New results on the prefix membership problem for one-relator groups
aa r X i v : . [ m a t h . G R ] N ov NEW RESULTS ON THE PREFIX MEMBERSHIP PROBLEMFOR ONE-RELATOR GROUPS
IGOR DOLINKA AND ROBERT D. GRAY
Abstract.
In this paper we prove several results regarding decidability ofthe membership problem for certain submonoids in amalgamated free prod-ucts and HNN extensions of groups. These general results are then appliedto solve the prefix membership problem for a number of classes of one-relatorgroups which are low in the Magnus–Moldavanski˘ı hierarchy. Since the pre-fix membership problem for one-relator groups is intimately related to theword problem for one-relator special inverse monoids in the E -unitary case(as discovered in 2001 by Ivanov, Margolis and Meakin), these results yieldsolutions of the word problem for several new classes of one-relator specialinverse monoids. In establishing these results, we introduce a new theory ofconservative factorisations of words which provides a link between the pre-fix membership problem of a one-relator group and the group of units of thecorresponding one-relator special inverse monoid. Finally, we exhibit the firstexample of a one-relator group, defined by a reduced relator word, that has anundecidable prefix membership problem. Introduction
From the early days of combinatorial group theory, algorithmic problems haveoccupied a central position in the course of its development, going back to thepioneering work of Dehn [8]. One of the most celebrated classical results in thisarea is the positive solution of the word problem for one-relator groups by Dehn’sstudent Magnus [29]. It is based on a previous important result by Magnus [28],the
Freiheitssatz , stating that if w is a cyclically reduced word then any subgroupof the one-relator group Gp h X | w = 1 i generated by a subset of X omitting atleast one letter that appears in w must be free. Magnus’s approach is applicableto an array of other algorithmic problems for one-relator groups and entails whatis today known as the Magnus method . The modern exposition of this methodstems from the paper of McCool and Schupp [33] (see also the monograph [27]),and is based on an observation due to Moldavanski˘ı [35] that if w is a reduced wordand has exponent sum zero for some letter from X , then Gp h X | w = 1 i is an HNNextension of a one-relator group with a defining relator shorter than w .Given the vigorous development of combinatorial algebra over a number ofdecades, it is quite striking that the following problem still remains open. Mathematics Subject Classification.
Primary 20F10; Secondary 20F05, 20M05, 20M18,68Q70.
Key words and phrases.
One-relator group; Prefix membership problem; Word problem; Spe-cial inverse monoid.The research of the first named author was supported by the Ministry of Education, Science,and Technological Development of the Republic of Serbia through the grant No.174019. Theresearch of the second named author was supported by the EPSRC grant EP/N033353/1 “Specialinverse monoids: subgroups, structure, geometry, rewriting systems and the word problem”.
Problem.
Is the word problem decidable for all one-relation monoids Mon h X | u = v i (where u, v are words over X )?This problem has received significant attention, with a number of special casesbeing solved. A strong early impetus was given by Adjan [1] who proved thatMon h X | u = v i has decidable word problem if either one of the words u, v areempty (this is the case of the so-called special monoids , with presentations of theform Mon h X | w = 1 i ), or both u, v are non-empty and have different initial anddifferent terminal letters. For both of these cases, Adjan exhibits a reduction ofthe monoid word problem to the word problem of an associated one-relator group,and then makes an appeal to Magnus’s result. Later on, Adyan and Oganessyan[2] showed that the word problem for Mon h X | u = v i can be reduced to the case ofmonoid presentations of the form Mon h X | asb = atc i , where a, b, c ∈ X , b = c , and s, t are arbitrary words over X .An entirely new approach to the problem was provided by the work of Ivanov,Margolis and Meakin [19] (which is also the central reference for the present paper).There, a crucial observation is made that the monoid Mon h X | asb = atc i (arisingfrom the reduction found in [2]) embeds into the inverse monoid defined by theinverse monoid presentation Inv h X | asbc − t − a − = 1 i ; consequently, the decid-ability of the word problem for special inverse monoid presentations Inv h X | w = 1 i (where w is a word over the alphabet X = X ∪ X − ) would immediately imply thepositive solution of the word problem for one-relator monoids. This strongly moti-vates the study of special inverse monoids and their word problems, which is alsointeresting in its own right, given the prevalence of inverse semigroups and theircombinatorial and geometric aspects in various areas of mathematics (see [24]).However, a recent surprising result of Gray [10] shows that the word problem forone-relator special inverse monoids in complete generality is undecidable .Given that the word problem for Inv h X | w = 1 i is undecidable in general, thekey problem that remains is to determine for which words w ∈ ( X ∪ X − ) ∗ it isdecidable? In particular, is it decidable if w is (i) a reduced word or (ii) a cyclicallyreduced word? A positive answer to the first of these questions would still, as aconsequence of the results from [19] described above, imply a positive answer todecidability of the word problem for arbitrary one-relator monoids Mon h X | u = v i .This motivates investigating the word problem in the cases that w is reduced orcyclically reduced.In the cyclically reduced case, the word problem for the one-relator inversemonoid Inv h X | w = 1 i is closely related to an algorithmic problem in the corre-sponding one-relator group Gp h X | w = 1 i called the prefix membership problem.For a one-relator group G = Gp h X | w = 1 i , let P w denote the submonoid of G generated by the elements of G represented by all prefixes of w . This is the prefixmonoid of G . Another crucial result from [19] (Theorem 3.1), shows that if theinverse monoid Inv h X | w = 1 i has the so-called E -unitary property (which is e.g.the case when w is cyclically reduced) then the word problem of Inv h X | w = 1 i is decidable whenever the membership problem for P w in G is decidable. This issignificant because it translates the word problem for a one-relator special inversemonoids into the realm of one-relator groups and associated decision problems.The connections between decision problems for monoids, inverse monoids andgroups just described highlight the importance of other, more general, algorithmic HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 3 problems. For example, it is still unknown whether the subgroup membership prob-lem —also called the generalised word problem —is decidable for one-relator groups.However, there exist one-relator groups in which the submonoid membership prob-lem (and thus the more general rational subset membership problem [25]) is un-decidable [10]. The one-relator group with undecidable submonoid membershipproblem given in [10] is an HNN extension of Z × Z with respect to an isomorphismmapping one of the natural copies of Z to the other. So in general the submonoidand rational subset membership problems are not well-behaved under the HNNextension construction, and similarly for free products with amalgamation. On theother hand, under the assumption of finiteness of edge groups, the decidability ofthe rational subset membership problem is preserved under the graph of groupsconstruction [21], which includes amalgamated free products and HNN extensions.We direct the reader e.g. to [12, 22, 25, 26] for a sampler of results in this broaderarea in which the present topic is couched.Motivated by the above discussion, both the word problem for one-relator inversemonoids and the prefix membership problem for one-relator groups, with cyclicallyreduced defining relator, have already received a great deal of attention in theliterature; see e.g. [14, 16, 19, 20, 32, 34] and [5, Question 13.10]. In this paper wewill make several new contributions towards resolving these open problems. Thenew approaches to these problems that we present in this article naturally divideinto two themes.Firstly, as mentioned above, the standard modern approach to proving resultsabout one-relator a one-relator group Gp h X | w = 1 i is by induction on the lengthof w using the McCool–Schupp [33, 27] Moldavanski˘ı [35] approach via HNN ex-tensions. The inductive step of this approach is based on the fact that the one-relator group embeds in a certain HNN extension of a one-relator group with ashorter defining relator. We shall refer to the steps in this induction as levels inthe Magnus–Moldavanski˘ı hierarchy. Given its utility in proving other results forone-relator groups, it is very natural to also attempt to use this approach to inves-tigate the prefix membership problem for Gp h X | w = 1 i . If the group happens tobe free then by a theorem of Benois [4] the prefix membership problem is decid-able (in fact, the more general rational subset membership problem is decidable forfree groups.). So the next natural step in this approach is to investigate the prefixmembership problem for one-relator groups that are one-step away from being freein this hierarchy. The general results we prove for HNN extensions in this paper aremotivated by this idea, and we will apply them in this paper to prove decidabilityof the prefix membership problem for several classes of one-relator groups whichare low in the Magnus–Moldavanski˘ı hierarchy.The second new viewpoint revealed by the results we prove in this paper isthat the word problem in Inv h A | w = 1 i can be often be shown to be decidable byanalysing decompositions of the word w ≡ w w . . . w k , where all the w i representinvertible elements of the monoid. We call this a unital decomposition of the word w . We shall identify several combinatorial conditions on unital decompositionswhich suffice to imply decidability of the word problem for the monoid. This givesa new approach to the word problem for one-relator inverse monoids which goesvia the group of units, in this sense. Something that makes this approach widelyapplicable is that the above decomposition of w does not need to be minimal inorder for our results to apply. That is, provided the words w i satisfying the needed IGOR DOLINKA AND ROBERT D. GRAY combinatorial properties, it is not important whether or not there is a finer decom-position of w as a product of units. This means that there are situations where wecan show the word problem is decidable without necessarily having an algorithmto compute the minimal invertible pieces of the defining relator word. Similarlyit means that the word problem can sometimes be shown to be decidable withouthaving to determine the structure of the group of units of the monoid. To make useof information about unital decompositions in the inverse monoid presentation tosolve the prefix membership problem in the corresponding group Gp h A | w = 1 i wedevelop a theory of, so-called, conservative factorisations of relator words in one-relator groups. This is another key new idea that we introduce in this paper, sinceit allows us to transform algebraic information about units in the inverse monoidinto corresponding algebraic information about submonoids of the maximal groupimage generated by prefixes of pieces of the relator. This allows us to state ourresults entirely in terms of one-relator groups and conservative factorisations, andthen apply them to solve the word problem for various families of one-relator inversemonoids.These new approaches give rise to results which, when expressed in their mostgeneral form, prove decidability of the membership problem in certain submonoidsof amalgamated free products of groups and HNN extensions of groups. In thispaper we prove four new general results of this kind. Specifically, we prove twogeneral theorems for amalgamated free products in Section 4, Theorems A andB, and then in Section 6 we prove two general theorems for HNN extensions ofgroups, namely Theorem C and Theorem D. Then, in Sections 5 and 7, respectively,we show how, via ideas summarised in the description of the two main themesabove, we can apply these general results to solve the prefix membership problemfor certain one-relator groups and, consequently, to solve the word problem forsome classes of special one-relator inverse monoids. As applications we recover newproofs of numerous results from the literature [6, 19, 20, 30, 32, 34] (bringing themunder a common framework), and at the same time we prove decidability of theprefix membership problem for many classes of one-relator groups (and special one-relator inverse monoids) not covered by previous results. In particular, our workwas inspired by the attempts to solve the word problem for the so-called O’Haremonoid (see [31, 34] and Example 3.2 below), which is eventually dealt with in thispaper, in Proposition 5.4. Other main applications of our general results includeTheorems 5.1, 5.7, 5.10, 7.2, 7.8 and 7.10.In the last section of the paper we present a result of a different flavour whichsays something about the limits of what we should hope to be able to prove aboutthe prefix membership problem in one-relator groups. Specifically, by modifying theconstruction from [10], we will show in Theorem 8.2 that there is a finite alphabet X and a reduced word w ∈ ( X ∪ X − ) ∗ such that Gp h X | w = 1 i has undecidableprefix membership problem. Hence if [5, Question 13.10] has a positive answer thenthe cyclically reduced hypothesis will need to be used.The paper is organised as follows. In the next preliminary section we gather thenotation and basic notions, aiming to make the paper reasonably self-contained.This is followed by Section 3 where we discuss the relationship between two typesof factorisations of a word w appearing as a relator in M = Inv h X | w = 1 i , namely, unital ones, decomposing w into pieces representing invertible elements (units) ofthe inverse monoid M , and conservative factorisations preserving, in a sense, the HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 5 prefix monoid P w . When M is E -unitary, these two types of factorisation coincide(see Theorem 3.3), and that, taken together with the Benois factorisation algorithmdevised by Gray and Ruˇskuc [11] (producing such factorisations in a manner finerthan the Adjan-Zhang overlap algorithm [1, 42]), is an important pre-requisite forsome of our decidability results. The main body of our results is then presented inSections 4–7. We finish the paper by few concluding remarks in Section 8.2. Preliminaries
We give some background definitions and results from combinatorial group andmonoid theory that will be needed later. For more background we refer the readerto [27] for groups, [18, 24, 38] for monoids and inverse semigroups, and [15, 39] forautomata and formal languages. In particular we refer the reader to [27] for basicnotions from the algorithmic theory of finitely generated groups.2.1.
