Lamplighter groups, bireversible automata and rational series over finite rings
aa r X i v : . [ m a t h . G R ] N ov LAMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATAAND RATIONAL SERIES OVER FINITE RINGS
RACHEL SKIPPER AND BENJAMIN STEINBERG
Abstract.
We realize lamplighter groups A ≀ Z , with A a finite abeliangroup, as automaton groups via affine transformations of power seriesrings with coefficients in a finite commutative ring. Our methods canrealize A ≀ Z as a bireversible automaton group if and only if the 2-Sylowsubgroup of A has no multiplicity one summands in its expression as adirect sum of cyclic groups of order a power of 2. Introduction
In the process of classifying all two-state automaton groups over a two-letter alphabet, Grigorchuk and ˙Zuk discovered that the lamplighter group Z / Z ≀ Z can be realized as an automaton group [GZ01]. Moreover, they usedthis realization to compute the spectral measure for a simple random walkon the group with respect to the generating set arising from the automaton.This in turn led to the first counterexample to the strong form of the Atiyahconjecture on ℓ -Betti numbers [GLSZ00]. The proof of Grigorchuk and ˙Zukthat their automaton generated a lamplighter groups was computational,making use of the wreath product representation of automaton groups. Amore conceptual proof, using affine transformations of the power series ringover the two-element field, appeared in [GNS00]. This paper also realizedlamplighter groups of the form ( Z /p Z ) n ≀ Z with p prime using series overfields. Here, by a lamplighter group we mean a group of the form F ≀ Z where F is a finite group.Dicks and Schick computed spectral measures for random walks on arbi-trary lamplighter groups F ≀ Z with respect to a set of generators inspired bythe automaton generators of Grigorchuk and ˙Zuk [DS02]. With respect tothis generating set, the Cayley graph of F ≀ Z depends (up to isomorphism) Date : December 2, 2019.2010
Mathematics Subject Classification.
Primary 20F65; Secondary 20E08, 20F10.
Key words and phrases.
Automata groups, lamplighter groups, reversible automata,bireversible automata.The first author was partially supported by a grant from the Simons Foundation( only on | F | and not the group structure of F . Their approach avoided au-tomata completely. The second author and Silva realized all lamplightergroups A ≀ Z with A a finite abelian group in [SS05] by using affine transfor-mations of power series rings over finite commutative rings. The automatongenerators in this case are exactly those considered by Dicks and Schickand so the second author, together with Kambites and Silva, exploited thisautomaton realization in [KSS06] to give a new proof of the results of Dicksand Schick using the original scheme of Grigorchuk and ˙Zuk. We mentionanother early paper realizing lamplighter groups was [BˇS06].Since then there has been further study of automaton group realizationsof lamplighter groups, particularly in connection with power series. For ex-ample, Savchuk and Sidki [SS16] have recently studied realizations of lamp-lighter groups as affine transformations over power series rings with coeffi-cients in Z /d Z and they provide a detailed study of all such representationsover the binary tree. The lamplighter group Z / Z ≀ Z is represented as abounded automaton group in [JSW17]. The recent paper [BS18] constructs A ≀ Z d as an automaton group for any finite abelian group A and d ≥ Z / Z ≀ Z . Their original proof was using wreath product representa-tions but Bondarenko and Savchuk have recently announced a constructionvia affine transformations of the ring of power series over Z / Z . Multipli-cation by a rational power series over a field can always be implementedby a finite state automaton. Bondarenko and Savchuk announced a com-plete description of all automaton groups generated by the states of suchautomata (they are always lamplighter groups if the series is invertible andnon-constant), as well as their dual automaton semigroups. In particular,they characterize when such automaton groups are bireversible. Such an AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 3 approach was also used by Ahmed and Savchuk [AS19] to realize ( Z / Z ) ≀ Z as a bireversible automaton group.In this paper, like in [SS05], we consider rational power series over a finitecommutative ring R . More specifically, we consider invertible rational seriesof the form f ( t ) = r (cid:18) − at − bt (cid:19) = r (1 − at ) · ∞ X n =0 b n t n (1.1)where r ∈ R × is a unit and a, b ∈ R . These are precisely the invertiblerational series, which together with their inverses, have corresponding re-currence of degree at most 1. We explicitly construct an initial automaton A f computing left multiplication by f ( t ). Here we identify R ω with R J t K inthe obvious way. We then consider the automaton group generated by thestates of A f . We prove that if a − b is a unit of R , then we obtain R + ≀ Z ,where R + is the additive group of R . The special case where r = 1, a = 0and b = 1 recovers the construction in [SS05]. Most likely if a − b is not aunit, one never gets the lamplighter group R + ≀ Z , but we prove this negativeresult explicitly only when R = Z /n Z .The main result is that A f is a bireversible automaton generating a groupisomorphic to R + ≀ Z if and only if a , b and a − b are units of R . This thenleads to the natural question of which finite abelian groups A can be ob-tained as the additive group of a finite commutative ring R with two unitswhose difference is a unit. Note that R has this property if and only if it hasno ideal of index 2. Not all finite abelian groups can be realized this way.For example Z /n Z with n even can never be realized as the additive groupof such a ring. We prove that A is isomorphic to the additive group of a ring R with no ideal of index 2 if and only if the 2-Sylow subgroup of A can beexpressed as a direct sum of cyclic groups of the form ( Z / i Z ) r i with r i ≥ A ≀ Z as bireversible automatongroups whenever the 2-Sylow subgroup of A satisfies this condition. Weare hopeful that the extra symmetry appearing in a bireversible automatonmight make it possible to perform spectral computations for these lamp-lighter groups with respect to other probability measures than the uniformmeasure on the automaton generating set.The paper is organized as follows. We begin by recalling some basicnotions concerning automata, automaton groups and actions on rooted trees.The next section discusses the action of multiplication by f ( t ) from (1.1) onpower series from this viewpoint and constructs its minimal automaton. Westudy the group generated by the states of this automaton and prove that itis a lamplighter group when a − b is a unit of R . Before doing this, we reviewsome of the basic theory of finite commutative rings, e.g., that they are directproducts of local rings. We also characterize reversibility and bireversibilityof the automaton. The following section characterizes the additive groups R. SKIPPER AND B. STEINBERG of finite commutative rings containing two units whose difference is a unit.The final section provides some examples of our construction.2.
