideal I from Proposition 4.1 is non-zero in this case, and thealgebra F [ G ] is not simple.4.5. Modules k G x . Let G be an ´etale minimal groupoid. Consider the space k G x of maps φ : G x −→ k with finite support, where G x = { g ∈ G : o ( g ) = x } . It iseasy to see that for every φ ∈ k G x and f ∈ k [ G ] the convolution f · φ is an elementof k G x , and that k G x is a left k [ G ]-module with respect to the convolution. Proposition 4.8.
Let S be an finite set of open compact G -bisections, and let V ⊂ k [ G ] be the linear span of their characteristic functions and G (0) . Then forevery n ≥ we have dim V n · δ x ≤ γ S ( x, n ) , where δ x ∈ k G x is the characteristic function of x ∈ G x , and γ S ( x, n ) is the growthof the Cayley graph based at x of the groupoid generated by the union of the elementsof S .If the isotropy group of x is trivial, then the module k G x is simple.Proof. The growth estimate is obvious, since for every g ∈ G x and S ∈ S we have1 S · δ g = δ Sg , if Sg = ∅ , and 1 S · δ g = 0 otherwise.Let us show that k G x is simple if the isotropy group of x is trivial. It is enoughto show that for every non-zero element φ ∈ k G x there exist elements f , f ∈ k [ G ]such that f · φ = δ x and f · δ x = φ .Let φ ∈ k G x , and let { g , g , . . . , g k } be the support of φ . Since the isotropygroup of x is trivial, t ( g i ) are pairwise different. Let U , U , . . . , U k be open compact G -bisections such that g i ∈ U i and t ( U i ) are disjoint. Then (cid:16)P ki =1 φ ( g i )1 U i (cid:17) · δ x = φ and φ ( g ) − U − φ = δ x . (cid:3) Example 4.6.
Let X be a finite alphabet, and let w ∈ X Z be a non-periodicsequence such that closure X w of the shift orbit of w is minimal. Let S be thegroupoid generated by the action of the shift on X w . Denote by T and T − thecharacteristic functions of the sets of germs of the shift and its inverse, and forevery x ∈ X , denote by D x the characteristic function of the cylindrical set { w ∈X w : w (0) = x } . Then k [ S ] is generated by T, T − and D x for x ∈ X . Note that we can remove one of the generators D x , since P x ∈ X D x = 1 = T T − . Considerthe set S w = { ( s n , w ) : n ∈ Z } and the corresponding module k S w . Its basis as a k -vector space consists of the delta-functions e n = δ ( s n ,w ) , n ∈ Z . In this naturallyordered basis left multiplication by T is given by the matrix T = . . . ... ... ... ... · · · · · ·· · · · · ·· · · · · ·· · · · · · ... ... ... ... . . . = ( t ij ) i ∈ Z ,j ∈ Z with the entries t m,n = δ m − ,n . The element T − is given by the transposed matrix,and an element D x is given by the diagonal matrix ( a ij ) with entries given by therule a nn = (cid:26) w ( n ) = x ,0 otherwise.It follows that the algebra k [ S ] is isomorphic to the algebra generated by suchmatrices. For example, if X = { , } , then the algebra is generated by the matrices T , T ⊤ , and the diagonal matrix with the sequence w on the diagonal. References [1] C. Anantharaman-Delaroche and J. Renault.
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