Algorithms and topology for Cayley graphs of groups
aa r X i v : . [ m a t h . G R ] J u l ALGORITHMS AND TOPOLOGY OF CAYLEY GRAPHSFOR GROUPS
MARK BRITTENHAM, SUSAN HERMILLER, AND DEREK HOLT
Abstract.
Autostackability for finitely generated groups is defined viaa topological property of the associated Cayley graph which can be en-coded in a finite state automaton. Autostackable groups have solvableword problem and an effective inductive procedure for constructing vanKampen diagrams with respect to a canonical finite presentation. Acomparison with automatic groups is given. Another characterizationof autostackability is given in terms of prefix-rewriting systems. Ev-ery group which admits a finite complete rewriting system or an asyn-chronously automatic structure with respect to a prefix-closed set ofnormal forms is also autostackable. As a consequence, the fundamentalgroup of every closed 3-manifold with any of the eight possible uniformgeometries is autostackable. Introduction
A primary motivation for the definition of the class of automatic groupsis to make computing the word problem for 3-manifold groups tractable;however, in their introduction of the theory of automatic groups, Epstein, et. al. [10] showed that the fundamental group of a closed 3-manifold havingNil or Sol geometry is not automatic. Brady [1] showed that there are Solgeometry groups that do not belong to the wider class of asynchronouslyautomatic groups. Bridson and Gilman [4] further relaxed the languagetheoretic restriction on the associated normal forms, replacing regular withindexed languages, and showed that every 3-manifold has an asynchronouscombing with respect to an indexed language. More recently, Kharlam-povich, Khoussainov, and Miasnikov [22] have defined the class of Cayleyautomatic groups, extending the notion of an automatic structure (preserv-ing the regular language restriction), but it is as yet unknown whether all Niland Sol 3-manifold groups are Cayley automatic. In this paper we define thenotion of autostackability for finitely generated groups using properties veryclosely related to automatic structures, that holds for 3-manifold groups ofall uniform geometries.Let G be a group with an inverse-closed finite generating set A , and letΓ = Γ( G, A ) be the associated Cayley graph. Let ~E be the set of directed Mathematics Subject Classification . 20F65; 20F10, 68Q42. edges; for each g ∈ G and a ∈ A , let e g,a denote the directed edge of Γwith initial vertex g , terminal vertex ga , and label a . Let N ⊂ A ∗ be aset of normal forms for G over A ; for each g ∈ G , we denote the normalform word representing g by y g . Note that whenever we have an equality ofwords y g a = y ga or y g = y ga a − , then there is a van Kampen diagram forthe word y g ay − ga that contains no 2-cells; in this case we call the edge e g,a degenerate . Let ~E N ,d = ~E d be the set of all degenerate directed edges, andlet ~E N ,r = ~E r := ~E \ ~E d ; we refer to elements of ~E r as recursive edges. Definition 1.1.
A group G with finite inverse-closed generating set A is autostackable if there are a set N of normal forms for G over A containingthe empty word, a constant k , and a function φ : N × A → A ∗ such that thefollowing hold: (1) The graph of the function φ , graph ( φ ) := { ( y g , a, φ ( y g , a )) | g ∈ G, a ∈ A } , is a synchronously regular language. (2) For each g ∈ G and a ∈ A , the word φ ( y g , a ) has length at most k and represents the element a of G , and: (2d) If e g,a ∈ ~E N ,d , then the equality of words φ ( y g , a ) = a holds. (2r) The transitive closure < φ of the relation < on ~E N ,r , defined by e ′ < e g,a whenever e g,a , e ′ ∈ ~E N ,r and e ′ is on the directedpath in Γ labeled φ ( y g , a ) starting at the vertex g is a strict well-founded partial ordering. Removing the algorithmic property in (1), the group G is called stackable over the inverse-closed generating set A if property (2) holds for some normalform set N (containing λ ), constant k , and function φ : N × A → A ∗ . In [6],the first two authors define and study the class of stackable groups. In [6,Lemma 1.5] they show that stackability implies that the finite set R c of wordsof the form φ ( y g , a ) a − (for g ∈ G and a ∈ A ) is a set of defining relators for G , and the set N of normal forms is closed under taking prefixes. Hence theset N uniquely determines a maximal tree in the Cayley graph Γ, consistingof the edges that lie on paths labeled by words in N .This leads to a topological description of the concept of autostackability.Let T be a maximal tree in Γ. For each g ∈ G and a ∈ A , we view the twodirected edges e g,a and e ga,a − of Γ to have a single underlying undirectededge in Γ. Let ~P be the set of all finite length directed edge paths in Γ.A flow function associated to T is a function Φ : ~E → ~P satisfying theproperties that:(a) For each edge e ∈ ~E , the path Φ( e ) has the same initial and terminalvertices as e .(b-d) If the undirected edge underlying e lies in the tree T , then Φ( e ) = e . LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 3 (b-r) The transitive closure < Φ of the relation < on ~E , defined by e ′ < e whenever e ′ lies on the path Φ( e ) and the undirectededges underlying both e and e ′ do not lie in T ,is a strict well-founded partial ordering.That is, the map Φ fixes the edges lying in the tree T and describes a “flow”of the non-tree edges toward the tree (or toward the basepoint). A flowfunction is bounded if there is a constant k such that for all e ∈ ~E , the pathΦ( e ) has length at most k .For each element g ∈ G , let y g be the unique word labeling a geodesic pathin the tree T from the identity element 1 of G to g , and let N T := { y g | g ∈ G } be the corresponding set of normal forms. Let β T : N T × A → ~E denotethe natural bijection defined by β T ( y g , a ) := e g,a , and let ρ : ~P → A ∗ be thefunction that maps each directed path to the word labeling that path in Γ.The composition ρ ◦ Φ ◦ β T : N × A → A ∗ is part of a stackable structurefor G over A , which we call the induced stacking function .Conversely, [6,Lemma 1.5] implies that given a stacking function φ : N × A → A ∗ from astackable structure, there is an induced flow function Φ : ~E → ~P , such thatΦ( e g,a ) is the path in Γ starting at the vertex g labeled by the word φ ( y g , a ).Thus we have the following characterizations. Proposition 1.2.
Let G be a group with a finite inverse-closed generatingset A . (1) The group G is stackable over A if and only if the Cayley graph Γ( G, A ) admits a maximal tree with an associated bounded flow function.(2) The group G is autostackable over A if and only if there exists a maximaltree in Γ( G, A ) with a bounded flow function such that the graph of theinduced stacking function is synchronously regular. In Section 2 of this paper, we give definitions and notation, and discussbackground on normal forms, van Kampen diagrams, and language theory.Section 3 contains a comparison of the definitions for autostackable groupsversus automatic groups. We contrast word problem solutions and van Kam-pen diagram constructions for these two classes of groups. In analogy withthe relationship between autostackable and stackable groups above, remov-ing the algorithmic Property (i) of Definition 3.1 of automaticity yields thedefinition of combable groups. We show how to modify the proof of [6,Propositions 1.7,1.12] to show the following.
Proposition 3.3.
Autostackable groups are finitely presented, have solv-able word problem, and admit a recursive algorithm to build a van Kampendiagram for each word representing the identity element.
The class of automatic groups is strictly contained in the class of asyn-chronously automatic groups; in Section 4, we consider this larger class.
Theorem 4.1.
Every group that has an asynchronously automatic structurewith a prefix-closed normal form set is autostackable.