Automata
For background on automata groups, the reader is referred to the bookof Nekrashevych [Nek05] or the survey paper [GNS00]. Let X be a finiteset called an alphabet and let X ∗ be the free monoid on X , that is, the setof finite words over the alphabet X including the empty word, denoted ∅ .The length of a word w ∈ X ∗ is denoted by | w | . The Cayley graph of X ∗ naturally has the structure of a regular rooted tree with ∅ as the root andtwo words u and v are connected by an edge if ux = v or u = vx for some x ∈ X , and so we denote the Cayley graph by T X .An automorphism of T X is a bijection from X ∗ to X ∗ which preservesedge incidences. The group of all automorphisms of T X is denoted Aut( T X ).Any automorphism of T X induces an action on the boundary of the treewhich can be identified with the set of infinite words over X and is denoted X ω . Conversely, any permutation of X ω which preserves the length of thelongest common prefix of any pair of infinite words is induced by the actionof a unique automorphism of T X .For v ∈ X ∗ and u ∈ X ω , an automorphism g ∈ Aut( T X ) acts on vu via g ( vu ) = g ( v ) g | v ( u )where g ( v ) ∈ X ∗ with | g ( v ) | = | v | and g | v ∈ Aut( T X ), depending only on v .We call g | v the state of g at v (some authors use the term “section”). Anautomorphism g of T X is said to be finite state if the set { g | v : v ∈ X ∗ } isfinite. The following is a straightforward computation, taking the action byautomorphisms to be a left action. Lemma 2.1.
Let g, h ∈ Aut( T X ) . Then the states of gh and g − at v ∈ X ∗ are given by ( gh ) | v = g | h ( v ) h | v ( g − ) | v = ( g | g − ( v ) ) − . It follows from the lemma that a composition of finite state automor-phisms is again finite state and the inverse of a finite state automorphism isfinite state.A subgroup G ≤ Aut( T X ) is said to be self-similar if, for all g ∈ G and v ∈ X ∗ , g | v is in G . For a vertex v , the stabilizer of G at v , denotedStab G ( v ), is the set of g ∈ G with g ( v ) = v . The group G is self-replicating if { g | v : g ∈ Stab G ( v ) } = G for all v ∈ X ∗ (actually, it is enough for all v ∈ X ). It is spherically transitive if for every v and w of same length over the alphabet, there exists g ∈ G AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 5 with g ( v ) = w . Note that some authors include spherical transitivity as partof the definition of self-replicating.A self-similar group is said to be an automaton group if it is finitely gen-erated and every element of the group is finite state. In this case, a finitestate Mealy automaton can be used to describe the generators.A (Mealy) automaton is A is a 4-tuple A = ( Q, X, δ, λ ) where Q is a finiteset of states, X is a finite alphabet, δ : Q × X → Q is the transition function ,and λ : Q × X → X is the output function . For each q ∈ Q and x ∈ X ,we will use the notation λ q ( x ) to mean λ ( q, x ) and will call λ q the statefunction corresponding to q . Similarly, for x ∈ X , we define δ x : Q → Q by δ x ( q ) = δ ( q, x ). When the set of states is finite, we say A is a finite stateautomaton .It is common to describe an automaton A by a directed, labeled graphwith vertices labeled by Q and edges q x | λ q ( x ) −−−−−→ δ ( q, x )for each q ∈ Q and x ∈ X .The function λ q can be extended uniquely to a function on both the sets X ∗ of finite words and X ω of infinite words over X , which, abusing notation,we shall also refer to as λ q . These extensions are described recursively asfollows: λ q ( x x · · · x n ) = λ q ( x ) λ δ ( q,x ) ( x · · · x n )and λ q ( x x · · · ) = lim n →∞ λ q ( x x · · · x n ) . Intuitively, the input word labels the left hand side of a unique path startingat q in the directed, labeled graph representing A and the output is the labelof the right hand side of this path.For each g ∈ Aut( T X ), there is a unique minimal automaton A g and state q such that g = λ q . The state set is given by { g | v : v ∈ X ∗ } , the transitionfunction is given by δ ( g | v , x ) = g | vx and the output function is given by λ ( g | v , x ) = g | v ( x ). One then has g = λ g | ∅ . Moreover, A g is finite if and onlyif g is finite state.An automaton A is invertible if, for each q , λ q is a permutation of thealphabet. Invertible automata are precisely the automata for which eachstate function describes an automorphism of the tree T X with vertex set X ∗ . For an automaton A = ( Q, X, δ, λ ), the inverse automaton A − isobtained by switching the input and output letters on the edge labels. Inthis case, the inverse to λ q is computed by the state corresponding to q in A − . Note that, for g ∈ Aut( T X ), the minimal automaton for g − is theinverse of the minimal automaton for g , as is easily seen from Lemma 2.1. R. SKIPPER AND B. STEINBERG