Words and free objects.
Let X be a finite alphabet. By X ∗ we denotethe free monoid on X , consisting of all words over X including the empty word 1.However, whenever we are concerned with groups and inverse monoids it is moreuseful to consider a ‘doubled’ alphabet X = X ∪ X − , where X − = { x − : x ∈ X } is a disjoint copy of X , with an obvious bijective correspondence between X and X − . Now the free monoid X ∗ has a natural involutory operation so that for a word w = x ε . . . x ε k k , x , . . . , x k ∈ X , ε , . . . , ε k ∈ {− , } , we have w − = x − ε k k . . . x − ε .When w ∈ X ∗ , we use the notation w ( x , . . . , x n ) to stress that the lettersoccurring in w are among x , . . . , x n , x − , . . . , x − n . In other words, an occurrenceof a letter x i in w can happen either as x i , or as x − i . Given w ( x , . . . , x n ) and asequence of (not necessarily distinct) words p , . . . , p n ∈ X ∗ , we write w ( p , . . . , p n )to denote the word obtained from w = w ( x , . . . , x n ) by replacing each letter x i by p i and each letter x − i by p − i .Given w ∈ X ∗ we denote by red( w ) the reduced form of w , which is obtainedfrom w by the confluent rewriting process of successively removing subwords of theform xx − and x − x , where x ∈ X . This notation is extended to sets, too, so thatfor A ⊆ X ∗ , red( A ) stands for the set of words obtained by reducing each word from A . As is well known, one can identify the elements of the free group F G ( X ) on X with the set of all reduced words from X ∗ , so that the result of the multiplicationof two such words u, v is red( uv ), and the inverse of u is simply u − .A monoid M is called inverse [24, 38] if for every a ∈ M , there is a unique element a − ∈ M , called the inverse of a , such that aa − a = a and a − aa − = a − . Inversemonoids form a variety in the sense of universal algebra, so free inverse monoids F IM ( X ) exist. A straightforward, albeit implicit description of F IM ( X ) is givenas a quotient of X ∗ by the so-called Vagner congruence : this is the congruenceof X ∗ generated by all pairs of the form ( u, uu − u ) and ( uu − vv − , vv − uu − ),where u, v ∈ X ∗ . Concrete descriptions of F IM ( X ) (and so the solutions of itsword problem) go back to Scheiblich [40] and Munn [36]: the element of F IM ( X )represented by a word w ∈ X ∗ can be identified with a birooted tree today calledthe Munn tree of w . This is obtained as a connected subtree of the Cayley treeof the free group F G ( X ) which arises by travelling along the path labelled by w .Hence, u, v ∈ X ∗ represent the same element of F IM ( X ) if and only if they giverise to the same Munn tree. Clearly, there is a natural surjective homomorphism F IM ( X ) → F G ( X ). IGOR DOLINKA AND ROBERT D. GRAY
Presentations.
We denote by G = Gp h X | w i = 1 ( i ∈ I ) i the group presented by generators X and relators w i , i ∈ I ; as usual, this iscanonically the quotient of the free group F G ( X ) by its smallest normal subgroup N containing all the elements (reduced words) w i , i ∈ I . Similarly, the monoiddefined by a monoid presentation M = Mon h X | u i = v i ( i ∈ I ) i is the quotient ofthe free monoid X ∗ by the congruence ρ generated by the pairs ( u i , v i ), i ∈ I .In an analogous fashion, inverse monoids can be given by inverse monoid pre-sentations M = Inv h X | u i = v i ( i ∈ I ) i , where M ∼ = F IM ( X ) /ρ for the inverse monoid congruence ρ of F IM ( X ) generatedby the pairs ( u i , v i ), i ∈ I . This is equivalent to saying that M as the quotient X ∗ /ρ ′ where ρ ′ is the smallest congruence containing the Vagner congruence andall the pairs ( u i , v i ), i ∈ I . When one of the sides of each defining relation isthe empty word, say v i is the empty word for all i ∈ I , we get the notion of a special inverse monoid and special inverse monoid presentations. The maximalgroup homomorphic image of M = Inv h X | u i = 1 ( i ∈ I ) i is the group defined bythe presentation Gp h X | u i = 1 ( i ∈ I ) i .For a monoid or inverse monoid M , we denote by U M the group of units of M .So U M is the set of all invertible elements of the monoid M . If G is a group and A ⊆ G , we denote by Gp h A i the subgroup generated by A , while Mon h A i is the submonoid of G generated by A .Throughout the paper, if M is an inverse monoid generated by a set X , givenany two words u, v ∈ X ∗ we say u = v in M to mean that that the two wordsrepresent the same element of the inverse monoid, and write u ≡ v to mean that u and v are identical as words in X ∗ . The same comments apply in particular whenwe are working with a group G generated by a set X . Also, in this situation, givenany subset A of X ∗ by the submonoid of G generated by A , we mean the submonoidgenerated by the set of all elements of G represented be words in A (that is, theimage of A in G ). We write this as Mon h A i ≤ G . Similarly we talk about thesubgroup Gp h A i of G generated by the set of words A .2.3. E -unitary inverse monoids. Let M be an inverse monoid and A ⊆ M . Thesubset A is said to be left unitary if a ∈ A , s ∈ M and as ∈ A imply s ∈ A . Thenotion of right unitary subset is defined dually. A subset is unitary if it is bothleft and right unitary. As is shown, for example, in [24, Proposition 2.4.3], when A is E = E ( M ), the set of idempotents of M , the properties of being left, right andtwo-sided E -unitary coincide, thus defining the class of E -unitary inverse monoids .Each inverse monoid M has the minimum group congruence σ , the smallestcongruence of M such that M/σ is a group (see [24, Theorem 2.4.1]). On the otherhand, on any inverse monoid M one can define the compatibility relation ∼ by a ∼ b if and only if ab − , a − b ∈ E ( M ) . Whenever M is E -unitary, the relation ∼ is an equivalence relation, and, further-more, a congruence of M . In general, σ is the congruence generated by the relation ∼ . In fact, the following characterisation holds (see [24, Theorem 2.4.6]). Proposition 2.1.
An inverse monoid M is E -unitary if and only if σ = ∼ . HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 7
Turning to the case of special inverse monoids M = Inv h X | u i = 1 ( i ∈ I ) i , wehave that M/σ = G = Gp h X | u i = 1 ( i ∈ I ) i , and σ is simply the kernel relation ofthe natural homomorphism M → G . Therefore, we immediately get the followingwell-known result. Lemma 2.2.
Assume that the inverse monoid M = Inv h X | u i = 1 ( i ∈ I ) i is E -unitary, and let u, v ∈ X ∗ be such that u = v holds in G = Gp h X | u i = 1 ( i ∈ I ) i .Then u ∼ v holds in M . Let us repeat the main result of [19], which confirmed a conjecture formulatedearlier in [31].
Theorem 2.3. ([19, Theorem 4.1])
If the word w ∈ X ∗ is cyclically reduced thenthe inverse monoid M = Inv h X | u = 1 i is E -unitary. In [19] one can find an example of a special inverse monoid with more than onedefining relation, and with both defining relators being cyclically reduced words,which is non- E -unitary, so the one-relator condition is essential here.2.4. Decision problems in finitely generated groups, finite state automata,the Benois Theorem.
Let G = h X i be a finitely generated group with canonicalhomomorphism π : X ∗ → G , let A be a finite subset of X ∗ and let M = Mon h A i be the submonoid of G generated by A . The membership problem for M in G isthe following decision problem:INPUT: A word w ∈ X ∗ .QUESTION: wπ ∈ M ?In other words, is w equal in G to some product of words from A ?Given a one-relator group G = Gp h X | w = 1 i we define the associated prefixmonoid P w to be the submonoid P w = Mon h pref( w ) i ≤ G, where pref( w ) denotes the set of all prefixes of the word w . We use suff( w ) todenote the set of all suffixes of w . We stress that the prefix monoid of G actually depends on the presentation of G —it can happen that two different presentationsdefine the same group, while the corresponding prefix monoids are different, asshown in the following simple example. Example 2.4.
Both groups G = Gp h a, b | aba = 1 i and G = Gp h a, b | baa = 1 i are infinite cyclic, that is, free groups of rank 1 generated by a . In this sense, thesetwo presentations define the same group G = G = G . However, the prefix monoidcorresponding to the first presentation is M = Mon h a, ab i = Mon h a, a − i = G , while the prefix monoid for the second presentation is M = Mon h b, ba i =Mon h a − , a − i = Mon h a − i = { , a − , a − , a − , . . . } , clearly a proper submonoidof M .Therefore, we are always going to refer to the prefix monoid of a one-relatorgroup defined by a explicitly stated presentation G = Gp h X | w = 1 i , or make surethat the presentation for G is clear from the context. Proceeding in this vein, wesay that the one-relator group G defined by the presentation Gp h X | w = 1 i hasdecidable prefix membership problem if the membership problem for P w in G isdecidable. The following crucial connection to the word problem of one-relator IGOR DOLINKA AND ROBERT D. GRAY special inverse monoids was made in [19]. It is an immediate consequence of [19,Theorem 3.3].
Theorem 2.5. ([19])
Let w ∈ X ∗ be a word such that the inverse monoid M = Inv h X | w = 1 i is E -unitary. If the prefix membership problem for G = Gp h X | w = 1 i is decidable,so is the word problem for M . Let M = Inv h X | w = 1 i and let R be the set of right units of M . Then R is a submonoid, but in general not an inverse submonoid, of M . Clearly, everyprefix of w represents an element of R . Conversely, by the geometric argumentgiven in the second paragraph of the proof of [19, Proposition 4.2], for every word u ∈ X ∗ representing an element of R there are prefixes p , . . . , p k of w such that u = p . . . p k holds in M . The statement of this proposition in [19] actually assumesthat w is cyclically reduced; but it is straightforward to check that the correspond-ing argument does not make use of that assumption. Hence, in general in a specialone-relator inverse monoid M = Inv h X | w = 1 i every right unit can be expressedas a product of prefixes of w . Therefore, the image of R under the natural homo-morphism M → G = Gp h X | w = 1 i is precisely P w .We use a standard model for finite state automata (FSA): this is a quintuple A = ( Q, Σ , E, I, T ), where Q is a finite set of states, Σ is the alphabet, I, T ⊆ Q are the initial and final states, respectively, and E ⊆ Q × Σ × Q are the transitions.The automaton A accepts the word w ∈ Σ ∗ if there is a path from an initial state toa final state labelled by w ; the set of all accepted words L ( A ) ⊆ Σ ∗ is the language of the FSA. By Kleene’s Theorem [15], the class of languages of FSA is preciselythe class of regular languages.Given a group G , the class of rational subsets of G is the smallest set containingall finite subsets of G that is closed with respect to union, product and submonoidgeneration. Note that it is immediate from this definition that every finitely gen-erated submonoid M of G is a rational subset of G . Combining this notion withKleene’s Theorem, it is immediate to arrive at the following result. Proposition 2.6.
Let G = Gp h X i be a finitely generated group and let π : X ∗ → G be the corresponding canonical homomorphism. A subset R ⊆ G is a rational if andonly if there is a FSA A over X such that R = L ( A ) π . We note that the empty set is a regular language, and the empty set is a rationalsubset of G for any group G .This proposition shows that FSA are convenient vehicles to define rational sub-sets in finitely generated groups by a finite amount of data. The rational subsetmembership problem [25] for a finitely generated group G = Gp h X i with the canon-ical homomorphism π : X ∗ → G is the following decision problem.INPUT: A FSA A over X and a word w ∈ X ∗ .QUESTION: wπ ∈ L ( A ) π ?A particularly pleasant algorithmic properties are enjoyed by free groups, as aconsequence of a key result due to Benois [4] (see also e.g. [3, 6]). Theorem 2.7. ([4]) If L ⊆ X ∗ is a regular language over X then red( L ) is also aregular language. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 9
Corollary 2.8.