M. BRITTENHAM, S. HERMILLER, AND D. HOLT
We note that although Epstein et. al. [10, Theorems 2.5.1,5.5.9] haveshown that every automatic group has an automatic structure with re-spect to a set of normal forms, and also an automatic structure with re-spect to a prefix-closed set of not necessarily unique representatives, it isan open problem [10, Open Question 2.5.20] whether there must be anautomatic structure on a prefix-closed set of normal forms. Gilman hasgiven other characterizations of groups that are automatic with respect toa prefix-closed normal form set in [12]. Groups known to have an auto-matic structure with respect to prefix-closed normal forms include finitegroups [10], virtually abelian (and hence Euclidean) groups and word hy-perbolic groups [10], Coxeter groups [5], Artin groups of finite type [7] andof large type [28],[19], and small cancellation groups satisfying conditions C ′′ ( p ) − T ( q ) for ( p, q ) ∈ { (3 , , (4 , , (6 , } [21]. The class of automaticgroups with respect to prefix-closed normal forms is closed under graphproducts [15, Theorem B] and finite extensions [10, Theorem 4.1.4].In Section 5, we give a purely algorithmic characterization of autostacka-bility, using another type of word problem solution, namely ‘prefix-sensitiverewriting’. A convergent prefix-rewriting system for a group G consists of afinite set A together with a subset R ⊂ A ∗ × A ∗ such that as a monoid, G ispresented by G = M on h A | u = v whenever ( u, v ) ∈ R i , and the rewritingoperations of the form uz → vz for all ( u, v ) ∈ R and z ∈ A ∗ satisfy: • Normal forms:
Each g ∈ G is represented by exactly one irreducible word (i.e. word that cannot be rewritten) over A . • Termination:
There does not exist an infinite sequence of rewritings x → x → x → · · · .A prefix-rewriting system is bounded if there exists a constant k such thatfor each pair ( u, v ) ∈ R , there are words s, t, w ∈ A ∗ with s and t of lengthat most k such that u = ws and v = wt . Theorem 5.3.
Let G be a finitely generated group.(1) The group G is stackable if and only if G admits a bounded convergentprefix-rewriting system.(2) The group G is autostackable if and only if G admits a synchronouslyregular bounded convergent prefix-rewriting system. As part of the proof of Theorem 5.3, in Proposition 5.2, we show that givenany synchronously regular bounded convergent prefix-rewriting system R for G , there is a subset Q ′ of R that is a synchronously regular bounded prefix-rewriting system for G such that for every ( u, v ) ∈ Q ′ , every proper prefix of u is irreducible over R , and no two distinct word pairs in Q ′ have the sameleft hand side.In contrast to these results, Otto [27, Corollary 5.3] has shown that agroup is automatic with respect to a prefix-closed set of normal forms over amonoid generating set A if and only if there exists a synchronously regular LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 5 convergent prefix-rewriting system such that for every ( u, v ) ∈ R , the word v is irreducible over R , and the word u is irreducible over all of the otherrewriting rules of R .Synchronously regular bounded convergent prefix-rewriting systems are ageneralization of the more widely studied concept of finite convergent (alsocalled complete) rewriting systems, which admit rewriting operations of theform wuz → wvz whenever ( u, v ) ∈ R and w, z ∈ A ∗ . Thus Theorem 5.3yields: Corollary 5.4.
Every group that admits a finite convergent rewriting systemis autostackable.
Groups known to have a finite convergent rewriting system include finitegroups, alternating knot groups [8], surface groups [23], virtually abeliangroups, polycyclic groups, and more generally constructible solvable groups [13],Coxeter groups of large type [14], and Artin groups of finite type [16] (seealso Le Chenadec’s [24] text for many more examples). This class of groupsis closed under graph products [15], extensions [13],[16], and certain amal-gamated products and HNN extensions [13].The iterated Baumslag-Solitar groups presented by h a , a , ..., a k | a a = a , ..., a a k k − = a k − i were shown by Gersten [11, Section 6] to have Dehnfunction asymptotic to a k-fold iterated exponential function, and also tohave a finite convergent rewriting system (see [17] for details). The followingis then an immediate consequence of the results above. Corollary 1.3.
The class of autostackable groups includes groups whoseDehn functions’ growth is asymptotically equivalent to an iterated exponen-tial function with arbitrarily many iteration steps.
This result is in strong contrast to the quadratic upper bound on the Dehnfunction for any automatic group [10, Theorem 2.3.12].Miller [26, p. 31] has shown that there exists a split extension of a finitelygenerated free group by another finitely generated free group that has un-solvable conjugacy problem. Since free groups admit finite complete rewrit-ing systems, the results above also give the following.
Corollary 1.4.
The class of autostackable groups includes groups with un-solvable conjugacy problem.
Finally, we return to the motivation of computing the word problemin 3-manifold groups. In [18], Hermiller and Shapiro show that if M isa closed 3-manifold with uniform geometry that is not hyperbolic, then π ( M ) has a finite convergent rewriting system. On the other hand, Epsteinet. al. [10] show that every word hyperbolic group, and hence every hyper-bolic 3-manifold fundamental group, is automatic with respect to a shortlex,and hence prefix-closed, set of normal forms. Hence we obtain the following. M. BRITTENHAM, S. HERMILLER, AND D. HOLT
Corollary 1.5.
Every fundamental group of a closed 3-manifold with uni-form geometry is autostackable. Notation and background
Throughout this paper, let G be a group with a finite generating set A that is closed under inversion, and let Γ be the associated Cayley graph.Let A ∗ denote the free monoid, i.e. the set of all finite words over A , and let π : A ∗ → G denote the canonical surjection. Whenever u and v lie in theset A ∗ of all words over A , we write u = v if u and v are the same word,and u = G v if u and v represent the same element of G ; i.e., if π ( u ) = π ( v ).Let 1 denote the identity element of G and let λ denote the empty word in A ∗ ; then π ( λ ) = 1.Given a word w ∈ A ∗ , let l ( w ) denote the length of w as a word over A .For each a ∈ A , the symbol a − represents another element of A , and so foreach word u = a · · · a m in A ∗ with each a i in A , there is a formal inverseword u − := a − m · · · a − in A ∗ .2.1. Normal forms and van Kampen diagrams.
A set N of normalforms for G over A is a subset of the set A ∗ such that the restriction of thecanonical surjection π : A ∗ → G to N is a bijection. As in Section 1, thesymbol y g denotes the normal form for g ∈ G ; by slight abuse of notation,we use the symbol y w to denote the normal form for π ( w ) whenever w ∈ A ∗ .Given a set R of defining relators for a group G , so that P = h A | R i is apresentation for G , then for an arbitrary word w in A ∗ that represents theidentity element 1 of G , there is a van Kampen diagram (or Dehn diagram)∆ for w with respect to P . That is, ∆ is a finite, planar, contractiblecombinatorial 2-complex with edges directed and labeled by elements of A ,satisfying the properties that the boundary of ∆ is an edge path labeled bythe word w starting at a basepoint vertex ∗ and reading counterclockwise,and every 2-cell in ∆ has boundary labeled (in some orientation) by anelement of R . See [3] or [25] for more details on the theory of van Kampendiagrams.Let N be a set of normal forms for G over A such that each word w ∈N labels a simple path. For example, this property holds if N is closedunder taking prefixes of words. The “seashell” method to construct a vanKampen diagram (with respect to the presentation G = h A | R i ) for anyword w = b · · · b n ∈ A ∗ that represents the identity of G is as follows. Foreach i we denote the normal form word y i := y b ··· b i . Let ∆ i be a van Kampendiagram for the word y i − b i y − i . By successively gluing these diagrams alongthe simple normal form paths along their boundaries, we obtain a planarvan Kampen diagram for w ; see Figure 1 for an idealized picture. (See forexample [10], [2], or [6] for more details.) LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 7 i ... . . . * y i y i-1 bbbb bbb b ... ... i Figure 1.
Van Kampen diagram built with seashell method2.2.
Regular languages.