The dual automaton ∂ A is given by ( X, Q, λ, δ ), i.e., the alphabet andstates are interchanged and the output and transition functions are inter-changed. An invertible automaton is called reversible if its dual is invertibleand bireversible if it is reversible and additionally its inverse is reversible.Note that some authors do not require A to be invertible in order to bereversible. The state functions of ∂ A are precisely the functions δ x , with x ∈ X , and hence A is reversible if and only if δ x is a permutation of thestate set for each x ∈ X .For an invertible finite state automaton A , the group generated by { λ q : q ∈ Q } under the operation of composition is called the automaton group G ( A ). It is a self-similar group and it is an automaton group in the sensedefined above. The group generated by a bireversible automaton enjoys theproperty that its action on X ω is always essentially free [SVV11], whereas theaction of a reversible automaton is sometimes essentially free and sometimesnot [KSS06]. The action on X ω is essentially free if the stabilizer of aninfinite word is almost surely trivial with respect to the Bernoulli measure.This is exactly the property that one needs for the spectral measure of thesimple random walk on the Cayley graph to be computable as the limit ofthe Kesten-von Neumann-Serre spectral measures [GZ04] for the randomwalks on the levels of the tree in the spherically transitive case, see [KSS06]for further details.3. Power series and lamplighter groups
By a lamplighter group we mean a restricted wreath product F ≀ Z = L Z F ⋊ Z with F a finite group. Some authors refer to only the partic-ular case ( Z / Z ) ≀ Z as the lamplighter group. In this section we use thelanguage of formal power series to produce automata which generate lamp-lighter groups of the form A ≀ Z with A a finite abelian group. This wasfirst done for A = Z / Z by Grigorchuk and ˙Zuk [GZ01] (where power se-ries were not used, but see [GNS00] for a proof using power series) and forarbitrary finite abelian groups by the second author and Silva [SS05]. Wefind conditions on the power series that will guarantee that the automatonis reversible or bireversible. Related work, via power series over fields, hasbeen announced by Bondarenko and Savchuk at various conferences. Theirwork was announced before ours, and considers rational power series withhigher order recurrences than ours. However, by working over rings we canrealize many more lamplighter groups as bireversible automata than can berealized over a field.Let R be a finite commutative ring with unity, R + its additive group, R × its multiplicative group of units, and R J t K the ring of formal power serieswith coefficients in R . Note that f ∈ R J t K is a unit if and only if its constantterm f (0) is a unit of R . We will identify R ω with R J t K via( r , r , r , . . . ) r + r t + r t + · · · AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 7 where on the left hand side we write elements of R ω as tuples to avoidconfusion between concatenation of words and the multiplication in R .For any power series f ( t ) in R J t K we define two mappings of R ω given by µ f : g ( t ) f ( t ) g ( t )and α f : g ( t ) f ( t ) + g ( t ) . Note that α f is invertible with inverse α − f and preserves the length of thelongest common prefix. Thus α f gives an automorphism of the tree T R withvertex set R ∗ . On the other hand, µ f is invertible precisely when f is a unitof R J t K . In this case ( µ f ) − = µ f − and both these mappings preserve thelength of the longest common prefix. It is well known to automata theoriststhat if f is a rational power series, that is, f ( t ) = p ( t ) /q ( t ) with p ( t ) , q ( t )polynomials and q (0) = 0, then α f , µ f are finite state. We shall compute theminimal automaton for µ f in the case that p ( t ) , q ( t ) are linear, momentarily.The following proposition illustrates the relationship between µ and α . Proposition 3.1.
For any n ∈ Z and f ( t ) , h ( t ) ∈ R J t K , µ f α h µ f − = α fh .Proof. If g ( t ) ∈ R J t K , then we compute µ f α h µ f − ( g ) = µ f α h ( f − g ) = µ f ( f − g + h ) = g + f h = α fh ( g ). (cid:3) For any power series f ( t ) = P ∞ i =0 c i t i we define also the shift of f by σ ( f ) = ∞ X i =0 c i +1 t i = c + c t + c t + · · · so that f ( t ) = c + σ ( f ) t .We wish to study rational power series of the form f ( t ) = r − r tr − r t . In order for f ( t ) to be a unit of R J t K , r and r must be units of R , and sowe will write f ( t ) as f ( t ) = r (cid:18) − at − bt (cid:19) for a and b in R and r ∈ R × . With this notation f ( t ) = r + r ( b − a ) t + rb ( b − a ) t + rb ( b − a ) t + rb ( b − a ) t + · · · as is easily checked. Lemma 3.2.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × and a, b ∈ R . Then σ ( f ) = bf − ra . R. SKIPPER AND B. STEINBERG
Proof.
Observe that bf − ra = b ( r + r ( b − a ) t + rb ( b − a ) t + rb ( b − a ) t + · · · ) − ra = br − ra + rb ( b − a ) t + rb ( b − a ) t + rb ( b − a ) t + · · · = r ( b − a ) + rb ( b − a ) t + rb ( b − a ) t + rb ( b − a ) t + · · · = σ ( f ) . (cid:3) Proposition 3.3.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × and a, b ∈ R . Then µ f is finite state with set of states { α − sra µ f α sb : s ∈ R } . Moreover, for any s ∈ R , the state α − sra µ f α sb permutes the degree zero terms via: ˜ s r (˜ s + ( b − a ) s ) . Proof.