Let X be a finite set. Then the free group F G ( X ) has decidablerational subset membership problem. In particular, F G ( X ) has decidable submonoidmembership problem and subgroup membership problem. Also, the rational subsetsof F G ( X ) are closed for intersection and complement. Note that, in general, rational subsets of (finitely generated) groups need not beclosed under intersection nor complement.2.5.
A theorem of Herbst on rational subsets of subgroups of groups.
Let G be a finitely generated group, let U be a subgroup of G , and let Q be a subsetof U . It is immediate from the definition of rational subset that if Q is a rationalsubset of U then Q is also a rational subset of G . The converse is also true, but it isfar less obvious. It was proved by Herbst in [13] that, under the above assumptions,if Q is a rational subset of G then Q is a also rational subset of U .In this subsection we will explain and give a proof of an effective version ofHerbst’s theorem which will be of crucial importance for us in this paper.Let us begin by recalling some basic facts about rational subsets and regularlanguages. Let M be a monoid. Just as we did for groups above, we can talkabout the rational subsets of the monoid M . The rational subsets of M are thesets that can be obtained from finite subsets using the operations of union, productand submonoid generation. If A is a set of generators for M then a subset of M isrational if and only if it is accepted by a FSA over A . Here a subset U of M is saidto be accepted by a FSA over A if, with π : A ∗ → M the canonical homomorphism,there is a FSA A such that L ( A ) π = U . It is a standard fact that a subset U ofthe free monoid A ∗ is the language of a finite state automaton if and only if U canbe described by a rational expression over A . A rational expression for a subset U of A ∗ is a formal expression which gives a description of a way of constructingthe set U from finite subsets using finitely many operations of union, product andthe Kleene star operation (which is submonoid generating in A ∗ ). For example,( ab ) ∗ ∪ ( ba ) ∗ ∪ a ( ba ) ∗ ∪ b ( ab ) ∗ , is a rational expression for the regular language ofall words with alternating a s and b s. Of course, different rational expressions candescribe the same regular language. It is easy to show that there is an algorithmwhich takes any rational expression over A and constructs a FSA recognising thelanguage that the rational expression defines, and conversely there is an algorithmwhich given a FSA A computes a rational expression defining L ( A ). It follows thatif M is a monoid generated by a set A , then a subset U of M is rational if and only ifthere is a rational expression over A which defines U . We will not be working withregular expressions much in this paper, but we shall need this notion when makingreference to proofs of Herbst below. For a formal definition of rational expressionwe refer the reader to Section 2 of the book [39]. Theorem 2.9.
Let G be a finitely generated group with finite generating set X ,and canonical homomorphism π : ( X ∪ X − ) ∗ → G . Suppose further that G hasa recursively enumerable word problem. Let U be a finitely generated subgroup of G with finite generating set Y and canonical homomorphism σ : ( Y ∪ Y − ) ∗ → U .Then for every subset Q of U we have Q ∈ Rat( G ) ⇔ Q ∈ Rat( U ) . Furthermore there is an algorithm which (1) for any FSA A over X ∪ X − such that L ( A ) π ⊆ U computes a FSA B over Y ∪ Y − such that L ( B ) σ = L ( A ) π , and there is an algorithm which (2) for any FSA B over Y ∪ Y − computes a FSA A over X ∪ X − such that L ( A ) π = L ( B ) σ .Proof. Throughout this proof we make use of the same notation and conventionsas in [13]. From the discussion above we know that there is an algorithm which willtake a FSA A over A recognising Q and use it to compute a rational expression ρ over A defining L ( A ).We now show how the argument used in the proof of [13, Proposition 5.2] canbe used to prove the following claim. Claim.
There is an algorithm which takes as input any rational expression T over X ∪ X − such that L ( T ) π ⊆ U and returns a rational expression T over Y ∪ Y − such that σ ( L ( T )) = π ( L ( T )) ⊆ U .Here L ( R ) denotes the language defined by a rational expression R . It followsfrom the discussion preceding the statement of the theorem that once this claimhas been established then (1) will follow. We prove Claim 2.5 by induction onthe starheight, denoted SH( T ). It is important to note that here by SH( T ) wemean the starheight of the rational expression T , which might well be larger thanthe minimum starheight of the language L ( T ) defined by this rational expression.That is, it is possible that there is a rational expression T ′ with SH( T ′ ) < SH( T )and with L ( T ′ ) = L ( T ).When SH( T ) = 0 then clearly there is an algorithm which replaces T by anexpression of the form w ∪ w ∪ . . . ∪ w m for a finite set { w , . . . , w m } ⊆ ( X ∪ X − ) ∗ and π ( w i ) ∈ U for 1 ≤ i ≤ m . So we mayassume that T has this form. Since G is recursive enumerable can apply Lemma 4.4(see below). By Lemma 4.4 there is an algorithm that computes words w , . . . , w m ∈ ( Y ∪ Y − ) ∗ such that σ ( w i ) = π ( w i ) for 1 ≤ i ≤ m . Hence σ ( { w , . . . , w m } ) = π ( { w , . . . , w m } ) and this deals with this case.For the induction step now let us assume SH( T ) >
0. We can write T = τ ∪ . . . ∪ τ q where each τ j is a rational expression of the form w T ∗ w T ∗ w . . . w n T ∗ n w n +1 , (2.1)where each w i ∈ ( X ∪ X − ) ∗ and each T i is a rational expression over ( X ∪ X − ) withSH( T i ) < SH( T ) for 1 ≤ i ≤ n . Consider such an expression (2.1) corresponding to τ k . Set S i = w w . . . w i T i w − i . . . w − w − for 1 ≤ i ≤ n noting that SH( S i ) = SH( T i ) < SH( T ) and, by [13, Lemma 5.1], wehave π ( L ( S i )) ⊆ U . By induction we can suppose that our algorithm takes S i andreturns a rational expression S i over Y ∪ Y − such that π ( L ( S i )) = σ ( L ( S i )). Itfollows from the argument given in the proof of [13, Proposition 5.2] that π ( L ( S ∗ S ∗ . . . S ∗ n w w . . . w n +1 )) = π ( L ( τ k )) . Then we instruct our algorithm to compute S ∗ S ∗ . . . S n ∗ w . . . w w n +1 , HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 11 which is a rational expression over Y ∪ Y − such that σ ( L ( S ∗ S ∗ . . . S n ∗ w . . . w w n +1 )) = π ( L ( S ∗ S ∗ . . . S ∗ n w w . . . w n +1 )) = π ( L ( τ k )) . Repeating this for all t k in T = τ ∪ . . . ∪ τ q our algorithm computes a rationalexpression T over Y ∪ Y − satisfying Claim 2.5. It follows from this argumentthat there is a recursive algorithm satisfying the requirements of Claim 2.5. Thiscompletes the proof of part (1).(2) By the comments preceding the statement of the theorem, there is an algo-rithm which takes B and computes a rational expression R over Y ∪ Y − such that L ( R ) σ = L ( B ) σ ⊆ U . For each y ∈ Y let ˆ y ∈ ( X ∪ X − ) ∗ such that yσ = ˆ yπ in U . Extend this notation to y ∈ Y − where d y − = ˆ y − , the formal inverse of theword. Let ξ : ( Y ∪ Y − ) ∗ → ( X ∪ X − ) ∗ be the unique homomorphism satisfying yξ = ˆ y for all y ∈ Y ∪ Y − . Let S = Rξ be the expression obtained by replacingevery word w in the rational expression w by the word wξ . Then S is a rationalexpression over X ∪ X − such that L ( S ) π = L ( R ) σ . Finally apply the algorithmthat takes S and returns a FSA A over X ∪ X − such that L ( A ) π = L ( S ) π . Thiscompletes the proof. (cid:3) Lemma 2.10.
Let G be a finitely generated group with finite generating set X , andcanonical homomorphism π : ( X ∪ X − ) ∗ → G . Furthermore suppose that G has arecursively enumerable word problem. Let A, B ≤ G be finitely generated subgroupswith φ : A → B is an isomorphism. Then there is an algorithm which takes anyFSA A over X ∪ X − such that L ( A ) π ⊆ A computes a FSA B over X ∪ X − suchthat L ( B ) π = ( L ( A ) π ) φ .Proof. Let Y be a finite generating set for A with canonical homomorphism σ :( Y ∪ Y − ) ∗ → A . Apply the algorithm from Theorem 2.9(1) to compute a FSA Q over Y ∪ Y − such that L ( Q ) σ = L ( A ) π . Since φ is an isomorphism it followsthat Y is a finite generating set for the group B with canonical homomorphism σφ . Now apply Theorem 2.9(2) to the pair G , B , with respect to the canonicalhomomorphism σφ to compute a FSA B over X ∪ X − such that L ( Q ) σφ = L ( B ) π .This completes the proof since L ( B ) π = L ( Q ) σφ = L ( A ) πφ. (cid:3) Remark . Note that the hypotheses of Lemma 2.10 hold in particular if G hasdecidable word problem.3. Unital and conservative factorisations
Let w ∈ X ∗ . A factorisation of w is a decomposition w ≡ w . . . w m where w , . . . , w k ∈ X ∗ . The words w i (1 ≤ i ≤ k ) are called the factors of thisfactorisation.Let u , . . . , u k be distinct words in X ∗ and let w ∈ X ∗ . If w belongs to thesubmonoid of X ∗ generated by u , . . . , u k , then there is a word w ′ ( x , . . . , x k ) overthe alphabet { x , . . . , x k } such that w ≡ w ′ ( u , . . . , u k ) . This expression gives a factorisation of w where each factor is equal to u j for some1 ≤ j ≤ k . A factorisation w ≡ v . . . v n is finer than a factorisation w ≡ w . . . w m if thereexist 0 ≤ k ≤ · · · ≤ k m − ≤ n such that w ≡ v . . . v k , w m ≡ v k m − +1 . . . v n and w i ≡ v k i − +1 . . . v k i for all 1 < i < m .We say that a factorisation w ≡ w . . . w m is unital if each of its factors representsa unit of the inverse monoid M = Inv h X | w = 1 i . In such a case, w , . . . , w m arecalled invertible pieces . It is an easy exercise to show that there is a unique finestunital factorisation of w ; this is a decomposition into minimal invertible pieces w ≡ w . . . w m , so that no proper prefix of any of the words w i represents anelement of U M . In fact, there is a strong connection between the minimal invertiblepieces of w and the group of units U M . Proposition 3.1.