For more details and proofs of the material inthis subsection, we refer the reader to [10] or [20].A language over a finite set A is a subset of the set A ∗ of all finite wordsover A . We also refer to subsets of ( A ∗ ) n as languages over A n . The set A + denotes the language A ∗ \ { λ } of all nonempty words over A , and A ≤ k denotes the finite language of all words over A of length at most k .The regular languages over A are the subsets of A ∗ obtained from thefinite subsets of A ∗ using finitely many operations from among union, in-tersection, complement, concatenation ( S · T := { vw | v ∈ S and w ∈ T } ),and Kleene star ( S := { λ } , S n := S n − · S and S ∗ := ∪ ∞ n =0 S n ). A finitestate automaton , or FSA, is a 5-tuple M := ( A, Q, q , P, δ ), where Q is afinite set called the set of states , q ∈ Q is the initial state , P ⊆ Q is theset of accept states , and δ : Q × A → Q is the transition function . The map δ extends to a function (often given the same label) δ : Q × A ∗ → Q byrecursively defining δ ( q, wx ) := δ ( δ ( q, w ) , x ) whenever q ∈ Q , w ∈ A ∗ , and x ∈ A . A word w ∈ A ∗ is in the language accepted by M if and only if thestate δ ( q , w ) lies in the set P . A language L over A is regular if and onlyif L is the language accepted by a finite state automaton.The class of regular languages is closed under both image and preimagevia monoid homomorphisms (see, for example, [20, Theorem 3.5]). The classof regular sets is also closed under quotients (see [20, Theorem 3.6]); we writeout a special case of this in the following lemma for use in later sections ofthis paper. Lemma 2.1. [20, Theorem 3.6] ) If A is a finite set, L ⊆ A ∗ is a regularlanguage, and w ∈ A ∗ , then the quotient language L/w := { x ∈ A ∗ | xw ∈ L } is also a regular language. Let $ be a symbol not contained in A . The set A n := ( A ∪{ $ } ) n \{ ($ , ..., $) } is the padded n -tuple alphabet derived from A . For any n -tuple of words u =( u , ..., u n ) ∈ ( A ∗ ) n , write u i = a i, · · · a i,j i with each a i,m ∈ A for 1 ≤ i ≤ n and 1 ≤ m ≤ j i . Let M := max { j , ..., j n } , and define ˜ u i := u $ M − j i , so that M. BRITTENHAM, S. HERMILLER, AND D. HOLT each of ˜ u , ..., ˜ u n has length M . That is, ˜ u i is a word over the alphabet( A ∪ { $ } ) ∗ , and we can write ˜ u i = c i, · · · c i,M with each c i,m ∈ A ∪ { $ } .The word µ ( u ) := ( c , , ..., c n, ) · · · ( c ,M , ..., c n,M ) is the padded word overthe alphabet A n induced by the n -tuple ( u , ..., u n ) in ( A ∗ ) n .A subset L ⊆ ( A ∗ ) n is called synchronously regular if the padded extension set µ ( L ) := { µ ( u ) | u ∈ L } of padded words associated to the elements of L isa regular language over the alphabet A n . The class of synchronously regularlanguages is closed under finite unions and intersections, since the paddedextension of a union [resp. intersection] is the union [resp. intersection] of thepadded extensions. We also include two lemmas on synchronously regularlanguages for use in later sections. The first lemma says that the “diagonal”of a regular set is regular. Lemma 2.2. If L is a regular language over an alphabet A , then the set ∆( L ) := { µ ( w, w ) | w ∈ L } is a regular language over the alphabet A =( A ∪ $) \ { ($ , $) } .Proof. Given an expression of the regular language L using letters of A together with the operations ∪ , ∩ , ( ) c , · , ( ) ∗ , replace every instance of aletter a ∈ A with the letter ( a, a ) ∈ A . (cid:3) Lemma 2.3. If L , ..., L n are regular languages over A , then their Cartesianproduct L × · · · × L n ⊆ ( A ∗ ) n is synchronously regular.Proof. For each 1 ≤ i ≤ n define the monoid homomorphism ρ i : A ∗ n → ( A ∪ $) ∗ by ρ i ( a , ..., a n ) := a i . Then the padded extension of the productlanguage L := L × · · · × L n satisfies µ ( L ) = ∩ ni =1 ρ − i ( L i $ ∗ ). Since each lan-guage L i $ ∗ is regular, and regular languages are closed under homomorphicpreimage and finite intersection, then µ ( L ) is regular. (cid:3) A (deterministic) asynchronous (two tape) automaton over A is a finitestate automaton M = ( A ∪ { } , Q, q , P, δ ) satisfying: (1) The state set Q isa disjoint union Q = Q ∪ Q ∪ Q ∪ Q ∪{ q f }∪{ F } of six subsets, the initialstate q lies in Q ∪ Q , and the set of accept states is P = { q f } . (2) Thetransition function δ : Q × ( A ∪ { } ) → Q satisfies δ ( q, a ) ∈ Q ∪ Q ∪ { F } if q ∈ Q ∪ Q and a ∈ A ; δ ( q, a ) ∈ Q ∪ { F } if either ( q ∈ Q and a = q ∈ Q and a ∈ A ); δ ( q, a ) ∈ Q ∪ { F } if either ( q ∈ Q and a = q ∈ Q and a ∈ A ); δ ( q, a ) ∈ { q f , F } if q ∈ Q ∪ Q and a = δ ( q, a ) = F if q = F and a ∈ A ∪ { } . As before, extend δ to a function δ : Q × ( A ∪ { } ) ∗ → Q recursively by δ ( q, wa ) := δ ( δ ( q, w ) , a ).This finite state automaton is viewed as reading from two tapes ratherthan one, by the interpretation that the words on each tape are to havean ending symbol M is in a state in Q i ∪ Q i , then M will read the next symbol from tape i . Then the automatonis in a state of Q i after M has finished reading the word on the other tape. LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 9
More precisely, given a pair of words ( u, v ) ∈ (( A ∪ { } ) ∗ ) , a shuffle of( u, v ) is a word u v · · · u j v j ∈ ( A ∪ { } ) ∗ such that each u i , v i ∈ ( A ∪ { } ) ∗ , u = u · · · u j , and v = v · · · v j . Let ( u, v ) ∈ (( A ∪ { } ) ∗ ) , and write u = a , · · · a ,m and v = a , · · · a ,m where each a i,j ∈ A ∪ { } . Given a state q ∈ Q and the pair ( u, v ), there is a unique word σ M,q ( u, v ) := c · · · c m + n ∈ ( A ∪ { , F } ) ∗ defined recursively, such that c := a i, if q ∈ Q i ∪ Q i (and1 ≤ m i ) and c := F if q ∈ { q f , F } , and whenever k ≤ m + n −
1, if c · · · c k is a shuffle of ( a , · · · a ,k , a , · · · a ,k ) with δ ( q, c · · · c k ) = q ′ ,then c k +1 := a i,k i +1 if q ′ ∈ Q i ∪ Q i (and k i < m i ) and c k +1 := F if q ′ ∈ { q f , F } ; and if c k = F then c k +1 := F .A pair ( u, v ) ∈ ( A ∗ ) is accepted by the asynchronous automaton M if andonly if σ M,q ( u , v u , v F . (Equivalently, ( u , u ) is in the language of M if and only if themachine M reads the next letter from the u i M is in a stateof Q i ∪ Q i , M starts in state q , and M ends in state q f when both tapeshave been read.) A subset of A ∗ × A ∗ is an asynchronously regular languageif it is the set of word pairs accepted by an asynchronous automaton.Again we include a closure property for asynchronously regular languagesfor later use. This result is proved by Rabin and Scott in [29, Theorem 16]. Lemma 2.4. [29] If L ⊂ ( A ∗ ) is an asynchronously regular language, thenthe projection on the first coordinate given by the set ρ ( L ) := { u | ∃ ( u, v ) ∈ L } is a regular language over A . Autostackable versus automatic: Word problems and vanKampen diagrams
We give a definition of automatic structures for groups that is equivalentto, but differs from, the original definition in [10], in order to illustrate morecompletely the close connection to Definition 1.1 of autostackable structuresabove. Both automaticity and autostackability utilize the concepts of a set N of normal forms for a group G over a generating set A , but in contrast tothe stacking function φ for autostackability which has a finite image set, thedefinition of automaticity relies on the normal form map nf N : N × A → A ∗ defined by nf N ( y g , a ) := y ga . Definition 3.1.