Let g ( t ) = P ∞ i =0 d i t i = d + σ ( g ) t . Then by Lemma 3.2 µ f ( g ( t )) = ( r + σ ( f ) t )( d + σ ( g ) t )= rd + f σ ( g ) t + d σ ( f ) t = rd + f σ ( g ) t + d ( bf − ra ) t = rd + [ − d ra + f σ ( g ) + f d b ] t = rd + [ − d ra + f ( σ ( g ) + d b )] t = rd + [ α − d ra µ f α d b ( σ ( g ))] t. Therefore, the state of µ f at the vertex d is given by α − d ra µ f α d b and isof the desired form and µ f ( d ) = rd .Now it just remains to show that the states of α − sra µ f α sb for words oflength 1 have the same form. Note that the states of α − sra and α sb at everyword of length greater than 0 are trivial since, for any d ∈ R , we have α d ( c + c t + c t + · · · ) = ( d + c ) + c t + c t + · · · . Applying Lemma 2.1, we see that, for any d ∈ R ,( α − sra µ f α sb ) | d = ( α − sra µ f ) | d + sb = ( α − sra ) | r ( d + sb ) ( µ f ) | d + sb = α − ( d + sb ) ra µ f α ( d + sb ) b . Therefore, the state of α − sra µ f α sb at d given by α − ( sb + d ) ra µ f α ( sb + d ) b is again of the desired form.Finally, it is straightforward to check that if g ( t ) is a power series withconstant term ˜ s then the constant term of α − sra µ f α sb ( g ) is r (˜ s + ( b − a ) s ).Indeed, since µ f acts on the root of T R via multiplication by r , we have that α − sra µ f α sb (˜ s ) = − sra + r (˜ s + sb ) = r (˜ s + ( b − a ) s ). (cid:3) AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 9
The above proposition tells us that we can associate to µ f a finite stateautomaton whose state functions are given by the states of µ f . In otherwords, define A f = ( Q, X, δ, λ ) with states Q = { α − sra µ f α sb : s ∈ R } andalphabet X = R . The transition function δ is given by δ ( α − sra µ f α sb , ˜ s ) = α − ( sb +˜ s ) ra µ f α ( sb +˜ s ) b (3.1)and the output function by λ ( α − sra µ f α sb , ˜ s ) = r (˜ s + ( b − a ) s ) . (3.2)The state function λ µ f is precisely µ f and, more generally, λ α − sra µ f α sb = α − sra µ f α sb for s ∈ R .We now review some basic properties of finite commutative rings. It is wellknown that if a commutative ring with unity is Artinian, then it is a finitedirect product of Artinian commutative local rings (cf. [Eis95, Cor. 2.16]).Since R is a finite ring it is clearly Artinian and so we shall write R = R × · · · × R n where R , . . . , R n are local rings. Let e i = (0 , . . . , , , , . . . , i -th position so that P ni =1 e i = 1. In fact, e , . . . , e n form acomplete set of orthogonal primitive idempotents of R .For any R -module M , let M i = e i M ; it is an R i -module. Then M = M × · · · × M n and the action of R is coordinatewise:( r , . . . , r n )( m , . . . , m n ) = ( r m , . . . , r n m n ) . It is clear that a subset A of M is linearly independent over R if and onlyif, for each i , its projection e i A into M i is linearly independent over R i .Recall now that, for a commutative Artinian local ring S , the uniquemaximal ideal m consists of the nilpotent elements of S and that S × = S \ m . Proposition 3.4.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × and a, b ∈ R . Then the set { f m : m ∈ Z } is linearly indepen-dent over R if and only if a − b ∈ R × .Proof. Write R = R × · · · × R n where each R i is local. For s ∈ R , let s i be the image of s in R i for 1 ≤ i ≤ n . Note that a − b ∈ R × if and only if a i − b i ∈ R × i for all i and so we may assume without loss of generality that R is local with maximal ideal m by the observation preceding the proposition.Suppose first that a − b / ∈ R × and so a − b ∈ m . Since m consists ofnilpotent elements, we can find k ≥ a − b ) k = 0 and ( a − b ) k − = 0.Since f ( t ) = r (cid:18) − at − bt (cid:19) = r (cid:18) − ( a − b ) t − bt (cid:19) we get that ( a − b ) k − f = r ( a − b ) k − . Therefore, ( a − b ) k − f − r ( a − b ) k − f =0 and ( a − b ) k − , r ( a − b ) k − = 0. Thus the powers of f are linearly dependentover R . Assume now that a − b ∈ R × . Then either a or b is not in m . Withoutloss of generality, assume a ∈ R × . Since f is a unit, if cf n = 0 with c ∈ R ,then c = 0. Suppose that c f n + · · · + c k f n k = 0 with n < n < · · · < n k , c i = 0 for all i , and k >