Let M = Inv h X | w = 1 i where w ∈ X ∗ . Then the minimalinvertible pieces w , . . . , w m of w generate the group U M .Proof. This follows from the argument given in the proof of [19, Proposition 4.2].Note that, as already mentioned above, while in the statement of [19, Proposition4.2] it is assumed that the words in the defining relators are all cyclically reduced,that assumption is not used anywhere in the proof, and the proposition holds withthat assumption removed. (cid:3)
We now briefly recall the Adjan overlap algorithm (as presented in [23]). Let X be an alphabet, and assume W is a finite set of words over X . Inductively,we define a sequence W i , i ≥
0, of sets of words as follows. Assuming that W k isdetermined, we let u ∈ W k +1 if one of the following conditions hold:(i) u ∈ W k and u pref( v ) ∪ suff( v ) for all v ∈ W k \ { u } ;(ii) there exist v, v ′ ∈ X ∗ not both empty such that uv, v ′ u ∈ W k ;(iii) there exist v, v ′ ∈ X ∗ such that v is non-empty and uv, vv ′ ∈ W k ;(iv) there exist v, v ′ ∈ X ∗ such that v ′ is non-empty and v ′ u, vv ′ ∈ W k .It is fairly easy to show that the sequence of sets of words W k stabilises after finitelymany steps, i.e. that there exists k such that W k = W k for all k ≥ k . DefineΓ = W k .This algorithm is applied to special monoid presentations by setting W to be theset of relator words in the considered presentation; in particular, for Mon h X | w = 1 i we put W = { w } . The same algorithm can be applied to special inverse monoidpresentation, and in particular to one-relator special inverse monoids Inv h X | w = 1 i by replacing X by X .It is easy to see that the words from the set Γ generated by the Adjan algorithmall represent invertible elements of the monoid. Adjan [1] proved that in the case ofone-relator special monoid presentations, this algorithm actually computes the de-composition of the defining relator word w into minimal invertible pieces. However,this is no longer true for arbitrary special monoid presentations. In fact, more thanthis, for finitely presented special monoids the problem of computing the minimalinvertible pieces is known to be undecidable. Indeed, in [37] it is shown that it isundecidable whether a finitely presented special monoid is a group, and if therewere an algorithm or computing the minimal invertible pieces then that algorithmcould be used to decide whether a special monoid is a group by testing whether allthe generators appear in at least one relator, and that the minimal invertible piecesall have size one.For special inverse monoids the Adjan algorithm fails to compute the minimalinvertible pieces even in the one-relator case. This is illustrated by the following HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 13 example, which appeared first in print in [31] (see also [19]), sometimes known asthe
O’Hare example (because it was constructed by Margolis and Meakin whilewaiting for a connecting flight at the O’Hare International Airport, Chicago).
Example 3.2.
Let M = Inv h a, b, c, d | abcdacdadabbcdacd = 1 i . Applying a geometric method called
Stephen’s procedure [41] it was shown in [31]that w ≡ ( abcd )( acd )( ad )( abbcd )( acd )is a unital factorisation of the relator word. In fact, the same methods show thatany subword of the relator word representing a unit of M must begin with either a or d − and end with either d or a − , so it follows that this is the decompositionof w into minimal invertible pieces. Thus it follows from the previous propositionthat { abcd, acd, ad, abbcd } is a generating set of U M ( abbcd can be shown to beredundant).On the other hand, this is not something Adjan algorithm would discover, asΓ = { abcdacdadabbcdacd } in this case. Indeed, this is the reason why the monoid Mon h a, b, c, d | abcdacdadabbcdacd = 1 i has a trivial group of units, in contrast tothe inverse monoid defined by the same presentation which has an infinite group ofunits.To address this, in [11] Gray and Ruˇskuc devised a new, finer pieces computingalgorithm better suited for special inverse monoid presentations M = Inv h X | w i = 1 ( i ∈ I ) i , called the Benois algorithm (because it relies on the Benois theorem and its conse-quences). Namely, let U = { pref( w i ) : i ∈ I } ∪ { pref( w − i ) : i ∈ I } . Observe that all the words in U represent right invertible elements of the inversemonoid M . Let V = Mon h U i be the submonoid of the free group F G ( X ) generatedby the words from U viewed as elements of the free group i.e. the submonoidof F G ( X ) generated by red( U ). Now by Corollary 2.8 of Benois’ Theorem thefree group F G ( X ) has decidable submonoid membership problem. So, we canalgorithmically test, for each prefix p of some word w i , whether p − represents anelement of V . If it does then p − is right invertible which implies p is left invertible,and hence since p is also right invertible being a prefix of w i , it follows that theword p represents an invertible element of the monoid M . Thus this algorithm givesa method for finding invertible pieces of relators. The collection of all such prefixesfor which the answer is ‘yes’ naturally gives rise to factorisations w i ≡ w i, . . . w i,k i for all i ∈ I , such that for every prefix p of w i we have p − ∈ V if and only if p ≡ w i, . . . w i,j for some j ∈ { , . . . , k i } . It is clear from the definition that for all i ∈ I the decomposition of w i into piecescomputed by the Benois algorithm is unital. In fact, it is shown in [11] that foreach word w i the factorisation computed by the Benois algorithm is a refinementof the decomposition computed by the Adjan algorithm. In the same way as for special monoid presentations, there is no algorithm whichtakes finitely presented special inverse monoids and computes the minimal piecesof the defining relator words. In the particular case of one-relator special inversemonoids, whether the Benois algorithm computes the minimal invertible pieces isan open problem. It may be shown (see [11]) that when applied to the O’Haremonoid the Benois algorithm does compute the minimal invertible pieces of thedefining relator, giving the unital factorisation of the defining relator describedabove in Example 3.2. This example shows that there are cases where the Benoisalgorithm preforms strictly better than the Adjan algorithm, in the sense that itgives a decomposition which is a strict refinement of the Adjan decomposition.While it is not known whether the Benois algorithm computes the minimal in-vertible pieces for one-relator special inverse monoids, one very important centraltheme of the present paper will be that it is often the case that it is sufficient tofind some suitable (not necessarily minimal) unital factorisation of w in order toprove that the monoid M = Inv h A | w = 1 i has decidable word problem. In thissense, the Benois algorithm will provide a key tool for solving the word problem forcertain examples and classes of one-relator special inverse monoids.Now we introduce another type of factorisation of a word that makes computingand handling the prefix monoid of a one-relator group presentation somewhat easierand more manageable. Let w ∈ X ∗ . Then for a factorisation w ≡ w . . . w k let P ( w , . . . , w k ) denote the submonoid of G = Gp h X | w = 1 i generated by ele-ments k [ i =1 pref( w i ) . It is quite easy to see that we always have P w ⊆ P ( w , . . . , w k ). In the case that P w = P ( w , . . . , w k ) then we say that the factorisation w ≡ w . . . w k is conser-vative . The next result establishes a connection between unital and conservativefactorisations. Theorem 3.3.
Let w ∈ X ∗ . (i) Any unital factorisation of w is conservative. (ii) If Inv h X | w = 1 i is E -unitary (in particular, if w is cyclically reduced) thenevery conservative factorisation of w is unital.Proof. (i) Assume w ≡ w . . . w k is a unital factorisation, so that each word w i ,1 ≤ i ≤ k , represents a unit of M = Inv h X | w = 1 i . Let p be a prefix of w i forsome i . Then p ′ ≡ w . . . w i − p is a prefix of w and thus in G = Gp h X | w = 1 i this word represents an element ofthe prefix monoid P w . Since p ′ is right invertible in M and w . . . w i − is invertiblein M it follows that p is right invertible in M . Thus by the remarks followingTheorem 2.5 it follows that p is equal in M to a product of prefixes of w . Applyingthe natural homomorphism from M to its maximal group image, it follows that p represents in G an element of the prefix monoid P w . It follows that P ( w , . . . , w k )must be contained in P w , so in fact we have an equality of these two submonoidsof G , confirming that the considered factorisation in conservative. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 15 (ii) Assume that M = Inv h X | w = 1 i is E -unitary and let w ≡ w . . . w k be aconservative factorisation of w , so that we have P ( w , . . . , w k ) = P w . Our aim isto show that w represents a unit of M , that is, we want to prove that w ∈ U M .Since w w . . . w k = 1 it is immediate that w is a right unit of M . Now we needto prove that w is a left unit which clearly is equivalent to proving that w − is aright unit of M . In G = Gp h X | w = 1 i we have w − = w . . . w k ∈ P ( w , . . . , w k ) = P w since the factorisation is conservative. Thus in G we have w − = p . . . p m for some p i ∈ pref( w ) for all 1 ≤ i ≤ m . Since M is E -unitary, it follows fromLemma 2.2 that we have w − ∼ p . . . p m in M . This implies that w p . . . p m ∈ E ( M ). Since the only right invertible idempotent in an inverse monoid is theidentity element it follows that w p . . . p m = 1 in M . From this we deduce that w p . . . p m w = w and p . . . p m w p . . . p m = p . . . p m and hence w − = p . . . p m in M . We conclude that w − represents an invertible element of M . This concludesthe proof that w represents an invertible element of the monoid M .Consequently, w . . . w k w = w − ( w . . . w k ) w = w − w = 1 , so now we can argue, by repeating the previous argument, that w ∈ U M . In thissay we can prove that each factor represents an invertible element of the monoid.We conclude that the considered factorisation is unital. (cid:3) Corollary 3.4.
Let w ∈ X ∗ . Then the Benois algorithm applied to w computes aconservative factorisation of w . This importance of this corollary will become in Section 5 where there are theo-rems whose hypotheses are that the defining relator admits a conservative factori-sation satisfying certain properties. This corollary will help us verify that thesehypotheses do hold in particular concrete examples.4.
Deciding membership in submonoids of amalgamated free products
As explained in the introduction, our aim in this section is to give two generaldecidability results concerning the membership problem for certain submonoids offree amalgamated products B ∗ A C of finitely generated groups. Then in the nextsection these results will be applied to the prefix membership problem for certainone-relator groups.Let us now fix the notation and conventions that will be in place throughout thissection. We refer the reader to [27] for more background and proofs of standardresults. Throughout this section we use B ∗ A C to denote the amalgamated freeproduct of two groups B and C over a group A . We shall always assume that allthree of these groups are finitely generated by, respectively, the finite sets X , Y and Z . We formalise this by fixing canonical homomorphisms π : X ∗ → B , θ : Y ∗ → C and ξ : Z → A .In addition, we have two injective homomorphisms f : A → B and g : A → C and B ∗ A C is the corresponding pushout in the category of groups. Then theamalgamated free product B ∗ A C is defined by the presentation obtained by takingthe disjoint union of presentations for B and C together with additional defining relations zf = zg for all z ∈ Z . To be more precise, let Z = { z , . . . , z m } . Thenthere are words α i ∈ X ∗ and β i ∈ Y ∗ , 1 ≤ i ≤ m , such that the mappings f : Z → X ∗ and g : Z → Y ∗ defined by z i f = α i and z i g = β i induce injectivehomomorphisms f : A → B and g : A → C . Note that f and g are used to denoteboth mappings from A into B and C , respectively, and also to define mappings onwords f : Z ∗ → X ∗ and g : Z ∗ → Y ∗ . Consequently, wf ≡ w ( α , . . . , α m ) and wg ≡ w ( β , . . . , β m )for any w ∈ Z ∗ . Recall that w ( α , . . . , α m ) is the word in X ∗ obtained by replacing z i by α i for each letter z i ∈ Z in the word w . Using this notation, we may speakabout the membership problem for the subgroup A in B and C respectively: thealgorithmic question is whether there is an algorithm which takes as input any word u over X (resp. Y ) and decides whether or not there exists a word w ∈ Z ∗ suchthat u = w ( α , . . . , α m ) holds in B (resp. u = w ( β , . . . , β m ) holds in C ). We willoften identify A with its image in B ∗ A C . In this way, each of A , B and C is viewedas a subset of the amalgamated free product B ∗ A C and B ∩ C = A . So, for any b ∈ B by saying that b belongs to A we mean that b ∈ Af , and analogously we talkabout an element c ∈ C belonging to A .We are now in a position to state the two main general results of this section. Theorem A.
Let G = B ∗ A C , where A, B, C are finitely generated groups suchthat both
B, C have decidable word problems, and the membership problem for A in both B and C is decidable. Let M be a submonoid of G such that the followingconditions hold: (i) A ⊆ M ; (ii) both M ∩ B and M ∩ C are finitely generated and M = Mon h ( M ∩ B ) ∪ ( M ∩ C ) i ;(iii) the membership problem for M ∩ B in B is decidable; (iv) the membership problem for M ∩ C in C is decidable.Then the membership problem for M in G is decidable. Definition 4.1. (Closed for rational intersections) Let G be a finitely generatedgroup, generated by a finite set Ω with canonical homomorphism τ : Ω → G , andlet H be a finitely generated subgroup of G . We say that H in G is closed forrational intersections if R ∩ H ∈ Rat( G ) for all R ∈ Rat( G ). We say that H in G is effectively closed for rational intersections if it is closed for rational intersectionsand moreover that there is an algorithm which given a FSA A over Ω computes aFSA A H over Ω such that L ( A H ) τ = ( L ( A ) τ ) ∩ H . Theorem B.