A group G with finite inverse-closed generating set A is automatic if there are a set N of normal forms for G over A and a constant k such that the following hold: (i) The graph of the function nf N : N × A → A ∗ , graph (nf N ) := { ( y g , a, y ga ) | g ∈ G, a ∈ A } , is a synchronously regular language. (ii) For each g ∈ G and a ∈ A , the pair of paths in Γ labeled y g and y ga beginning at the identity vertex and ending at the endpoints of e g,a must k -fellow travel ; that is, for any natural number i , if w and w ′ are the length i prefixes of the words y g and y ga , then there must bea path in Γ of length at most k between the vertices of Γ labeled by w and w ′ . In fact, the definition of automaticity given in [10, Defn. 2.3.1,Thm. 2.5.1]requires only property (i) above; indeed, in [10, Thm. 2.3.5,Thm. 3.3.4] Ep-stein et. al. show that the geometric property (ii) follows from the algorith-mic property (i). Moreover, it is immediate from the properties of regularlanguages discussed in Section 2.2 that the set graph (nf N ) is a synchronouslyregular language if and only if the sets L a := { ( y g , y ga ) | g ∈ G } ⊂ ( A ∗ ) are synchronously regular for each a ∈ A ∪ { λ } , giving the equivalence ofproperty (i) above with the definition in [10].Comparing Definitions 1.1 and 3.1, the automatic property (i) requires afinite state automaton that can recognize the tuple ( y g , a, z ) where the thirdcoordinate is the normal form z = y ga , but the autostackable property (1)requires only a FSA that recognizes such a tuple in which z is a boundedlength word giving information toward eventually finding the normal form y ga . (We make this more precise below.) In analogy with the autostackableproperty (2) of Definition 1.1, the automatic group property (ii) naturallydivides into degenerate and recursive cases, in that if the directed edge e g,a is degenerate, we have the stronger property that the paths y g , y ga G is called combable over A ifthe geometric property (ii) of Definition 3.1 holds for some set N of normalforms and some constant k . Note that combability, and hence also auto-maticity, imply finite presentability; in particular, the set R of all words oflength up to 2 k + 2 that represent the identity are a set of defining relatorsfor the group.If G is a combable group satisfying the further property that the words ofthe normal form set N label simple paths in the Cayley graph Γ, for examplein the case that N is closed under taking prefixes, then the “seashell” methoddiscussed in Section 2.1 extends to the following procedure to constructa van Kampen diagram (with respect to the presentation induced by thecombable structure) for any word that represents the identity of G . Givena word w = b · · · b n representing the identity of G , with each b i ∈ A , let y i := y b ··· b i for each i . Property (ii) shows that for each i there is a vanKampen diagram ∆ i labeled by y i − b i y − i that is “ k -thin” as illustrated inFigure 2. Gluing these k -thin diagrams along their y i boundaries results ina planar van Kampen diagram for w ; see Figure 1. In the case that thegroup is automatic, this yields a solution of the word problem. (See [10] forfull details.) LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 11 yy <_ k<_ k * b i-1i i Figure 2. “ k -thin” van Kampen diagramIn [6], the first two authors of this paper show that every stackable group G also has a finite presentation and admits a procedure for building van Kam-pen diagrams. Moreover, in the case that the set S φ := { ( w, a, φ ( e π ( w ) ,a )) | w ∈ A ∗ , a ∈ A } obtained from the stacking function φ is decidable, theyshow that the procedure is an effective algorithm and the group has solv-able word problem. In part (1) of Definition 1.1 above, however, the syn-chronously regular (and hence recursive) set graph ( φ ) is a subset of S φ ,namely graph ( φ ) = S φ ∩ ( N × A × A ∗ ). We alter the stacking reduction pro-cedure of [6] to solve the word problem for autostackable groups as follows.For a group G with a stackable structure given by a set N of normal formsover a inverse-closed generating set A and a stacking function φ : N × A → A ∗ , the stacking reduction algorithm on words over A is a prefix-rewritingsystem given by R φ := { ( ya, yφ ( y, a )) | y ∈ N , a ∈ A, ya / ∈ N ∪ A ∗ a − a }∪ { ( yaa − , y ) | ya ∈ N , a ∈ A } . Recall that starting from any word w in A ∗ , whenever we can decompose w as w = ux for some rule ( u, v ) ∈ R φ and word x ∈ A ∗ , then we canrewrite w → vx . Each of these rewritings consists either of free reductionor φ -reduction . Lemma 3.2. If G is a group with inverse-closed generating set A and astackable structure consisting of a normal form set N and a stacking function φ , then the prefix-rewriting system R φ is a convergent prefix-rewriting systemfor G .Proof. Let w be any word in A ∗ , and write w = b · · · b m with each b i in A . Suppose that w ′ = c · · · c n , with each c j in A , is obtained from w byrepeated applications of free and φ -reductions, and that w ′ → w ′′ is a singleinstance of another R φ rewriting operation. If the rewriting w ′ → w ′′ is afree reduction, then two letters of w ′ are removed, and if this rewriting isa φ -reduction, then a single letter of w ′ is replaced by a bounded lengthword. Inductively this shows that each letter of the word w ′′ is the result ofsuccessive rewritings from a specific letter b i of the original word w . Viewingthis topologically, if the rewriting operation w ′ → w ′′ is free reduction, thenthe directed path in the Cayley graph Γ( G, A ) starting at 1 and labeled w ′′ is obtained from the path labeled by w ′ via the removal of two edges, and if the rewriting is φ -reduction, then a single edge e ′ of the w ′ path isreplaced by the path Φ( e ′ ), where Φ is the flow function induced by thestacking function φ . In the latter case, there is a specific recursive edge e i := e π ( b ··· b i − ) ,b i ∈ ~E N ,r for some index 1 ≤ i ≤ m on the path labeled w from 1 in Γ such that e ′ was obtained from e i via successive applications ofthe flow function, and for each recursive edge e ′′ along the path Φ( e ′ ), wehave e ′′ < φ e ′ , where < φ is the strict well-founded partial ordering given inDefinition 1.1(2r). Since at each application of the flow function a boundednumber of recursive edges are added to the path, K¨onig’s Infinity Lemma(see, for example, [9, Lemma 8.1.2]) shows that at most finitely many Φ-reductions can be applied starting from each of the finitely many edges ofthe original path labeled w . Hence only finitely many φ -reductions can beapplied in any sequence of rewritings starting from the word w . Betweenthese φ -reductions, only finitely many free reductions can occur. Hence afterfinitely many R φ rewriting operations, we must obtain an irreducible word y w , and so the prefix-rewriting system R φ is terminating.Now suppose that y is any irreducible word with respect to R φ . Write y = a · · · a n with each a i in A and y i := a · · · a i for each i , and supposethat y j is the shortest prefix of y that does not lie in N . Since the emptyword λ lies in the normal form set N of the stackable structure, we have j ≥
1. Now y j − ∈ N , and either y j − = y j − a − j , in which case a freereduction rule of R φ applies to y , or else y j − does not end with the letter a − j , in which case a φ -reduction rule applies to y . However, this contradictsthe irreducibility of y . Therefore every prefix of the word y , including theword y itself, must lie in N . Thus the set of irreducible words with respectto R φ is contained in the set N of normal forms for G .Next suppose that w is any word in the normal form set N . By thetermination proof above, there is a finite sequence of rewritings from w toan irreducible word y w . Since every pair of words in the prefix-rewritingsystem R φ represents the same element of the group G , then w = G y w . Bythe previous paragraph, the irreducible word y w must lie in N . But sinceeach element of G has exactly one representative in N , this implies that w = y w . Hence the set N of normal forms for the stackable structure isequal to the set of irreducible words with respect to the prefix-rewritingsystem R φ . Note that this shows both that the set N is prefix-closed, andthat the set of R φ -irreducible words are a set of normal forms. Hence R φ isconvergent.Finally, since A is a monoid generating set for G , and the rules of R φ define relations of G that give a set of normal forms for G , the convergentprefix-rewriting system R φ gives a monoid presentation of G . (cid:3) Recall that the normal form of the identity element in an autostackablegroup must be the empty word. Decidability of the set graph ( φ ) implies LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 13 that for any word w ∈ A ∗ , one can determine whether or not a φ -reductionapplies, and so Lemma 3.2 completes the word problem solution in that case.An immediate consequence of Lemma 3.2 is that the stacking presentation G = h A | { φ ( y g , a ) a − | g ∈ G, a ∈ A }i is a (group) presentation for the stackable group G ; property (2) of thedefinition of stackable implies that this presentation is finite. In [6, Proposi-tion 1.12], the first two authors of this paper show how to use computabilityof the set S φ to obtain an algorithm for constructing van Kampen dia-grams over this presentation; a similar alteration of the proof shows thatthis algorithm applies in the case that graph ( φ ) is recursive. However, in [6,Proposition 1.12], another hypothesis was included, that the generating set A of the stackable structure did not include a letter representing the iden-tity element of the group. We note that given any autostackable structurefor a group G , with inverse-closed generating set A , normal forms N , andstacking function φ , if A ′ ⊂ A is the set of letters in A representing 1, thensince the normal form set is prefix-closed, no element of N can contain aletter from A ′ . It can then be shown that G is also autostackable over theinverse-closed generating set B := A \ A ′ , with the same normal form set N ,and the stacking function φ ′ : N × B → B ∗ given by setting φ ′ ( y, b ) equalto the word φ ( y, b ) with all instances of letters in A ′ removed.Hence we have the following. Proposition 3.3.