1. Multiplying the expression by f − n if necessary,we may assume n = 0. Now multiply by (1 − bt ) n k to get c (1 − bt ) n k + c r n (1 − bt ) n k − n (1 − at ) n + · · · + c k r n k (1 − at ) n k = 0 . Putting t = a , we get that c (1 − ba ) n k = 0 and hence c ( a − ba ) n k = 0. Since a − b and a are both units we get that c = 0, a contradiction. Therefore, { f m : m ∈ Z } is linearly independent over R . (cid:3) Proposition 3.5.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × and a, b ∈ R . Then the set B = { ( − ar + bf ) f m : m ∈ Z } is linearly independent over R if and only if a − b ∈ R × . Moreover, if a − b / ∈ R × , then there exists s ∈ R \ { } with sB = 0 .Proof. First note that − ar + bf = − ar (1 − bt ) + br (1 − at )1 − bt = − r ( a − b )1 − bt . Thus B = (cid:26) − r ( a − b ) (cid:18) − bt (cid:19) f m : m ∈ Z (cid:27) . (3.3)If a − b is not a unit of R , then since R is a finite, there exists s ∈ R \ { } with s ( a − b ) = 0. It follows that sB = 0 and hence B is not linearlyindependent over R . Conversely, if a − b ∈ R × , then − r ( a − b ) (cid:16) − bt (cid:17) is aunit of R J t K . The linear independence of B over R is then immediate fromthe linear independence of { f m : m ∈ Z } from Proposition 3.4. (cid:3) We remark that if f ( t ) = 1 − t in ( Z / Z ) J t K , then f = 1 and so G ( A f )is in fact a finite group. In this case a = 2, b = 0 and a − b is not a unit. Theorem 3.6.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × and a, b ∈ R . If a − b ∈ R × , then G ( A f ) = h α − sra µ f α sb : s ∈ R i ∼ = R + ≀ Z .Proof. Since α − sra µ f α sb = α − sra + sbf µ f = α s ( − ar + bf ) µ f by Proposition 3.1,we can take { α s ( − ar + bf ) , µ f : s ∈ R } as a generating set for G ( A f ).By Proposition 3.5, the set { ( − ar + bf ) f m : m ∈ Z } is a linearly indepen-dent set over R and forms an infinite dimensional basis for a free module AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 11 over R . Therefore, by Proposition 3.1, N = h α s ( − ar + bf ) f m : m ∈ Z i = h ( µ f ) m α s ( − ar + bf ) ( µ f ) − m : m ∈ Z i ∼ = M Z R + . Moreover, N is clearly normal in G ( A f ) and contains α s ( − ar + bf ) for all s ∈ R . Furthermore, N intersects h µ f i trivially (as Proposition 3.4 implies µ f has infinite order) and µ f acts on N via a shift. We conclude that G ( A f ) ∼ = M Z R + ⋊ Z = R + ≀ Z as desired. (cid:3) Remark . Note that if R = Z /n Z and a, b ∈ R with a − b not a unit, theneither µ f has finite order (and hence G ( A f ) is finite) or, by Proposition 3.5,the torsion subgroup of G ( A f ), that is, the subgroup h α ˜ s ( − ar + bf ) f m : m ∈ Z , ˜ s ∈ R i is annihilated by some 0 < s < n hence is not isomorphic to L Z R + . Thus G ( A f ) is not isomorphic to R + ≀ Z in this case. Lemma 3.8.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × , a, b ∈ R and a − b is a unit. Then A f has | R | states.Proof. Note that, for s ∈ S , we have α − sra µ f α sb = α − sra + sbf µ f by Proposi-tion 3.1. Since a − b is a unit and µ f is invertible, we see from Proposition 3.5that distinct elements s ∈ R give rise to distinct states of A f . (cid:3) Theorem 3.9.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × , a, b ∈ R and a − b is a unit. Then (i) A f is reversible if and only if b is a unit. (ii) ( A f ) − is reversible if and only if a is a unit. (iii) A f is bireversible if and only if both a and b are units.Proof. Inspection of (3.1) shows that A f is reversible if and only if, for each˜ s ∈ R , the mapping α − sra µ f α sb α − ( sb +˜ s ) ra µ f α ( sb +˜ s ) b is one-to-one as this is the output function δ ˜ s for the dual automaton ∂ A f at state ˜ s . Suppose first that b is a unit and α − ( sb +˜ s ) ra µ f α ( sb +˜ s ) b = α − ( s ′ b +˜ s ) ra µ f α ( s ′ b +˜ s ) b for some s, s ′ ∈ R . By Proposition 3.1, this can be rewritten as α − ( sb +˜ s ) ra +( sb +˜ s ) bf µ f = α − ( s ′ b +˜ s ) ra +( s ′ b +˜ r ) bf µ f . Since µ f is an invertible function this implies that α ( sb +˜ s )( − ra + bf ) = α ( s ′ b +˜ s )( − ra + bf ) and hence, by Proposition 3.5, we get that( sb + ˜ s ) b = ( s ′ b + ˜ s ) b. Now since b ∈ R × , we deduce that s = s ′ . Therefore, ∂ A f is invertible and A f is reversible.Conversely, if b is not a unit, then since R is finite, sb = 0 for some s = 0in S . Taking ˜ s = 0, we then have α − sra µ f α sb µ f and µ f µ f under δ . Thus δ is not a permutation of the state set (as the states µ f and α − sra µ f α sb are distinct by Lemma 3.8) and so A f is not reversible. Thisproves (i).Now observe that ( A f ) − = A f − and that f − = r − (cid:18) − bt − at (cid:19) . Therefore, by the same argument as above ( A f ) − is reversible if and only if a is a unit, establishing (ii). Item (iii) follows directly from (i) and (ii). (cid:3) Note that if a − b is a unit, then we can identify the states of A f bijectivelywith R via s α − sra µ f α sb by the above proof. Under this identificationthe the edges of A f become s ˜ s | r (˜ s +( b − a ) s ) −−−−−−−−−→ sb + ˜ s. (3.4)We now prove that when a − b is a unit, the automaton group acts spher-ically transitively. This is a necessary condition in order to perform spectralcomputations using the automaton group representation. Proposition 3.10.