Let G = B ∗ A C , where A, B, C are finitely generated groups. Let M be a submonoid of G such that both M ∩ B and M ∩ C are finitely generatedand M = Mon h ( M ∩ B ) ∪ ( M ∩ C ) i . Assume further that the following conditionshold: (i) B and C have decidable rational subset membership problems; (ii) A ≤ B is effectively closed for rational intersections; (iii) A ≤ C is effectively closed for rational intersections.Then the membership problem for M in G is decidable. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 17
Comparing these two theorems, Theorem A shows that the membership problemfor suitably nice submonoids in relatively general amalgamated free products isdecidable, while Theorem B shows that under stronger assumptions on the groupsof the amalgamated free product, we get a much broader family of submonoids inwhich we can decide membership.As is well known, see e.g. [27, pp. 186-187], each element g ∈ G = B ∗ A C canbe written as g = b c . . . b n c n with b i ∈ B and c i ∈ C for all 1 ≤ i ≤ n . The above representation is said to be in reduced form if • If n > b i does not belong to A for all i = 1, b is either equal to 1or else does not belong to A , c i does not belong to A for all i = n , and c n is either equal to 1 or else does not belong to A . • If n = 1 then if both b and c belong to A then exactly one of b = 1 or c = 1 holds.Moreover, a word w ≡ u v . . . u k v k where u i ∈ X ∗ and v i ∈ Y ∗ , 1 ≤ i ≤ k , is saidto be a word in reduced form if and only if ( u π )( v θ ) . . . ( u k π )( v k θ ) is a reducedform. The following result is standard and can be proved e.g. by applying [27,Theorem IV.2.6]. Lemma 4.2.
An equality of two reduced forms b c . . . b n c n = p q . . . p k q k holds in G = B ∗ A C if and only if the following conditions are satisfied: (i) n = k ; (ii) there exist a , a , . . . , a n − , a n = 1 ∈ A such that for all ≤ i ≤ n wehave p i = a − i − b i a i − and q i = a − i − c i a i . Before embarking on the proofs of Theorems A and B, we first collect severaluseful lemmas.
Lemma 4.3.
Let G = B ∗ A C and let M be a submonoid of G such that M = Mon h ( M ∩ B ) ∪ ( M ∩ C ) i . Then every element g ∈ M can be written in reduced form g = p q . . . p n q n where p i ∈ M ∩ B and q i ∈ M ∩ C for ≤ i ≤ n .Proof. By assumption we can write g = r s . . . r k s k for some r i ∈ M ∩ B and s i ∈ M ∩ C , 1 ≤ i ≤ k . If this is in reduced form we aredone. Otherwise, there is some term, say r i , with r i ∈ A . Since r i ∈ ( M ∩ B ) ∩ A = M ∩ A ⊆ M ∩ C , we have s ′ i − = s i − r i s i ∈ M ∩ C , so upon relabelling r ′ j = r j +1 , s ′ j = s j +1 for i ≤ j < k we get g = r s . . . r i − s ′ i − r ′ i s ′ i . . . r ′ k − s ′ k − , a shorter alternating product of elements of M ∩ B and M ∩ C . A similar argumentapplies if there is a term s i with s i ∈ A . At the end of this process we arrive at a reduced form p q . . . p n q n for g whose terms alternatively belong to M ∩ B and M ∩ C , completing the proof of the lemma. (cid:3) A crucial algorithmic aspect is settled by the following observation. The proofis routine and so it is omitted.
Lemma 4.4.
Let G be a group finitely generated by Ω and suppose that G has arecursively enumerable word problem. Let w , . . . , w n ∈ Ω ∗ and H = Gp h w , . . . , w k i ≤ G. Suppose that the membership problem for H in G is decidable. Then there existsan algorithm which, given any word w ∈ Ω ∗ for which the algorithm for testingmembership in H returns ‘yes’, outputs a word u ( t , . . . , t k ) ∈ T ∗ , where T = { t , . . . , t k } , such that w = u ( w , . . . , w k ) holds in G . The following key lemma identifies conditions under which the process in Lemma4.3 can be performed algorithmically.
Lemma 4.5.
Let G = B ∗ A C and assume that the following conditions hold: • B and C both have recursively enumerable word problem; • the membership problem for A in B is decidable; and • the membership problem for A in C is decidable.Then there is an algorithm which takes as input any word w ≡ u v . . . u k v k where u i ∈ X ∗ and v i ∈ Y ∗ , ≤ i ≤ k , and returns a word p q . . . p n q n in reduced formwhere p i ∈ X ∗ and q i ∈ Y ∗ , ≤ i ≤ k , such that w = p q . . . p n q n holds in G .Proof. It follows by the assumptions that there is algorithm which decides for eachof the terms u i and v j whether or not that term represents an element of A . If noneof the terms represents an element of A then u v . . . u k v k is a word in reduced formand the algorithm terminates and outputs this word. Otherwise, suppose that some u i or v j does represent an element of A . Let u i be the first term that the algorithmdetects as belonging to A . Since B has recursively enumerable word problem andthe membership problem for A within B is decidable, we can apply Lemma 4.4.This tells us that there is an algorithm which takes any such u i as input and returnsa word u ′ ∈ Z ∗ such that we have u i = u ′ ( α , . . . , α m ) , in B , where α i = z i f for 1 ≤ i ≤ m . The algorithm then computes the word v i − u ′ ( β , . . . , β m ) v i ∈ Y ∗ . Let v ′ i − denote this word. Then, since u ′ ( α , . . . , α m ) = u ′ ( β , . . . , β m ) in G , it follows that u v . . . u i − v ′ i − u ′ i v ′ i . . . u ′ k − v ′ k − , is equal to w in G , where u ′ j = u j +1 and v ′ j = v j +1 for i ≤ j < k . This word hasstrictly fewer terms than the input word w . If v i is the first term that the algorithmdetects as belonging to A then a similar argument applies, working within C . Inthis way, in a finite number of steps the algorithm eventually terminates outputtinga word in reduced form. (cid:3) The following straightforward consequence of Theorem 2.9 will be important forthe proof of Theorem B.
HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 19
Lemma 4.6.
Let G = B ∗ A C where A, B, C are finitely generated groups withfinite generating sets X , Y , and Z respectively, and canonical homomorphisms π : X ∗ → B , θ : Y ∗ → C , ξ : Z ∗ → A . Then we have the following. (i) There is an algorithm which takes any FSA P over X such that [ L ( P )] π ⊆ A as input and returns a FSA P ′ over Z , and a FSA P ′′ over Y , such that [ L ( P ′′ )] θ = [ L ( P ′ )] ξ = [ L ( P )] π. (ii) There is an algorithm which takes any FSA Q over Y such that [ L ( Q )] θ ⊆ A as input and returns a FSA Q ′ over Z , and a FSA Q ′′ over X , such that [ L ( Q ′′ )] π = [ L ( Q ′ )] ξ = [ L ( Q )] θ. Proof of Theorem A.
A key ingredient in our proof is summarised in the fol-lowing auxiliary result, which strengthens Lemma 4.3 under the stronger conditionsof Theorem A.
Proposition 4.7.
Assuming all the notation and conditions from Theorem A, let g = b c . . . b n c n be an element of G = B ∗ A C written in reduced form. Then g ∈ M if and only if b i ∈ M ∩ B and c i ∈ M ∩ C for all ≤ i ≤ n .Proof. ( ⇒ ) By Lemma 4.3, we can write g = p q . . . p m q m in reduced form such that p i ∈ M ∩ B and q i ∈ M ∩ C for 1 ≤ i ≤ m . Now, byLemma 4.2 we have m = n and b i = a − i − p i a i − and c i = a − i − q i a i for some a j ∈ A , 0 ≤ j ≤ n , 1 ≤ i ≤ n . But this implies b i ∈ A ( M ∩ B ) A ⊆ M ∩ B and c i ∈ A ( M ∩ C ) A ⊆ M ∩ C , as A ⊆ ( M ∩ B ) ∩ ( M ∩ C ) by condition (i) inTheorem A.( ⇐ ) This is trivial, as M ∩ B and M ∩ C are both subsets of M . (cid:3) Proof of Theorem A.
To prove the theorem we must show that there is an algorithmwhich takes any word w from ( X ∪ Y ) ∗ as input and decides whether or not theword represents an element of the submonoid M . The hypotheses of Lemma 4.5are satisfied since by the assumptions B and C both have decidable word problem,and the membership problem for A in each of B and C is decidable. Applying thislemma we conclude that there is an algorithm that given such a word w returns aword p q . . . p n q n in reduced form where p i ∈ X ∗ and q i ∈ Y ∗ , 1 ≤ i ≤ k , suchthat w = p q . . . p n q n holds in G . It follows from Proposition 4.7 that w ∈ M ifand only if p i ∈ M ∩ B and q i ∈ M ∩ C for all 1 ≤ i ≤ n , which can be decided byconditions (iii) and (iv). (cid:3) With applications in mind, it is worthwhile to record a consequence of TheoremA for free products of groups, arising from the case when the amalgamated subgroup A is trivial. Corollary 4.8.
Let G = B ∗ C , where B, C are finitely generated groups such thatboth
B, C have decidable word problems. Let M be a submonoid of G such that thefollowing conditions hold: (i) both M ∩ B and M ∩ C are finitely generated and M = Mon h ( M ∩ B ) ∪ ( M ∩ C ) i ;(ii) the membership problem for M ∩ B in B is decidable; (iii) the membership problem for M ∩ C in C is decidable.Then the membership problem for M in G is decidable. Proof of Theorem B.
The following result which gives necessary and suffi-cient conditions for an element in reduced form to belong to M , will be essentialfor the proof of Theorem B. Proposition 4.9.
Let G = B ∗ A C and let M be a submonoid of G such that M = Mon h ( M ∩ B ) ∪ ( M ∩ C ) i . Let g = b c . . . b n c n be an element of G = B ∗ A C written in reduced form. For i ∈ { , . . . , n − } define Q i = Q i ( b , c , . . . , b n , c n ) in the following way: Q = { } ,Q k − = ( M ∩ B ) − Q k − b k ∩ A, for ≤ k ≤ n , Q k = ( M ∩ C ) − Q k − c k ∩ A, for ≤ k ≤ n − .Then g ∈ M if and only if c n ∈ Q − n − ( M ∩ C ) . Proof. ( ⇒ ) Assume that g ∈ M . Then by assumption the hypotheses of Lemma 4.3are satisfied and thus the element g can be written in reduced form g = p q . . . p n q n such that p i ∈ M ∩ B and q i ∈ M ∩ C , 1 ≤ i ≤ n . Now, by Lemma 4.2, we musthave b = p a c = a − q a ,b = a − p a , c = a − q a , ... ... b n = a − n − p n a n − , c n = a − n − q n , for some a , . . . , a n − ∈ A . Solving alternatively for a ’s with odd indices from thefirst and with even ones from the second column of equations, we obtain a = p − b ∈ ( M ∩ B ) − b ∩ A = Q ,a = q − a c ∈ ( M ∩ C ) − Q c ∩ A = Q ,a = p − a b ∈ ( M ∩ B ) − Q b ∩ A = Q , ... a n − = p − n a n − b n ∈ ( M ∩ B ) − Q n − b n ∩ A = Q n − . Therefore, from the last equation of the second column we conclude that c n = a − n − q n ∈ Q − n − ( M ∩ C ) , HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 21 as required.( ⇐ ) Assume that g = b c . . . b n c n is such that c n ∈ Q − n − ( M ∩ C ). Then c n = ξ − n − γ n for some ξ n − ∈ Q n − and γ n ∈ M ∩ C . The fact that ξ n − ∈ Q n − = ( M ∩ B ) − Q n − b n ∩ A implies that we can write ξ n − = β − n ξ n − b n for some β n ∈ M ∩ B and ξ n − ∈ Q n − . Continuing this process yields elements β n − , . . . , β ∈ M ∩ B , γ n − , . . . , γ ∈ M ∩ C and ξ i ∈ Q i , 0 ≤ i ≤ n −
1, such that ξ j − = β − j ξ j − b j for 1 ≤ j ≤ n (where ξ = 1), and ξ j = γ − j ξ j − c j for 1 ≤ j ≤ n −
1. Solving each of these equations for b j and c j , substituting intothe reduced form of g , and cancelling the ξ k ’s gives b c . . . b n c n = β γ . . . β n γ n , which belongs to M since β j , γ j ∈ M for all 1 ≤ j ≤ n . (cid:3) Lemma 4.10.