Autostackable groups are finitely presented, have solvableword problem, and admit a recursive algorithm which, upon input of a word w ∈ A ∗ with π ( w ) = 1 , builds a van Kampen diagram for w over the stackingpresentation. We include a few more details here to illustrate the difference between thevan Kampen diagrams built from an autostackable structure and those builtfrom a prefix-closed automatic structure. For an autostackable group, sincethe set of normal forms is prefix-closed, each normal form word must label asimple path in the Cayley graph Γ, and as in the case of automatic groups,we extend the “seashell” method described in Section 2.1 to a diagram-building algorithm. Given a word w = b · · · b n with each b i ∈ A and suchthat π ( w ) = 1, and letting y i := y b ··· b i for each i , this method requires analgorithm for building van Kampen diagrams ∆ i for the words y i − b i y − i ,which then can be glued as in Figure 1 to obtain the diagram for w . However,in this case the van Kampen diagram ∆ i will not be “thin” in general, butinstead is built by recursion using property (2) of Definition 1.1. If thedirected edge e y i − ,b i of Γ is degenerate, then the van Kampen diagram ∆ i is homeomorphic to a line segment, containing no 2-cells; this is picturedin Figure 3. On the other hand, if the edge e y i − ,b i is recursive, and wewrite φ ( y i − , b i ) = a · · · a m with each a j ∈ A , then by Noetherian induction(using the well-founded strict partial ordering < φ ) we may assume that for b * y y i-1 i i b * y y i i-1 i Figure 3.
Degenerate van Kampen diagrams yy a j * aa yy c( ) ij ‘ m y i-1 i , ‘ i-1 j ‘ i b b Figure 4.
Recursive van Kampen diagrameach 1 ≤ j ≤ m we have already built a van Kampen diagram ∆ ′ j for theword y ′ j − a j y ′− j , where y ′ j denotes the normal form word representing theelement y i − a · · · a j for each j . Successively gluing these diagrams ∆ ′ j , or stacking them, along their common (simple) boundary paths y ′ j , we obtaina planar diagram with boundary word y i − φ ( y i − , b i ) y − i . Finally, glue on asingle 2-cell whose boundary is labeled by the word φ ( y i − , b i ) − b i to obtainthe required van Kampen diagram ∆ i . This process is illustrated in Figure 4.4. Asynchronously automatic groups
A group G with finite inverse-closed generating set A is asynchronouslyautomatic if there is a regular language N = { y g | g ∈ G } of normal formsfor G over A such that for each a ∈ A the subset L a := { ( y g , y ga ) | g ∈ G } of A ∗ × A ∗ is an asynchronously regular language. Every automatic groupis also asynchronously automatic. This section is devoted to the proof ofTheorem 4.1. Theorem 4.1.
Every group that has an asynchronously automatic structurewith a prefix-closed normal form set is autostackable.Proof.
Let G be an asynchronously automatic group with finite inverse-closed generating set A and prefix-closed normal form set N , and for each a ∈ A let M a = ( A ∪ { } , Q a , q , P a , δ a ) be an asynchronous automatonaccepting the language L a . By [10, Theorem 7.2.4], we may also assumethat the asynchronously automatic structure is bounded . That is, there is a LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 15 constant C such that for each pair ( u, v ) ∈ L a , the shuffle σ M a ,q ( u , v u , v σ M a ,q ( u, v ) = u v · · · u m v m where u = u · · · u m , v = v · · · v m ,each u i , v i ∈ ( A ∪ { } ) ∗ , and the lengths of these subwords satisfy 0 ≤ l ( u ) ≤ C , 1 ≤ l ( u i ) ≤ C for all 2 ≤ i ≤ m , 1 ≤ l ( v i ) ≤ C for all1 ≤ i ≤ m −
1, and 0 ≤ l ( v m ) ≤ C .By increasing this constant if necessary, we may also assume that C isgreater than 3 and greater than the cardinality | Q a | of the set of states ofthe automaton M a for every a ∈ A . We can view M a as a finite graph withvertex set Q a and a directed edge labeled b ∈ A ∪ { } from ˆ q to ˜ q whenever δ a (ˆ q, b ) = ˜ q . Let Q gooda be the set of all states q in Q a, ∪ Q a, such that thereis a path in M a from the initial state q to q and there also is a path in M a from q to the accept state q f . For each q ∈ Q gooda , by eliminating repetitionof vertices along the path to q f , there must also be a directed edge path in M a from q to q f of length less than C . Let W q ∈ ( A ∪ { } ) ∗ be a fixedchoice of such a word, for each such q . Note that this word must containtwo instances of the letter W q = σ M a ,q ( p q , r q p q , r q ∈ A ∗ satisfying l ( p q ) + l ( r q ) ≤ C − φ : N × A → A ∗ , we first set φ ( y g , a ) := a wheneverthe edge e g,a lies in the set ~E d = ~E N ,d of degenerate edges of the Cayleygraph Γ( G, A ) with respect to the set N of normal forms; i.e. whenever either y g a = y ga or y ga a − = y g , as required for property (2d) of Definition 1.1.Now suppose that e g,a is recursive. If l ( y g ) + l ( y ga ) ≤ C + 3 C , then define φ ( y g , a ) := y − g y ga .On the other hand, suppose that l ( y g ) + l ( y ga ) > C + 3 C . The pair( y g , y ga ) is accepted by the asynchronous automaton M a , and so the word w := σ M a ,q ( y g , y ga y g , y ga δ a ( q , w ) = q f , the accept state of M a . The bounded property above implies that l ( y ga ≤ Cl ( y g l ( y g Cl ( y g > C + 3 C + 2, which gives l ( y g ) > C + 1. Write w = w ′ w ′′ where w ′′ is the shortest suffix of w containing exactly C + 1 letters from y g (i.e., C + 2 letters from y g w ′ = σ M a ,q ( u, v ) and w ′′ = σ M a ,q ( s , t q = δ a ( q , w ′ ) ∈ Q gooda , u, v, s, t ∈ A ∗ , us = y g , vt = y ga , l ( s ) = C + 1, and 1 ≤ l ( t ≤ C + 2 C . Returning to the view of M a as a finite graph, the word w ′ labels a path from q to q and w ′′ labels apath from q to q f . Thus the word ˜ w := w ′ W q also labels a path from q to q f , and so the pair ( up q , vr q ) lies in the language L a . Note that usa = G vt and up q a = G vr q , and so s − p q ar − q t = G a . Moreover, the pair ( s, t ) andthe state q are uniquely determined by ( y g , y ga ); i.e., by g and a . In thiscase we define φ ( y g , a ) := s − p q ar − q t . With this definition of the stacking function φ , the length of the word φ ( y g , a ) is at most C + 3 C for all g ∈ G and a ∈ A , and in each case φ ( y g , a ) = G a .Now suppose that e = e g,a and e ′ = e g ′ ,a ′ are recursive edges such that e ′ lies on the directed path in the Cayley graph Γ starting at the vertex g and labeled by the word φ ( y g , a ). Now when l ( y g ) + l ( y ga ) ≤ C + 3 C the path φ ( y g , a ) = y − g y ga follows only degenerate edges, so we must have l ( y g ) + l ( y ga ) > C + 3 C . In this case, the path starting at g and labeledby the word φ ( y g , a ) := s − p q ar − q t defined above follows only degenerateedges along the subpaths labeled by s − p q and r − q t , since us = y g , vt = y ga ,and ( up q , vr q ) ∈ L a , and so the words vr q , vt ∈ N as well. So the edge e ′ must be the edge labeled a ′ = a with initial vertex g ′ = G gs − p q = G up q .That is, we have normal forms y g = us with l ( s ) = C + 1 and y g ′ = up q with l ( p q ) ≤ C , and so the normal form to the initial vertex of e ′ is strictlyshorter than the normal form to the initial vertex of e . Hence the relation < φ defined in property (2r) of Definition 1.1 strictly increases the length ofthe normal form of the initial vertex of the edges, and so is a