Let f ( t ) = r (cid:18) − at − bt (cid:19) where r ∈ R × and a, b ∈ R . If a − b is a unit of R , then G ( A f ) is sphericallytransitive on T R . If, in addition, r = 1 and a = 0 or b = 0 , then G ( A f ) isself-replicating.Proof. The action of G ( A f ) on level d + 1 of T R can be identified with its ac-tion on R J t K / ( t d +1 ) by affine mappings for d ≥
0. The proof of Theorem 3.6shows that G ( A f ) contains the translations by s ( − ra + bf ) f m for all s ∈ R and m ∈ Z or, equivalently by (3.3), the R -span of all translations − bt f m with m ∈ Z (since r and a − b are units).Proposition 3.5 shows that the | R | d +1 power series11 − bt (cid:16) c + c f + · · · + c d f d (cid:17) (3.5) AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 13 with c , . . . , c d ∈ R are distinct. Multiplying (3.5) through by the invertiblepower series (1 − bt ) d +1 yields that the | R | d +1 polynomials c (1 − bt ) d + c (1 − at )(1 − bt ) d − + · · · + c d (1 − at ) d (3.6)of degree at most d are all distinct. Since (3.6) consists of polynomials ofdegree at most d , we obtain that the | R | d +1 cosets in R J t K / ( t d +1 ) c (1 − bt ) d + c (1 − at )(1 − bt ) d − + · · · + c d (1 − at ) d + ( t d +1 )are all distinct and hence multiplying by the unit (1 − bt ) − ( d +1) , we obtainthat the | R | d +1 cosets11 − bt (cid:16) c + c f + · · · + c d f d (cid:17) + ( t d +1 )are distinct. But these are then all the cosets in R J t K / ( t d +1 ) and so itfollows that the action of G ( A f ) on R J t K / ( t d +1 ) contains all the translationsand hence is transitive. This completes the proof that G ( A f ) is sphericallytransitive when a − b is a unit.Let G = G ( A f ). Suppose now that r = 1 and a or b is zero. Replacing f by its inverse, we may assume that b = 0. Note that a must then be aunit. We already have that G acts spherically transitively and so it sufficesto show that { g | : g ∈ Stab G (0) } = G by a standard argument. Recall thatby Proposition 3.3, for b = 0 and r = 1, we have G = h α − sa µ f : s ∈ R i .First note that µ f ∈ Stab G (0) as r = 1. Also note that ( µ f ) | s = α − sa µ f .In particular, α − sa ∈ G for all s ∈ S . Since a is a unit, for any c ∈ R , wecan find s ∈ R with − sa = c . Thus α c ∈ G for all c ∈ R . Then we have α − s µ f α s ∈ Stab G (0) and ( α − s µ f α s ) | = ( µ f ) | s = α − as µ f by Lemma 2.1 forall s ∈ R . Thus { g | : g ∈ Stab G (0) } = G . This completes the proof that G is self-replicating. (cid:3) Rings with a , b , and a − b in R × In this section, we characterize which finite abelian groups A can be theadditive group of a finite commutative ring with two units whose differenceis a unit so that we may realize A ≀ Z as a bireversible automaton group viaTheorem 3.6 and Theorem 3.9.We will write ( R, m ) to denote a finite commutative local ring with max-imal ideal m . We remind the reader now of the following well-known propo-sition, whose proof we include for completeness. Proposition 4.1. If ( R, m ) is a finite commutative local ring and | R/ m | = q ,then | R | = q i for some i ≥ . In particular, R has prime power order.Proof. Since R is finite, hence Artinian, and m is the Jacobson radical of R ,it follows that m is nilpotent and hence m k +1 = 0 for some k ≥
0, which wetake to be minimal. Thus we have a filtration R ) m ) · · · ) m k +1 = 0 and m j / m j +1 is a finite dimensional vector space over the residue field R/ m for 0 ≤ j ≤ k . Thus | m j / m j +1 | = q d j with d j = dim m j / m j +1 and hence | R | = q d + ··· + d k by repeated application of Lagrange’s theorem. (cid:3) Proposition 4.2.
Let ( R, m ) be a finite commutative local ring such that char( R/ m ) = p and suppose | R/ m | = p r . If R + ∼ = ( Z /p Z ) a ⊕ ( Z /p Z ) a ⊕ · · · ⊕ ( Z /p t Z ) a t , then r | a .Proof. Consider the ideal p R ⊆ m . Then ( R/p R, m /p R ) is a finite localcommutative ring with residue field R/ m and( R/p R ) + ∼ = ( Z /p Z ) a ⊕ ( Z /p Z ) k where k = a + · · · + a t . Note that it is possible that k = 0. Without lossof generality, we may assume that p R = 0 and R + ∼ = ( Z /p Z ) a ⊕ ( Z /p Z ) k .Then | R | = p a +2 k = ( p r ) n for some n by Proposition 4.1. Thus r | a + 2 k .Let I be the ideal I = { s ∈ R : ps = 0 } ; it is proper since R hascharacteristic p . Then I + ∼ = ( Z /p Z ) a ⊕ ( Z /p Z ) k and | I | = p a + k . Moreover, I ⊆ m and ( R/I, m /I ) is again a finite commutative local ring with residuefield R/ m . So | R/I | = ( p r ) m for some m by Proposition 4.1. Thus | I | = | R || R/I | = p rn p rm = p r ( n − m ) = p a + k and so r | a + k . We conclude that r | a + k ) − ( a + 2 k ) = a . (cid:3) The following theorem is presumably well known, but we could not finda reference.
Theorem 4.3.
Let ( R, m ) be a finite commutative local ring with R/ m ofcharacteristic p and | R/ m | = p r . Let R + ∼ = ( Z /p Z ) a ⊕ ( Z /p Z ) a ⊕ · · · ⊕ ( Z /p t Z ) a t . Then r | a i for ≤ i ≤ s .Proof. We already have that r | a by Proposition 4.2. Assume that, in-ductively, r | a , . . . , a k with 1 ≤ k < t . Let I be the ideal { s ∈ R : p k s = 0 } . It’s a proper ideal since k < t and so I is contained in m .Thus ( R/I, m /I ) is a finite commutative ring with residue field R/ m . But( R/I ) + ∼ = ( Z /p Z ) a k +1 ⊕ · · · ⊕ ( Z /p t − k Z ) a t and so r | a k +1 by Proposition 4.2.The result follows by induction. (cid:3) Let p be a prime and m, r ≥
1. Then there is a unique (up to isomorphism)finite commutative ring R = GR ( p m , r ) of characteristic p m , called a Galoisring , such that | R | = p mr and R/pR ∼ = F p r . It is a local ring with maximalideal pR . We do not prove the uniqueness of Galois rings, as we do not needit. The interested reader is referred to [BF02] for details. AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 15
Theorem 4.4.