Under the assumptions of Theorem B, in the statement of Proposi-tion 4.9 every set Q i ( ≤ i ≤ n − ) is a rational subset of A .Proof. By assumption both M ∩ B and M ∩ C are finitely generated submonoids of B and C , respectively. Hence ( M ∩ B ) − is a rational subset of B , and ( M ∩ C ) − is a rational subset of C . The lemma follows from this combined with conditions(ii) and (iii) in the statement of Theorem B and the definition of Q i . (cid:3) It is very important to note that the sequence of rational subsets Q i given inProposition 4.9 depends on the reduced form b c . . . b n c n . Proof of Theorem B.
Similarly to the proof of Theorem A, to prove the theoremwe must show that there is an algorithm which takes any word w from ( X ∪ Y ) ∗ asinput and decides whether or not the word represents an element of the submonoid M . By assumption (i) it follows that the groups B and C both have decidable sub-group membership problem, and in particular both have decidable word problem.Condition (i) also implies that the membership problem for A within B is decidable,and for A within C is decidable. Hence, the hypotheses of Lemma 4.5 are satisfied.Applying this lemma we conclude that there is an algorithm that given any suchword w returns a word p q . . . p n q n in reduced form where p i ∈ X ∗ and q i ∈ Y ∗ ,1 ≤ i ≤ k , such that w = p q . . . p n q n holds in G .Set b i = p i π and c i = q i θ for 1 ≤ i ≤ n , and let g = b c . . . b n c n notingthat this is a reduced form for the element g . For each i ∈ { , . . . , n − } let Q i = Q i ( b , c , . . . , b n , c n ) be defined as in the statement of Proposition 4.9. Thenby Lemma 4.10 each of these sets Q i is a rational subset of A , and therefore also arational subset of both B and C . Claim.
There exists an algorithm which for each i ∈ { , . . . , n − } computes • a finite state automaton A i over Z with [ L ( A i )] ξ = Q i , • a finite state automaton B i over X with [ L ( B i )] π = Q i , and • a finite state automaton C i over Y with [ L ( C i )] θ = Q i . Proof of claim.
The algorithm iteratively constructs the triples ( A i , B i , C i ) in thefollowing way. When i = 0 we have Q i = { } and it is clear that an appropriatetriple ( A , B , C ) can be computed e.g. by taking automata that accept only the empty word in each case. Now consider a typical stage i with i >
0. There are twocases depending on the parity of i .First suppose that i is odd, and write i = 2 k −
1. Then by definition Q i =( M ∩ B ) − Q i − b k ∩ A . Since M ∩ B is assumed to be finitely generated, thereis a fixed FSA (depending only on M ) over X , which we denote by B , satisfying[ L ( B )] π = ( M ∩ B ) − . Using B and B i − the algorithm then produces, in theobvious way, a FSA B ( i ) over X such that [ L ( B ( i ) )] π = ( M ∩ B ) − Q i − b k . Thealgorithm given by assumption (ii), in the statement of the theorem, is then appliedto the automaton B ( i ) which yields an automaton B i over X satisfying[ L ( B i )] π = ( M ∩ B ) − Q i − b k ∩ A = Q i . The algorithm then calls as a subroutine the algorithm given in Lemma 4.6 tocompute automata A i and C i the properties given in the statement of the claim.If i is even the procedure is analogous but with the roles of B and C interchanged. (cid:3) To complete the proof, by Proposition 4.9, we have g ∈ M if and only if c n ∈ Q − n − ( M ∩ C ). Using the automata C n − and C , the algorithm produces, in theobvious way, a FSA C ( w ) over Y such that [ L ( C ( w ))] θ = Q − n − ( M ∩ C ).Therefore, in summary we have shown that there is an algorithm which givenany word w ∈ ( X ∪ Y ) ∗ computes a word p q . . . p n q n in reduced form, equal to w in G , and also computes a FSA C ( w ) over Y such that w represents an elementof M if and only if q n θ ∈ [ L ( C ( w ))] θ . This is decidable by condition (i) of thetheorem. (cid:3) Corollary 4.11.
Let X and Y be finite alphabets, and let G = F G ( X ) ∗ A F G ( Y ) such that A is finitely generated. let M be a submonoid of G such that both M ∩ F G ( X ) and M ∩ F G ( Y ) are finitely generated and M = Mon h ( M ∩ F G ( X )) ∪ ( M ∩ F G ( Y )) i . Then the membership problem of M within G is decidable. Applications of amalgamated free product results to the prefixmembership problem
In this section we present several applications of the general results from theprevious section to the prefix membership problem for one-relator groups and theword problem for one-relator inverse monoids.We fix some terminology that will be in place throughout the section. Let v ∈ X ∗ and let x ∈ X . We say that the letter x appears in the word v if either v ≡ v xv or v ≡ v x − v for some words v , v ∈ X ∗ . Given two words w , w ∈ X ∗ we saythat w and w have no letters in common if there is no x ∈ X which appears bothin w and in w . Furthermore, let z ∈ X ∗ and let x ∈ X . We say that z contains x if z ≡ z xz for some z , z ∈ X ∗ .5.1. Unique marker letter theorem.Theorem 5.1.
Let G = Gp h X | w = 1 i and let u = u ( y , . . . , y k ) ∈ Y ∗ , with Y = { y , . . . , y k } , be such that the decomposition w ≡ u ( w , . . . , w k ) determinesa conservative factorisation of w , where w , . . . , w k ∈ X ∗ . Suppose that for all i ∈ { , . . . , k } there is a letter x i ∈ X that appears exactly once in w i and does not HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 23 appear in any w j for j = i . Then the group G = Gp h X | w = 1 i has decidable prefixmembership problem.Consequently, if the above conditions are satisfied, and the one-relator inversemonoid Inv h X | w = 1 i is E -unitary (in particular, if w is cyclically reduced) then Inv h X | w = 1 i has decidable word problem.Proof. Denote X = { x , . . . , x k } and set X = X \ X . So for each 1 ≤ i ≤ k theletter x i appears exactly once in the word w i (either as x i or x − i ) and x i does notappear in any of the words w j with j = i . Therefore, for all 1 ≤ i ≤ k , we can write w i ≡ p i x ε i i q i where ε i ∈ { , − } and p i , q i ∈ X ∗ . Let us now apply Tietze transformationsto the initial presentation of G by introducing new letters Z = { z , . . . , z k } withthe aim of replacing the factors w , . . . , w k . The conditions on the words w i thenallow us to apply further Tietze transformations showing that the generators x i areredundant and thus can be eliminated giving a presentation just in terms of thegenerators X ∪ Z . This gives G = Gp h X ∪ X ∪ Z | z i = w i (1 ≤ i ≤ k ) , u ( w , . . . , w k ) = 1 i = Gp h X ∪ X ∪ Z | z i = w i (1 ≤ i ≤ k ) , u ( z , . . . , z k ) = 1 i = Gp h X ∪ X ∪ Z | x ε i i = p − i z i q − i (1 ≤ i ≤ k ) , u ( z , . . . , z k ) = 1 i = Gp h X ∪ Z | u ( z , . . . , z k ) = 1 i . Therefore, G = F G ( X ) ∗ H , where H = Gp h Z | u ( z , . . . , z k ) = 1 i is a one-relatorgroup.We now turn to considering the prefix monoid P w = Mon h pref w i ≤ G .Without loss of generality, we may suppose that the letters of the alphabet Y = { y , . . . , y k } are ordered in such a way that the following conditions hold: • y , . . . , y r appear in u while none of y r +1 , . . . y k appears in u ; • y − s , . . . , y − k appear in u while none of y − , . . . , y − s − appears in u ,where r ∈ { , . . . , k } and s ∈ { , . . . , k + 1 } and s ≤ r + 1. The condition s ≤ r + 1 comes from the fact that all of the letters y , . . . , y k appear in the word u = u ( y , . . . , y k ) either as y j or y − j .Since the given factorisation is assumed to be conservative, P w = Mon h pref( w ) ∪ · · · ∪ pref( w r ) ∪ pref( w − s ) ∪ · · · ∪ pref( w − k ) i . Clearly, in the group G we have the following equalities of setspref( w i ) = pref( p i ) ∪ p i x ε i i · pref( q i )= pref( p i ) ∪ z i q − i · pref( q i )= pref( p i ) ∪ z i · pref( q − i ) . In particular, z i is among the elements represented by prefixes of w i . Similarly,pref( w − i ) = pref( q − i ) ∪ z − i · suff( p − i ) − = pref( q − i ) ∪ z − i · pref( p i ) , which includes the element z − i . Thus P w is equal to the submonoid of F G ( X ) ∗ H generated by the set { z , . . . , z r , z − s , . . . , z − k } ∪ [ ≤ i ≤ k (cid:0) pref( p i ) ∪ pref( q − i ) (cid:1) . Next observe that for any 1 ≤ j ≤ r we can write u ≡ u ( z , . . . , z k ) ≡ u ′ z j u ′′ , whichmeans that z − j = u ′′ u ′ holds in H , since u ( z , . . . , z k ) = 1 in H . But both u ′ , u ′′ are products of letters from { z , . . . , z r , z − s , . . . , z − k } , which shows that z − j ∈ P w .Similarly, z j ∈ P w for all s ≤ j ≤ k . Since r ≤ s + 1 this proves that Z ⊆ P w andhence the entire group H is contained in P w . Thus P w is equal to the submonoidof F G ( X ) ∗ H generated by the set H ∪ Q where Q = [ ≤ i ≤ k (cid:0) pref( p i ) ∪ pref( q − i ) (cid:1) ⊆ F G ( X ) . To complete the proof it will suffice to show that the conditions of Corollary 4.8 aresatisfied for the submonoid P w of the group G = F G ( X ) ∗ H . The groups F G ( X )and H both have decidable word problem by Magnus’ Theorem.We have P w ∩ H = H which is finitely generated. We claim that P w ∩ F G ( X )equal to the submonoid of F G ( X ) generated by Q , and hence is finite generated.Indeed, we have that P w is equal to the submonoid of F G ( X ) ∗ H generated bythe set H ∪ Q . Let g ∈ P w ∩ F G ( X ). Since g ∈ P w we can write g = h t h . . . h l t l where h i ∈ H and t i ∈ Mon h Q i , and, furthermore, h i = 1 for 1 ≤ i ≤ l − t i = 1 for all i . In the free product F G ( X ) ∗ H this is a reduced form. Sincewe are assuming that g ∈ F G ( X ) it follows by Lemma 4.2 that we must have g = t ∈ Mon h Q i . This proves that P w ∩ F G ( X ) is contained in Mon h Q i . Theopposite containment is trivial. Hence condition (i) holds.Condition (iii) holds again since P w ∩ H = H , while condition (ii) holds byBenois’ Theorem as P w ∩ F G ( X ) is a finitely generated submonoid of the freegroup F G ( X ). This completes the proof of the theorem. (cid:3) Example 5.2.