well-foundedstrict partial ordering. Therefore G is stackable over A .By hypothesis the normal form set N is a regular language, and so for each a ∈ A , an application of Lemma 2.1 shows that the language J a := { y | ya ∈N } is regular. Lemma 2.3 then shows that the languages J a × { a } × { a } and( N ∩ A ∗ a − ) × { a }× { a } are synchronously regular. The finite union of thesesets for a ∈ A is the subset of graph ( φ ) corresponding to the application of φ to degenerate edges, and therefore this set is synchronously regular.The subset L smallrec := { ( y g , a, φ ( y g , a )) | g ∈ G, a ∈ A, e g,a ∈ ~E r and l ( y g )+ l ( y ga ) ≤ C +3 C } of graph ( φ ) is finite, and therefore also is synchronously regular. For use inavoiding overlapping sets later, denote J a,smallrec := { y g | ( y g , a, φ ( y g , a )) ∈ L smallrec } .For each a ∈ A and q ∈ Q gooda , let K a,q := { ( u, v ) | u, v ∈ A ∗ , δ a ( q , σ M a ,q ( u, v )) = q } , and note that by definition of Q gooda the set K a,q is nonempty. This sub-set of A ∗ × A ∗ is asynchronously regular; in particular, if q ∈ Q a, , then K a,q is the accepted language of the asynchronous automaton f M = ( A ∪{ } , e Q, q , P a , ˜ δ ) where e Q = Q a, , e Q = Q a, , e Q = ∅ , e Q = { ˜ q } , and˜ δ ( q ′ , b ) = δ ( q ′ , b ) for all q ′ ∈ e Q a, ∪ e Q a, and b ∈ A , ˜ δ ( q, q , ˜ δ (˜ q, q f ,and ˜ δ ( q ′ , b ) = F otherwise. The case that q ∈ Q a, is similar. ThenLemma 2.4 shows that the set ρ ( K a,q ) = { u | ∃ ( u, v ) ∈ K a,q } is a regu-lar language. LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 17
Let S a,q := { ( s, t ) | s, t ∈ A ∗ , l ( s ) = C + 1 , and δ ( q, σ M a ,q ( s , t q f } . The boundedness of the asynchronously automatic structure implies that l ( t ) < C +2 C , and the set S a,q is finite. Moreover, note that if ( s, t ) , ( s, t ′ ) ∈ S a,q , then if we let ( u, v ) be an element of the nonempty set K a,q , we have( us, vt ) , ( us, vt ′ ) ∈ L a , and so vt, vt ′ are both normal form words represent-ing the same element π ( usa ) of G ; hence t = t ′ . Thus for each s ∈ A C +1 ,there is at most one word t q,s such that the pair ( s, t q,s ) ∈ S a,q .Next for each a ∈ A , q ∈ Q gooda , and ( s, t ) ∈ S a,q , let L a,q,s := ρ ( K a,q ) s ∩ [ A ∗ \ ( J a ∪ ( N ∩ A ∗ a − ) ∪ J a,smallrec )] . This is the set of words us ∈ A ∗ s such that us ∈ N , the edge e π ( us ) ,a isrecursive, l ( us ) + l ( y usa ) > C + 3 C , and the path labeled σ M a ,q ( us, y usa )goes from q through q to q f in M a . Closure properties of regular sets showsthat this language is regular. Applying Lemma 2.3 again, the language L a,q,s × { a } × { s − p q ar − q t q,s } is a synchronously regular subset of graph ( φ )corresponding to these recursive edges.We can now write the graph of the stacking function φ as the finite union graph ( φ ) = [ ∪ a ∈ A (( J a ∪ ( N ∩ A ∗ a − ) × { a } × { a } )] ∪ L smallrec ∪ [ ∪ a ∈ A,q ∈ Q gooda , ( s,t q,s ) ∈ S a,q ( L a,q,s × { a } × { s − p q ar − q t q,s } )] . Closure of the class of synchronously regular sets under finite unions thenshows that graph ( φ ) is synchronously regular. Thus G is autostackable. (cid:3) Rewriting systems
In this section we prove the characterization of autostackable groups interms of synchronously regular bounded convergent prefix-rewriting systems,and conclude with a discussion of finite convergent rewriting systems. Webegin by discussing a process for minimizing prefix-rewriting systems.
Definition 5.1.
A convergent prefix-rewriting system R ⊂ A ∗ × A ∗ for agroup G is processed if: (a) For each a ∈ A there is a letter in A , which we denote a − , suchthat π ( a ) − = π ( a − ) (where π : A ∗ → G is the canonical map). (b) For each pair ( u, v ) ∈ R , every proper prefix of u is irreducible withrespect to the rewriting operations of R . (c) Whenever ( u, v ) , ( u, v ) ∈ R , then v = v . For any prefix-rewriting system R over A , let Irr( R ) denote the set ofirreducible words with respect to the rewriting operations ux → vx whenever( u, v ) ∈ R and x ∈ A ∗ . Note that every prefix of a word in Irr( R ) must alsolie in Irr( R ). Proposition 5.2.
If a group G admits a synchronously regular boundedconvergent prefix-rewriting system R over a monoid generating set B , then G also admits a processed synchronously regular bounded convergent prefix-rewriting system Q over the generating set A := B ∪ B − , such that Irr( R )= Irr( Q ).Proof. Let R be a bounded convergent prefix-rewriting system over B for G , and let π : B ∗ → G be the associated surjective monoid homomorphism.For each element b ∈ B , let the symbol b − denote another letter, and let A := B ∪ { b − | b ∈ B } . For each b ∈ B , let z b denote the unique word inIrr( R ) representing the element π ( b − ) of G .Let R ′ := { ( yb, v ) | ( yb, v ) ∈ R, y ∈ Irr( R ) , b ∈ B } ;i.e., the set of all rules of R whose left entry has every proper prefix irre-ducible; i.e., that satisfies property (b) of Definition 5.1.Let k be the constant associated to the bounded property of the prefix-rewriting system R . Then by expressing the finite set W := { ( s, t ) ∈ B ≤ k × B ≤ k | the first letters of s and t are distinct } , as W = { ( s , t ) , ..., ( s n , t n ) } , we can write each element r := ( u, v ) of R inthe form r = ( ws i ( r ) , wt i ( r ) ) for a unique index i ( r ) ∈ { , ..., n } and word w ∈ B ∗ . For each 1 ≤ i ≤ n , let R ′ i := { r ∈ R ′ | i ( r ) = i } . Then R ′ is the disjoint union R ′ = ∪ ni =1 R ′ i . Note that if there are two pairs r = ( u, v ) , r = ( u, v ) ∈ R ′ that have the same left hand entry but v = v on the right, then the indices i ( r ) = i ( r ) must also be distinct. Let Q ′ := { r = ( u, v ) ∈ R ′ |6 ∃ ˜ r = ( u, ˜ v ) ∈ R ′ with i (˜ r ) < i ( r ) } . That is, the subset Q ′ of R satisfies properties (b) and (c) of Definition 5.1.We then define the prefix-rewriting system Q := Q ′ ∪ Q ′′ where Q ′′ := { ( yb − , yz b ) | y ∈ Irr( R ) \ B ∗ b } ∪ { ( ybb − , y ) | yb ∈ Irr( R ) , b ∈ B } . Suppose that w ∈ A ∗ is rewritten by a sequence of applications of rewrit-ing operations using the prefix-rewriting system Q . Since the only occur-rences of letters of B − in Q appear in left hand sides of pairs in Q ′′ , atmost l ( w ) of the rewritings in this sequence involve a rule of Q ′′ . The rulesin Q ′ all lie in the convergent prefix-rewriting system R , which satisfies thetermination property, and so only finite sequences of applications of Q ′ rulescan occur. Hence there can be at most finitely many rewritings in any suchrewriting of w ; that is, the prefix-rewriting system Q is terminating.Suppose that w is any word in Irr( R ). Then w ∈ B ∗ , so w can’t berewritten using a pair from Q ′′ , and since Q ′ ⊆ R , the word w also can’t bereduced using Q ′ . Hence Irr( R ) ⊆ Irr( Q ). LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 19