Let p be a prime and m, r ≥ . Then there is a finitecommutative local ring R = GR ( p m , r ) with maximal ideal pR such that R/pR ∼ = F p r and R + ∼ = ( Z /p m Z ) r .Proof. First we give an elementary construction. Let α be a primitive el-ement of F p r , so that F p r = F p ( α ). Let q ( x ) ∈ ( Z /p Z )[ x ] be the minimalpolynomial of α ; it is a monic polynomial of degree r . We can choose amonic polynomial Q ( x ) ∈ ( Z /p m Z )[ x ] of degree r which reduces to q ( x )by simply identifying the coefficients of q ( x ) with integers between 0 and p −
1. Set R = ( Z /p m Z )[ x ] / ( Q ( x )). First note that since Q ( x ) is monic, itfollows that ( Q ( x )) consists of polynomials of degree at least r . Thus thecosets 1+( Q ( x )) , x +( Q ( x )) , . . . , x r − +( Q ( x )) are linearly independent over Z /p m Z . Also, as Q ( x ) is monic, x r + ( Q ( x )) is in the Z /p m Z -span of thecosets x j + ( Q ( x )) with 0 ≤ j ≤ r − x t + ( Q ( x )) with t ≥ r . We conclude that R + ∼ = ( Z /p m Z ) r . Also pR is a nilpotent ideal of R , as ( pR ) m = p m R = 0, and so pR is contained inevery maximal ideal of R . But R/pR ∼ = ( Z /p Z )[ x ] / ( q ( x )) ∼ = F p r is a field,and so pR is a maximal ideal and hence the unique maximal ideal of R . Thiscompletes the proof.We now give a more conceptual construction. Let Q p be the p -adic ra-tionals and Z p the p -adic integers. Take ζ to be a primitive p r − Q p . Then Q p ( ζ ) is the unique unramifiedextension of Q p of degree r . The ring of integers in Q p ( ζ ) is O = Z p [ ζ ].Moreover, O is a complete discrete valuation ring with maximal ideal p O and residue field O /p O = F p r . Details can be found in [FT93, Chpt. III,Thm. 25] and [FT93, Equation (3.3), Page 136]. As a Z p -module, O is iso-morphic to ( Z p ) r and thus R = O /p m O is a finite commutative local ringwith additive group isomorphic to ( Z /p m Z ) r and maximal ideal pR with R/pR ∼ = F p r . (cid:3) Finally, we are ready to classify which finite abelian groups can be theadditive group of a ring with two units whose difference is a unit.
Theorem 4.5.
Let A be a finite abelian group. Then there is a finite com-mutative ring R with R + ∼ = A and two elements a, b ∈ R × with a − b ∈ R × if and only if A ∼ = A ⊕ A where A has odd order and A ∼ = ( Z / Z ) a ⊕ ( Z / Z ) a ⊕ · · · ⊕ ( Z / t Z ) a t with a i = 1 for all ≤ i ≤ t .Proof. Note that if the ring R decomposes as R ∼ = R × · · · × R n , then thereexist a and b in R × with a − b also in R × if and only if there exist a i and b i in R × i with a i − b i in R × for all 1 ≤ i ≤ n . Therefore, to show existenceit suffices to find Z /p k Z for an odd prime p and ( Z / m Z ) r for r ≥ m ≥ For the odd prime case, Z /p k Z , considered as a ring in the standard way,already serves our purpose since 1 , − − ( −
1) are all units in thisring.For the even case, for r ≥
2, let R = GR (2 m , r ) as per Theorem 4.4. Then R + ∼ = ( Z / m Z ) r , R is local with maximal ideal 2 R and R/ R ∼ = F r . Sowe can choose ¯ a and ¯ b non-zero in R/ R with ¯ a − ¯ b = 0. Let a and b bepre-images of ¯ a and ¯ b in R . Then a , b , and a − b are all units of R .We now show that no other finite abelian groups can be the additive groupof a ring with two units whose difference is a unit. Suppose A = A ⊕ A with A and A as in the statement of the theorem. Suppose R is a ring with A ∼ = R + . Write R = R × · · · × R n with R i local for 1 ≤ i ≤ n . If a i = 1for some i , then there exists an R j of even order with Z / i Z as a directsummand in R + j with multiplicity 1. Then by Theorem 4.3, the residue fieldof R j must be F . Let m j be the maximal ideal of R j . If a = ( a , . . . , a n )and b = ( b , . . . , b n ) are in R × then a j + m j = b j + m j as the residue field of R j is F . Thus a j − b j ∈ m j and is not a unit. Consequently, a − b is not aunit. So R does not have two units whose difference is a unit. (cid:3) Corollary 4.6.