Let X = { a, b, x, y } , let w = axbaybaybaxbaybaxb and set G =Gp h X | w = 1 i and M = Inv h X | w = 1 i . Since axb is both a prefix and a suffixof w it follows that this word represents an invertible element M . It follows thatthe word ( ayb ) aybaxb ( ayb ) also represents an invertible element of M and hence sodoes the word ayb . We conclude that w = ( axb )( ayb )( ayb )( axb )( ayb )( axb )is a unital factorisation and thus also a conservative factorisation by Theorem 3.3.Notice that x occurs exactly once in axb but not in ayb , and conversely, y occursjust once in ayb but not in axb . So, the above factorisation of w satisfies the uniquemarker letter condition of Theorem 5.1. Also note that w is a cyclically reducedword. Therefore applying the theorem we conclude that the group defined by thepresentation Gp h a, b, x, y | axbaybaybaxbaybaxb = 1 i has decidable prefix membership problem and the inverse monoidInv h a, b, x, y | axbaybaybaxbaybaxb = 1 i has decidable word problem.Many other similar examples to which Theorem 5.1 can be applied may beconstructed. In the example above in order to deduce that the inverse monoid hasdecidable word problem we just used the fact that it is E -unitary since the definingrelator is a cyclically reduced word. It was not important that the defining relator HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 25 was a positive word. For example, in much the same way we can show that theinverse monoid M ′ = Inv h a, b, x, y | a − xbab − ayb − b − ayb − a − b − x − ab − ayb − a − xba = 1 i . has decidable word problem.Next we shall see that Theorem 5.1 can also be applied in certain situationswhere the given defining relator does not immediately satisfy the unique markerletter condition. Example 5.3.
Let M be the “O’Hare inverse monoid”Inv h a, b, c, d | abcdacdadabbcdacd = 1 i . and let G be the group with the same presentation. Recall in Example 3.2 wherewe defined the “O’Hare inverse monoid” we saw that w ≡ ( abcd )( acd )( ad )( abbcd )( acd )is a unital factorisation and thus also a conservative one. Note that these invertiblepieces do not satisfy the unique marker letter property. However, as we shall nowsee, this monoid admits a one relator special inverse monoid presentation such thatthe defining relator does satisfy the hypotheses of Theorem 5.1. In fact, we shallidentify an infinite family of examples, which includes the O’Hare monoid, for whichthis approach is possible. Proposition 5.4.
Let M = Inv h X | au i dau i d . . . au i m d = 1 i , where a, d ∈ X and u i k ∈ Y ∗ is a reduced word for ≤ k ≤ m where Y = X \{ a, d } .Assume further that the following conditions hold: (i) for some ≤ j ≤ m , u i j is the empty word; (ii) for each x ∈ X \ { a, d } there exist r, s such that x ≡ red( u i r u − i s ) ; (iii) each word au i k d represents an invertible element of M .Then the group defined by the presentation G = Gp h X | au i dau i d . . . au i m d = 1 i , has decidable prefix membership problem, and the inverse monoid M has decidableword problem.Proof. By (iii) all of the words au i k d with 1 ≤ k ≤ m all represent invertibleelements of M . It follows that the inverse words ( au i k d ) − ≡ d − u − i k a − with1 ≤ k ≤ m also all represent invertible elements of M . Since the product oftwo invertible elements is invertible, it follows that for all 1 ≤ r, s ≤ m the word( au i r d )( d − u − i s a − ) represents an invertible element of M .We then use the following well-known fact from inverse semigroup theory: if bc isa right invertible element of an inverse monoid, then bcc − = b . Since every prefixof the word ( au i r d )( d − u − i s a − ) is right invertible, applying the above general factwe conclude that ( au i r d )( d − u − i s a − ) = a red( u i r u − i s ) a − holds in M for all 1 ≤ r, s ≤ m . By conditions (ii) it follows that for every letter x ∈ Y we have that axa − represents an invertible element of M . Also, by condition (i) and (iii) the word ad represents an invertible element of M .For each 1 ≤ r ≤ m write au i r d ≡ ab ,r . . . b t r ,r d where b i,r ∈ Y for 1 ≤ i ≤ t r . Using the observations from the previous paragraph,and the general observation above about cancelling inverse pairs in right invertiblewords we conclude that in M we have au i r d = ab ,r . . . b t r ,r d = ( ab ,r a − ) . . . ( ab t r ,r a − )( ad ) . (5.1)for all 1 ≤ r ≤ m .Let v r ≡ ( ab ,r a − ) . . . ( ab t r ,r a − )( ad ) for all 1 ≤ r ≤ m and then set w ′ ≡ v v . . . v k . We claim that the presentations Inv h X | w = 1 i and Inv h X | w ′ = 1 i areequivalent in the sense that the identity map on X induces an isomorphism betweenthe inverse monoids defined by these presentations. To prove this it suffices to showthat w ′ = 1 holds in the monoid M = Inv h X | w = 1 i , and, conversely, that w = 1holds in the monoids M ′ = Inv h X | w ′ = 1 i .The fact that w ′ = 1 holds in M follows immediately from Equation 5.1. Con-versely, in the inverse monoid M ′ = Inv h X | w ′ = 1 i each prefix of w ′ arising as aproduct of factors of the form ab j,r a − represents a right invertible element of M ′ .This observation along with the general fact above about cancelling inverse pairsin right invertible words makes it possible to delete from w ′ all factors of the form a − a without changing the value of w ′ in M ′ . In other words, w = 1 holds in M ′ .This shows that the presentations for M and M ′ are equivalent, and in particularthat M and M ′ are isomorphic via the identity map on X . It follows that for anyword γ ∈ X ∗ we have that γ represents a right invertible element of M if and onlyif γ represents a right invertible element of M ′ . Let R be the submonoid of rightinvertible elements of M , and let R ′ be the submonoid of right invertible elementsof M ′ . Let φ : M → G and φ ′ : M ′ → G be the maps to the maximal group imageinduced by the identity map on X . Then we have P w = Rφ = R ′ φ ′ = P w ′ . However, the relator word w ′ from the presentation of M ′ has a unital and thusconservative factorisation into factors of the form axa − , x ∈ X \ { a, d } , and ad .Picking x as the unique marker letter from axa − , and d from ad , shows that theinverse monoid M ′ has a presentation which satisfies the conditions of Theorem5.1. Hence, the membership problem for P w ′ = P w in G is decidable. Hence thegroup presentation Gp h X | w = 1 i has decidable prefix membership problem andthe inverse monoid M has decidable word problem. (cid:3) In particular the above proposition applies the O’Hare monoidInv h a, b, c, d | ( abcd )( acd )( ad )( abbcd )( acd ) = 1 i since, as already observed, the displayed decomposition is unital, and hence inparticular ad represents an invertible element of the monoid, and, moreover, weclearly have b = red(( bc ) c − ) and c = red( c − ). Hence all the hypotheses of theproposition are satisfied and we conclude that the O’Hare monoid has decidableword problem. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 27
Remark . It was pointed out to us by Jim Howie (Heriot-Watt University, Ed-inburgh) [17] that the “O’Hare group” Gp h a, b, c, d | abcdacdadabbcdacd = 1 i is infact a free group of rank 3 (albeit in a rather non-obvious way). Therefore, in thiscase P w has decidable membership in this group as a direct consequence of Benois’Theorem, and so the word problem for the O’Hare inverse monoid is decidable. Onthe other hand, not every group satisfying the hypotheses of Proposition 5.4 is free.5.2. Disjoint alphabets theorem.Lemma 5.6.
Let G = B ∗ A C and let U be a finite subset of B ∪ C such that M = Mon h U i , the submonoid of G generated by U , contains A . Then M ∩ B isgenerated by ( U ∩ B ) ∪ A and M ∩ C is generated by ( U ∩ C ) ∪ A . Consequently,if A is finitely generated, then so are the monoids M ∩ B and M ∩ C .Proof. Let g ∈ M ∩ B . Then, since g ∈ M , we may write g = c b c . . . b l c l for some b i ∈ Mon h U ∩ B i and c i ∈ Mon h U ∩ C i such that c i = 1 for 1 ≤ i ≤ l − b i = 1 for all i . This expression is either a reduced form in G , whence l = 1, c = c = 1 and g = b ∈ Mon h U ∩ B i ⊆ Mon h ( U ∩ B ) ∪ A i by Lemma 4.2,or, otherwise, at least one of the terms belongs to A . For example, suppose that b j ∈ A . Then c ′ j − = c j − b j c j ∈ Mon h ( M ∩ C ) ∪ A i , so upon relabelling b ′ k = b k for all 1 ≤ k ≤ j − c ′ k = c k for all 0 ≤ k ≤ j − b ′ k = b k +1 for all j + 1 ≤ k ≤ l − c ′ k = c k +1 for all j ≤ k ≤ l −
1, we get a new expression g = c ′ b ′ c ′ . . . b ′ l − c ′ l − , where b ′ k ∈ Mon h ( U ∩ B ) ∪ A i and c ′ k ∈ Mon h ( U ∩ C ) ∪ A i for all k , for which theprevious argument can be repeated. We proceed similarly if c j ∈ A for some j .Continuing in this fashion, we eventually arrive at a reduced form for g , leading tothe conclusion that g ∈ Mon h ( U ∩ B ) ∪ A i , as required.Analogously, if g ∈ M ∩ C we have that g ∈ Mon h ( U ∩ C ) ∪ A i , thus provingthe lemma. (cid:3) Here is our second application, whose proof makes an appeal to Theorem A.
Theorem 5.7.