On the other hand, suppose that x is a word in Irr( Q ) \ Irr( R ). If x ∈ B ∗ ,then write x = x ′ bx ′′ where x ′ ∈ B ∗ , b ∈ B , and x ′ b is the shortest prefixof x that does not lie in Irr( R ). But then there must be a pair ( u, v ) ∈ R and a word z ∈ B ∗ such that x ′ b = uz . Since x ′ is irreducible over R , then x ′ b = u and z = λ , and the rule ( u, v ) also lies in the subset R ′ of R . Hencethere also is a rule ( u, v ′ ) in the subset Q ′ of Q , since the sets of left handsides of rules of R ′ and Q ′ are the same. But then x is reducible over Q .This contradiction implies that B ∗ ∩ Irr( Q ) ⊆ Irr( R ), and so we must have x / ∈ B ∗ . In this case we can write x = x ′ ax ′′ with a ∈ B − and x ′ ∈ B ∗ ,where a is the first occurrence of a letter of A \ B in x . Since the set ofirreducible words over a prefix-rewriting system is prefix-closed, the word x ′ lies in Irr( Q ) ∩ B ∗ , and hence also in Irr( R ). But then the word x canbe reduced using an element of Q ′′ , another contradiction. Therefore wehave Irr( R ) = Irr( Q ). Since the set Irr( R ) is a set of normal forms for thegroup G , and whenever ( u, v ) ∈ Q we have u = G v , this shows that Q is aconvergent prefix-rewriting system for the group G .Since the convergent prefix-rewriting system R is bounded with constant k , the rules of the prefix-rewriting system Q are also bounded, with constantgiven by the maximum of k , 2, and max { l ( z b ) | b ∈ B } .Note that the prefix-rewriting system Q has been chosen to satisfy prop-erties (a), (b), and (c) of the Definition 5.1, and so Q is a processed boundedconvergent prefix-rewriting system.If moreover the set R is also synchronously regular, then the paddedextension set µ ( R ) = { µ ( u, v ) | ( u, v ) ∈ R } is a regular language overthe alphabet B = ( B ∪ $) \ { ($ , $) } . Define the monoid homomorphism ρ : B ∗ → B ∗ by ρ (( b , b )) := b if b ∈ B and ρ (( b , b )) := λ if b = $. Theset Irr( R ) is the language Irr( R ) = A ∗ \ ( ρ ( µ ( R )) A ∗ ); using closure of regularlanguages under homomorphic image, concatenation, and complement (seeSection 2.2 for more on regular languages), then Irr( R ) is a regular set overthe alphabet B . But then Irr( R ) is also regular over any alphabet containing B , including A .For each b ∈ B , Lemma 2.1 says that the set L b := { y | yb ∈ Irr( R ) } also isregular. Also recall from Lemma 2.2 that whenever L is a regular languageover B , then the diagonal set ∆( L ) := { µ ( y, y ) | y ∈ L } is a regular languageover B . Now the padded extension of the subset Q ′′ of the prefix-rewritingsystem Q has the decomposition µ ( Q ′′ ) = ∪ b ∈ B [(∆(Irr( R ) \ B ∗ b ) · µ ( b − , z b )) ∪ (∆( L b ) · µ ( bb − , λ ))] . Again applying closure properties (in particular under finite unions) of reg-ular languages, this shows that Q ′′ is synchronously regular.Analyzing the subset Q ′ of Q requires a few more steps. First we notethat the padded extension of the set of rules in R satisfying property (b) in Definition 5.1 is µ ( R ′ ) = µ ( R ) ∩ ρ − (Irr( R ) · B ), and so µ ( R ′ ) is a regularset.Next for each 1 ≤ i ≤ n (where n = | W | ), let L i := ρ ( µ ( R ′ ) ∩ (∆( B ∗ ) · µ ( s i , t i ))) be the set of left hand entries of all of the rules r in R ′ i . Againclosure properties show that L i is a regular language over B . Then the set L ′ i := L i \ ( ∪ i − j =1 L j )is the set of all left hand entries of elements q in Q ′ such that the index i ( q ) = i . Now Lemma 2.1 shows that the set L ′′ i := { y | ys i ∈ L ′ i } is regular.Putting all of these together, the padded extension of the set Q ′ has thedecomposition µ ( Q ′ ) = ∪ ni =1 ∆( L ′′ i ) · µ ( s i , t i ) . Thus µ ( Q ′ ) is a regular language over the alphabet B , and hence also overthe set A . Hence Q ′ also is synchronously regular.Finally the closure of synchronously regular sets under finite unions showsthat the bounded convergent prefix-rewriting system Q is synchronouslyregular, as required. (cid:3) Note that whenever R is a processed convergent prefix-rewriting systemover an alphabet A and w ∈ A ∗ is a reducible word, then there exists exactlyone rewriting operation (of the form w = ux → vx for some ( u, v ) ∈ R ) thatcan be applied to w . Hence for each w ∈ A ∗ , we can define the prefix-rewriting length prl R ( w ) to be the number of rewriting operations requiredto rewrite w to its normal form via R . Theorem 5.3.
Let G be a finitely generated group.(1) The group G is stackable if and only if G admits a bounded convergentprefix-rewriting system.(2) The group G is autostackable if and only if G admits a synchronouslyregular bounded convergent prefix-rewriting system.Proof. Suppose first that the group G is stackable over an inverse-closedgenerating set A , with normal form set N , constant k , and stacking function φ : N × A → A ∗ such that the length of φ ( y, a ) is at most k for all ( y, a ) ∈N × A . In Lemma 3.2, we show that R φ := { ( ya, yφ ( y, a )) | y ∈ N , a ∈ A, ya / ∈ N ∪ A ∗ a − a }∪ { ( yaa − , y ) | ya ∈ N , a ∈ A } . is a convergent prefix-rewriting system for the group G . (Moreover, theirreducible words are the normal forms from the stackable structure; i.e.,Irr( R φ ) = N .) The bound k on lengths of words in the image of φ impliesthat R φ is a bounded convergent prefix-rewriting system.If moreover G is autostackable, so that the set graph ( φ ) is synchronouslyregular, let µ ( graph ( φ )) := { µ ( y g , a, φ ( y g , a )) | g ∈ G, a ∈ A } be the regularlanguage of padded words over A = ( A ∪ $) \ { ($ , $ , $) } associated to the LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 21 elements of the set graph ( φ ). Define the monoid homomorphism ρ : A ∗ → A ∗ by ρ (( a , a , a )) := a if a ∈ A and ρ (( a , a , a )) := λ if a = $, andthe monoid homomorphism ρ , : A ∗ → (( A ∪ $) ) ∗ by ρ , (( a , a , a )) :=( a , a ), for each ( a , a , a ) ∈ A . The normal form set Irr( R φ ) = N = ρ ( µ ( graph ( φ ))) is the image of a regular set, and so is regular. For each a ∈ A , the set J a := { y ∈ A ∗ | ya ∈ N } is regular, applying Lemma 2.1.Also with this notation, for each a ∈ A and u ∈ A ≤ k we can write the set L a,u of all normal form words y ∈ N such that the stacking function φ maps( y, a ) to the word u as L a,u := ρ ( µ ( graph ( φ )) ∩ ρ − , ( µ ( a, u ) · ($ , $) ∗ )) . Recalling the fact that the class of regular languages is closed under finiteintersections and homomorphic image and preimage, then since the language µ ( a, u ) · ($ , $) ∗ over ( A ∪ $) is regular, the set L a,u is regular. Using the nota-tion ∆( L ) = { µ ( w, w ) | w ∈ L } for any language L , we can now decomposethe padded extension of the prefix-rewriting system as R φ = [ ∪ a ∈ A,u ∈ A ≤ k ,a = u ∆( L a,u ) · µ ( a, u )] ∪ [ ∪ a ∈ A ∆( J a ) · µ ( aa − , $)] . From Lemma 2.2, the languages ∆( L a,u ) and ∆( J a ) over ( A ∪ $) are regular.Since singleton sets are regular, and the class of regular languages is alsoclosed under concatenation and finite unions, this decomposition shows thatthe set µ ( graph ( φ )) is regular. Therefore R φ is a synchronously regularbounded convergent prefix-rewriting system for the autostackable group G .Conversely, suppose that the group G admits a bounded convergent prefix-rewriting system. From the proof of Proposition 5.2, there exists a processedbounded convergent prefix-rewriting system R , over an inverse-closed alpha-bet A , for the group G . Let k be the constant associated to the boundedproperty of this prefix-rewriting system. Let N be the set Irr( R ) of wordsthat are irreducible with respect to the rewriting operations ux → vx when-ever ( u, v ) ∈ R and x ∈ A ∗ . Since the prefix-rewriting system is convergent,then N is a set of normal forms for G . Note that the empty word and anyprefix