Let A ∼ = A ⊕ A be a finite abelian group where A hasodd order and A ∼ = ( Z / Z ) a ⊕ ( Z / Z ) a ⊕ · · · ⊕ ( Z / t Z ) a t with a i = 1 forall ≤ i ≤ t . Then there is a bireversible automaton over an | A | -elementalphabet with | A | states generating a group isomorphic to A ≀ Z .Remark . Note that if n is even, then there is no finite commutative ring R with R + ∼ = Z /n Z containing two units a, b with a − b a unit. So we cannotrealize the lamplighter group ( Z /n Z ) ≀ Z using bireversible automata via ourmethods when n is even. 5. Examples
Recall that for a fixed r , a , and b in our finite commutative ring R , thestates of A f , for f = r (cid:18) − at − bt (cid:19) , are { α − sra µ f α sb | s ∈ R } . If a − b is a unit of R , then different s ∈ R yielddifferent states by Lemma 3.8. For this reason, in all of the following figuresand tables we use s to denote the state α − sra µ f α sb . With this notation,transitions are given as in (3.4). Example . As a first example we show that the power series method can beused to recreate the bireversible automaton given in [BDR16] for the group Z / Z ≀ Z . This was already observed by Bondarenko and Savchuk in theirwork on rational series over fields. We use f ( t ) = 2 (cid:16) − t − t (cid:17) in ( Z / Z ) J t K .With r = 2 , a = 2, and b = 1, we see that the states of A f are of the form α − s µ f α s . The output function is given by λ ( s, ˜ s ) = − s + 2( s + ˜ s ) and the AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 17 transition function is given by δ ( s, ˜ s ) = α − ( s +˜ s ) µ f α s +˜ s for any ˜ s ∈ Z / Z .As was observed by Bondarenko, D’Angeli, and Rodaro, this automaton isequivalent to its dual. See Figure 1. | | | | | | | | | Figure 1.
The bireversible automata given by Bodarenko,D’Angeli, Rodaro for Z / Z ≀ Z and corresponding to f ( t ) =2 (cid:18) − t − t (cid:19) . Example . Let R = Z / Z with the standard ring structure and take a = 3, b = 2, and r = 1 so that neither a nor b is a unit. Therefore, for f = 1 − t − t , both A f and A f − are not reversible by Theorem 3.9. Here, thestates are given by α s µ f α s . The transition function is given by δ ( α s µ f α s , ˜ s ) = α s +˜ s ) µ f α s +˜ s ) = α s µ f α s +˜ s and the output function is given by λ ( α s µ f α s , ˜ s ) = ˜ s + 5 s = ˜ s − s. It is straightforward to check that δ x ( α s µ f α s ) = δ x ( α s ′ µ f α s ′ ) whenever s ≡ s ′ mod 3, which verifies that A f is not reversible. See Figure 2. Example . Our next example is that of a bireversible automaton thatgenerates A ≀ Z where A is the additive group of a ring but not of a field. Wetake A = Z / Z endowed with the standard ring structure. Taking r = 2, a = 1, and b = 2, we have that a − b = −
1, and so a , b , and a − b are allunits. For these values of r , a , and b , the states of A f are α − s µ f α s . Thetransition and output functions are given by δ ( α − s µ f α s , ˜ s ) = α − s +˜ s ) µ f α s +˜ s )8 R. SKIPPER AND B. STEINBERG | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Figure 2.
An automaton which generates Z / Z ≀ Z that isnot reversible and whose inverse is also not reversible.and λ ( α − s µ f α s , ˜ s ) = 2( s + ˜ s ) . Rather than drawing the A f for f = 2 (cid:18) − t − t (cid:19) which has 9 states anda 9 letter alphabet, we describe the transition and output functions usingTables 1 and 2. state \ letter 0 1 2 3 4 5 6 7 80 0 1 2 3 4 5 6 7 81 2 3 4 5 6 7 8 0 12 4 5 6 7 8 0 1 2 33 6 7 8 0 1 2 3 4 54 8 0 1 2 3 4 5 6 75 1 2 3 4 5 6 7 8 06 3 4 5 6 7 8 0 1 27 5 6 7 8 0 1 2 3 48 7 8 0 1 2 3 4 5 6 Table 1.
The transition table for f = 2 (cid:18) − t − t (cid:19) over Z / Z . Example . As a final example, we construct a bireversible automatongenerating ( Z / Z ) ≀ Z using the ring described in Theorem 4.4. In otherwords, we take O = Z [ ζ ] with ζ a third root of unity and R = O / O ∼ = AMPLIGHTER GROUPS, BIREVERSIBLE AUTOMATA AND RATIONAL SERIES 19 state \ letter 0 1 2 3 4 5 6 7 80 0 2 4 6 8 1 3 5 71 2 4 6 8 1 3 5 7 02 4 6 8 1 3 5 7 0 23 6 8 1 3 5 7 0 2 44 8 1 3 5 7 0 2 4 65 1 3 5 7 0 2 4 6 86 3 5 7 0 2 4 6 8 17 5 7 0 2 4 6 8 1 38 7 0 2 4 6 8 1 3 5 Table 2.
The output table for f = 2 (cid:18) − t − t (cid:19) over Z / Z . Z / Z [ ζ ]. Using the isomorphism R ∼ = ( Z / Z [ x ]) / (1 + x + x ) and taking r = 1, a = 1, and b = 2 + ζ , we find that a , b , and a − b are all units withinverses 1, 3 + ζ , and ζ respectively. With this choice of r , a , and b , thetransition and output tables are given by Tables 3 and 4 respectively for f = r (cid:18) − at − bt (cid:19) . R . S K I PPE R AN D B . S T E I N B E R G state \ letter 0 1 2 3 ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ Table 3.
The transition table for f = r (cid:18) − at − bt (cid:19) with r = 1, a = 1, and b = 2 + ζ . A M P L I G H T E R G R O U P S , B I R E V E R S I B L E AU T O M A T AAN D R A T I O NA L S E R I E S state \ letter 0 1 2 3 ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ Table 4.
The output table for f = r (cid:18) − at − bt (cid:19) with r = 1, a = 1, and b = 2 + ζ . Acknowledgments.
We would like to thank Marcin Mazur for his manyhelpful suggestions.
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Department of Mathematics, The Ohio State University, Columbus, OH43210
E-mail address : [email protected] Department of Mathematics, City College of New York, New York, NY10031
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