of an irreducible word are irreducible, and so N is a prefix-closedlanguage of normal forms for G over A that contains the empty word.Define the function φ : N × A → A ∗ as follows. For each y ∈ N and a ∈ A ,define φ ( y, a ) := a if either ya ∈ N or y ∈ A ∗ a − , as required for property(2d) of Definition 1.1. If neither of these conditions hold, then the word ya is reducible. Since the maximal prefix y is irreducible, any rule of the prefix-rewriting system that applies to the word ya must have the entire word ya as its left entry. Because this prefix-rewriting system is processed, there isexactly one element of R of the form ( ya, v ) for some v ∈ A ∗ . Moreover,there are words s, t ∈ A ≤ k and w ∈ X ∗ such that ya = wsa , v = wt , and (bytaking w to be as long as possible) the words s and t do not start with thesame letter. In this case we define φ ( y, a ) := s − t , where s − is the formalinverse of s in A ∗ . For every y ∈ N and a ∈ A , then, the length of the word φ ( y, a ) is at most 2 k , and since wsa = G wt in the rewriting presentation of G , we have φ ( y, a ) = G a .Let Γ be the Cayley graph for the group G with generating set A , and let ~E r = ~E N ,r denote the set of recursive edges with respect to the normal formset N . Given any directed edge e g,a of the Cayley graph Γ( G, A ) with g ∈ G and a ∈ A , let prl R ( e g,a ) := prl ( y g a ) denote the prefix-rewriting length over R of the associated word y g a , where y g is the irreducible normal form for g .Suppose that e g,a is any edge in ~E r , and that e ′ is an edge on the directedpath in Γ labeled by the word φ ( y g , a ) and starting at the vertex g . Thenthe word y g a is not in normal form, and there is a rule y g a = wsa → wt in the prefix-rewriting system R such that φ ( y g , a ) = s − t . Since the word y g is in normal form, the prefix s − of the word φ ( y g , a ) labels a path in Γstarting at the vertex g that follows only degenerate edges, in the maximaltree defined by the normal form set N . Writing the word t = b · · · b n with each b i in A , then e ′ = e gs − b ··· b i − ,b i = e wb ··· b i − ,b i for some i . Nowthe sequence of rewriting operations with respect to the prefix-rewritingsystem R of the word y g a has the form y g a = wsa → wt = wb · · · b n → ∗ y gs − b ··· b i − b i · · · b n → ∗ y ga , where → ∗ denotes a finite number (possibly0) of applications of rewriting rules, since no rewriting operation over theprocessed prefix-rewriting system R can be applied affecting the letter b i in these words until the prefix to the left of that letter has been rewritteninto its irreducible normal form. Hence the number of rewritings neededto obtain an irreducible word starting from the word y g a is strictly greaterthan the number required to obtain a normal form starting from the word y gs − b ··· b i − b i . That is, prl R ( e ′ ) < prl R ( e ). Then the usual strict well-founded partial ordering on the natural numbers implies that the relation < φ of property (2r) in Definition 1.1 is a strict well-founded partial ordering.Hence property (2) of the Definition 1.1 of autostackable holds, and so thegroup G is stackable.If moreover G has a bounded convergent prefix-rewriting system that issynchronously regular, then Proposition 5.2 says that there is a processedsynchronously regular bounded convergent prefix-rewriting system R over ainverse-closed generating set A for the group G . Synchronous regularity of R means that the set µ ( R ) = { µ ( u, v ) | ( u, v ) ∈ R } of padded words is aregular language over the set A = ( A ∪ $) \ { ($ , $) } . Let ρ : A ∗ → A ∗ be the monoid homomorphism defined by ρ ( a , a ) := a if a ∈ A and ρ ( a , a ) := λ if a = $. The set N of irreducible words with respect to R can then be written as N = A ∗ \ ( ρ ( µ ( R )) A ∗ ) , and so N is a regular language.For each a ∈ A , an application of Lemma 2.1 shows that the language L a := { y | ya ∈ N } is regular. Lemma 2.3 then shows that the languages LGORITHMS AND TOPOLOGY OF CAYLEY GRAPHS FOR GROUPS 23 L a × { a } × { a } and ( N ∩ A ∗ a − ) × { a } × { a } are synchronously regular. Thusthe subset of graph ( φ ) corresponding to the application of φ to degenerateedges is synchronously regular.Given a ∈ A , let W a be the finite set of all pairs ( s, t ) such that s, t ∈ A ≤ k , s and t begin with different letters of A , and s does not end with the letter a − . Let ∆( A ∗ ) := { ( w, w ) | w ∈ A ∗ } ; by Lemma 2.2, this language over A is regular. For each ( s, t ) ∈ W a , let P a,s,t := ρ ( µ ( R ) ∩ (∆( A ∗ ) · µ ( sa, t ))) , which is again regular using the closure properties of regular languages.Then the set of all words w such that the rule ( wsa, wt ) lies in R is L a,s,t := { w | wsa ∈ P a,s,t } , which is also regular (by Lemma 2.1). Applying Lemma 2.3 once more showsthat the subset ( L a,s,t · s ) × { a } × { s − t } of graph ( φ ) corresponding to theserecursive edges is also synchronously regular.We can now write the graph of the stacking function φ as graph ( φ ) = ∪ a ∈ A [( L a × { a } × { a } ) ∪ (( N ∩ A ∗ a − ) × { a } × { a } )] ∪ a ∈ A, ( s,t ) ∈ W a ( L a,s,t · s ) × { a } × { s − t } . Closure of the class of synchronously regular languages under finite unionsthen implies that graph ( φ ) is synchronously regular. Hence property (1) ofDefinition 1.1 of autostackability also holds in this case. (cid:3) Rewriting systems that are not “prefix-sensitive”, allowing rewriting rulesto be applied anywhere in a word, have been considerably more widelystudied and applied in the literature than prefix-rewriting systems. A finiteconvergent rewriting system for a group G consists of a finite set A togetherwith a finite subset R ⊆ A ∗ × A ∗ such that as a monoid, G is presented by G = M on h A | u = v whenever u → v ∈ R i , and the rewritings xuz → xvz for all x, z ∈ A ∗ and ( u, v ) in R satisfy: • Normal forms:
Each g ∈ G is represented by exactly one irreducible word (i.e. word that cannot be rewritten) over A . • Termination:
There does not exist an infinite sequence of rewritings x → x → x → · · · .The key difference here is that a rewriting system allows rewritings xuz → xvz for all x, z ∈ A ∗ and ( u, v ) ∈ R , but a prefix-rewriting system onlyallows rewritings uz → vz for all z ∈ A ∗ and ( u, v ) ∈ R . However, everyfinite convergent rewriting system gives rise to a bounded convergent prefix-rewriting system, yielding the following. Corollary 5.4.
Every group that admits a finite complete rewriting systemis autostackable.
Proof.
Given a finite convergent rewriting system R for a group G over agenerating set A , the prefix-rewriting system over A defined byˆ R := { ( wu, wv ) | ( u, v ) ∈ R, w ∈ A ∗ } allows exactly the same rewriting operations as the original finite conver-gent rewriting system, and therefore is a convergent prefix-rewriting system.Since the set R is finite, this prefix-rewriting system ˆ R is also bounded. Fi-nally, the padded extension of the set ˆ R can be written as µ ( ˆ R ) = ∪ ( u,v ) ∈ R ∆( A ∗ ) · µ ( u, v ), and so this set is synchronously regular. Theorem 5.3(2) now com-pletes the proof. (cid:3) Acknowledgments
This work was partially supported by a grant from the Simons Foundation(Grant Number 245625 to the second author).
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