Let G = Gp h X | w = 1 i where w ∈ X ∗ is a cyclically reduced word.Suppose that there is a finite alphabet Y = { y , . . . , y k } with k ≥ and a word u ∈ Y ∗ such that w ≡ u ( w , . . . , w k ) , all letters from Y appear in u (either inpositive or inverted form), and that this determines a conservative factorisationof w . Suppose that for any pair of distinct i, j ∈ { , . . . , k } the words w i and w j have no letters in common. Then the group G = Gp h X | w = 1 i has decidable prefixmembership problem and thus Inv h X | w = 1 i has decidable word problem.Proof. For 1 ≤ i ≤ k , let X i ⊆ X denote the content of w i , namely the set of allletters x ∈ X that appear in w i . Then the conditions given in the theorem statethat i = j implies X i ∩ X j = ∅ . We also let X = X \ S ≤ i ≤ k X i . Notice thatsince w is cyclically reduced it follows that u must also be cyclically reduced.Let t X be a new letter. Then an easy application of Tietze transformationsgives G = Gp h X, t | w t − = 1 , u ( t , w , . . . , w k ) = 1 i . Let G = Gp h X \ X , t | u ( t , w , . . . , w k ) = 1 i noting that by the assumptions theword u ( t , w , . . . , w k ) is written over the alphabet ( X \ X ) ∪{ t } and is a cyclically reduced word. Since by assumption k ≥
2, it follows by Magnus’ Freiheitssatz thatthe subgroup A ′ = Gp h t i of G generated by t is an infinite cyclic group. Onthe other hand, since w ∈ X ∗ is a non-empty reduced word, it follows that thesubgroup A = Gp h w i of free group F G ( X ) generated by w is also infinite cyclic.Thus we can form the amalgamated free product F G ( X ) ∗ A G . there A and A ′ are identified via the isomorphism sending w to t . This gives G = Gp h X , ( X \ X ) , t | w = t , u ( t , w , . . . , w k ) = 1 i = F G ( X ) ∗ A G . The same reasoning as above can be then applied to G : upon introducing anew letter t , one can decompose G as the free product of F G ( X ) and G =Gp h X \ ( X ∪ X ) , t , t | u ( t , t , w , . . . , w k ) = 1 i amalgamated over the infinitecyclic subgroups A and A ′ generated by w and t , respectively.Continuing in this way, after a finite number of steps we obtain the followingtower of amalgamated free products: G = F G ( X ) ∗ A ( F G ( X ) ∗ A ( . . . ( F G ( X k ) ∗ A k G k ) . . . )) , where A i is generated by t i = w i and G k = Gp h X , t , . . . , t k | u ( t , . . . , t k ) = 1 i = F G ( X ) ∗ H where H = Gp h t , . . . , t k | u ( t , . . . , t k ) = 1 i . For each 0 ≤ i ≤ k we shall now define a submonoid M i of G i inductively. Thesequence of monoids we define will have the property that for all i we have M i +1 = M i ∩ G i +1 , and also that M = P w . We will prove using Theorem A that the membershipproblem for M i in G i is decidable for all i , and since M = P w this will suffice tocomplete the proof of the theorem.For all 1 ≤ i ≤ k set W i = pref( w i ) if u ( t , . . . , t k ) contains t i but not t − i , pref( w − i ) if u ( t , . . . , t k ) contains t − i but not t i , pref( w i ) ∪ pref( w − i ) if u ( t , . . . , t k ) contains both t i , t − i . Then set M k = Mon h t εi : 1 ≤ i ≤ k and ε ∈ { , − } such that t εi is contained in u ( t , . . . , t k ) i which is a submonoid of G k = F G ( X ) ∗ H , and define inductively M i − = Mon h M i ∪ W i i ≤ G i − = F G ( X i ) ∗ A i G i for 1 ≤ i ≤ k .It may be shown that in fact M k = H using a similar argument as in the proofof Theorem 5.1. Indeed, if, for example, t i occurs in u ( t , . . . , t k ) but not t − i , thenone can write u ( t , . . . , t k ) ≡ u ′ t i u ′′ . Therefore, in G k we have t − i = u ′′ u ′ ∈ M k .This shows that t i , t − i ∈ M k for all 1 ≤ i ≤ k , so the required conclusion M k = H follows. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 29
Since the word u defines a conservative factorisation of w , it follows that theprefix monoid P w is equal to the submonoid of G = G generated by ∪ ≤ i ≤ k W i .Now by definition we have M = Mon h Mon h . . . Mon h Mon h H ∪ W k i ∪ W k − i . . . i ∪ W i . From this, using the natural embeddings of G i − into G i for all i it may be verifiedthat in G = G we have M = Mon h W ∪ W ∪ · · · ∪ W k i = P w . So, it remains to argue by induction that the memberhsip problem for M i withing G i is decidable for all i . Clearly, each M i is finitely generated, say M i = Mon h U i i forsome finite subset U i ⊂ M i . Since both F G ( X ) and the one-relator group H havedecidable word problems, the latter by Magnus’ Theorem, and M k ∩ F G ( X ) = { } and M k ∩ H = H , we can apply Corollary 4.8 to deduce that the membershipproblem for M k in G k is decidable.Now assume inductively that the membership problem for M i +1 in G i +1 is de-cidable for some i < k . The latter is a one-relator group, so it has decidable wordproblem, as does F G ( X i +1 ). Furthermore, since F G ( X i +1 ) is a free group, it fol-lows from Benois’ Theorem that the membership problem for A i +1 = Gp h w i +1 i in F G ( X i +1 ) is decidable. Since t i +1 is one of its generators, it follows by [27, The-orem IV.5.3] that the membership problem for A ′ i +1 = Gp h t i +1 i in the one-relatorgroup G i +1 is decidable.We claim that M i ∩ G i +1 = M i +1 . Indeed, we have G i = F G ( X i +1 ) ∗ A i +1 G i +1 ,while M i = Mon h M i +1 ∪ W i +1 i = Mon h U i +1 ∪ W i +1 i . Therefore, by Lemma 5.6, M i ∩ G i +1 is generated by (( U i +1 ∪ W i +1 ) ∩ G i +1 ) ∪ A ′ i +1 = U i +1 ∪ A ′ i +1 and thus M i ∩ G i +1 = Mon h U i +1 ∪ { t i +1 , t − i +1 }i = M i +1 because t i +1 , t − i +1 ∈ H ⊆ M i +1 .Similarly, Lemma 5.6 yields that M i ∩ F G ( X i +1 ) is finitely generated by W i +1 .This means that M i is indeed generated by ( M i ∩ G i +1 ) ∪ ( M i ∩ F G ( X i +1 )), and,furthermore, the membership problem for M i +1 = M i ∩ G i +1 in G i +1 is decidableby the inductive hypothesis, while the membership problem for M i ∩ F G ( X i +1 ) in F G ( X i +1 ) is decidable by by Benois’ Theorem.Hence the hypotheses of Theorem A are satisfied, and applying this theorem weconclude that the membership problem for M i in G i is decidable. In particular,the membership problem for M = P w in G = G is decidable, so the theoremfollows. (cid:3) Example 5.8.
Consider the inverse monoid M = Inv h a, b, c, d | ababcdcdababcdcdcdcdabab = 1 i . Then, if we denote α ≡ abab and β ≡ cdcd , the relator word becomes w ≡ αβαββα ,and we conclude, just as in Example 5.2, that both α and β represent invertibleelements of M . Hence, w ≡ ( abab )( cdcd )( abab )( cdcd )( cdcd )( abab )is a conservative factorisation that satisfies the conditions of Theorem 5.7. It followsthat G = Gp h a, b, c, d | αβαββα = 1 i has decidable prefix membership problem andthat the word problem of M is decidable, too. In more detail, G = F G ( a, b ) ∗ A ( F G ( c, d ) ∗ A Gp h t, s | tstsst = 1 i ) , where A and A are infinite cyclic groups generated by t = abab and s = cdcd ,respectively, and the prefix monoid is generated by t, t − = stsst, s, s − = tsstt (thus containing the whole group Gp h t, s | tstsst = 1 i ) and a, ab, aba, c, cd, cdc (here aba and cdc are obviously redundant). Example 5.9.
We finish the subsection by a non-example, showing the significanceof the disjoint content condition in Theorem 5.7. Namely, let M = Inv h a, b, c | ababbcbcbbababbcbcbbcbcbbababb = 1 i . Just as in previous examples, it is easy to see that w = ( ababb )( cbcbb )( ababb )( cbcbb )( cbcbb )( ababb )is a unital factorisation of the relator word, but the pieces do not have disjointcontent (they have the letter b in common). As w is cyclically reduced, the wordproblem for M reduces to the prefix membership problem for the group G definedby the presentation Gp h a, b, c | αβαββα = 1 i , where α ≡ ababb and β ≡ cbcbb . Wecan now replace its sole relation by ababbs − = 1 and scbcbbscbcbbcbcbbs = 1 toobtain the free amalgamated decomposition G = F G ( a, b ) ∗ A Gp h c, d, s | scbcbbscbcbbcbcbbs = 1 i , where A is a joint free subgroup of rank 2 generated by b and ababb = s . (Notethat the second factor satisfies the requirements for applying Theorem 5.7.)However, the prefix submonoid P w of G is generated (after removing some obvi-ous redundancies) by s, s − , a, ab, c, cb, cbcbb . It also contains ( cbcbb ) − = ( scbcbb ) and thus b − = ( cbcbb ) − ( cb ) . The problem is that we don’t know if A ⊆ P w : forthis we would need b ∈ P w (which seems likely not to hold). Therefore, at presentit seems that Theorem A cannot be applied to this case.5.3. Cyclically pinched presentations.
Following e.g. [7, 9] we say that a one-relator group G is cyclically pinched if it is defined by a presentation of the form h X ∪ Y | u = v i where u ∈ X ∗ and v ∈ Y ∗ are nonempty reduced words, with the alphabets X and Y disjoint. Clearly, the defining relation is equivalent to uv − = 1, and in this sensewe are going to refer to Gp h X ∪ Y | uv − = 1 i as a cyclically pinched presentation . Theorem 5.10.
The prefix membership problem is decidable for any group definedby a cyclically pinched presentation Gp h X ∪ Y | uv − = 1 i . Consequently, the word problem is decidable for all one-relator inverse monoids ofthe form
Inv h X ∪ Y | uv − = 1 i with u ∈ X ∗ and v ∈ Y ∗ both reduced words.Proof. As is well-known, a cyclically pinched group G is a free product of freegroups F G ( X ) and F G ( Y ) amalgamated over an infinite cyclic group A such that Af is generated by u and Ag is generated by v . Hence, it suffices to check if theconditions of Theorem A are satisfied for the prefix monoid P w where w ≡ uv − .Indeed, the latter monoid is generated in G bypref( u ) ∪ u · pref( v − ) . HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 31
Note that the set u · pref( v − ) is in G actually equal to pref( v ). Since the generatingset of P w contains u (and thus v ), the monoid P w contains the whole amalgamatedsubgroup. Since this subgroup is finitely generated, Lemma 5.6 implies that soare the monoids P w ∩ F G ( X ) = Mon h pref( u ) i and P w ∩ F G ( Y ) = Mon h pref( v ) i .Hence, the conditions (i) and (ii) of Theorem A are satisfied, and the remainingconditions hold by Benois’ Theorem. Therefore, the membership problem for P w in M is decidable. (cid:3) Example 5.11.
Both the orientable surface groupGp h a , . . . , a n , b , . . . , b n | [ a , b ] . . . [ a n , b n ] = 1 i and the non-orientable surface groupGp h a , . . . , a n | a . . . a n = 1 i of genus n ≥ u ≡ [ a , b ] . . . [ a n − , b n − ] and v ≡ [ a n , b n ] − in the orientable case, and for u ≡ a . . . a n − and v ≡ a − n for thenon-orientable case). Hence, the corresponding prefix membership problems aredecidable. For the first family of presentations above there are already several proofsin the literature that the prefix membership problem is decidable; see [19, 32, 34].6. Deciding membership in submonoids of HNN extensions
Our aim in this section is to give two general decidability results concerning themembership problem for certain submonoids of HNN extensions of finitely gener-ated groups. Then in the next section these results will be applied to the prefixmembership problem for certain one-relator groups.Before stating the main results we first recall some definitions and fix somenotation which will remain in place for the rest of the section. Let G be a groupfinitely generated by X with canonical homomorphism π : X ∗ → G . Let A and B be two isomorphic finitely generated subgroups of G and let φ : A → B be anisomorphism. Moreover, let Y = { y , . . . , y k } and Z = { z , . . . , z k } be, respectively,finite generating sets for A and B , with canonical homomorphisms θ : Y ∗ → A , ξ : Z ∗ → B such that y i θφ = z i ξ for 1 ≤ i ≤ k . Set a i = y i θ and b i = z i ξ for1 ≤ i ≤ k . Observe that { a , . . . , a k } is a finite subset of A which generates A , andsimilarly { b , . . . , b k } is a finite subset of B generating B .We use G ∗ = G ∗ t,φ : A → B to denote the HNN extension of G with a stable letter t X and associated finitelygenerated subgroups A, B . So G ∗ is the group obtained by taking a presentationfor G with respect to X , adding t as a new generator, and adding the relations t − a i t = b i for 1 ≤ i ≤ k . Formally these relations should be written over the alphabet X ∪ { t } ,that is, as t − u i t = v i where u i , v i ∈ X ∗ satisfy u i π = a i and v i π = b i . Throughoutthis section u i and v i will denote words with these properties. Note that for anyword w ( u , . . . , u k ) ∈ X ∗ representing the element a ∈ A , the word w ( v , . . . , v k ) ∈ X ∗ represents the element aφ ∈ B .We now state the two main results of this section. Theorem C.
Let G ∗ = G ∗ t,φ : A → B be an HNN extension of a finitely generatedgroup G such that A, B are also finitely generated. Assume that G has decidableword problem and that the membership problems of A and B in G are decidable.Let M be a submonoid of G ∗ such that the following conditions hold: (i) A ∪ B ⊆ M ; (ii) M ∩ G is finitely generated, and M = Mon h ( M ∩ G ) ∪ { t, t − }i ;(iii) the membership problem for M ∩ G in G is decidable.Then the membership problem for M in G ∗ is decidable. Theorem D.
Let G ∗ = G ∗ t,φ : A → B be an HNN extension of a finitely generatedgroup G such that A, B are also finitely generated. Assume that the following con-ditions hold: (i) the rational subset membership problem is decidable in G ; (ii) A ≤ G is effectively closed for rational intersections.Then for any finite W , W , . . . , W d , W ′ , . . . , W ′ d ⊆ G , d ≥ , the membership prob-lem for M = Mon h W ∪ W t ∪ W t ∪ · · · ∪ W d t d ∪ tW ′ ∪ · · · ∪ t d W ′ d i in G ∗ is decidable.Remark . The conclusion of the previous theorem also holds if we replace t by t − in the generating set of the monoid M . Namely, it is straightforward to seethat Mon h W ∪ W t − ∪ W t − ∪ · · · ∪ W d t − d ∪ t − W ′ ∪ · · · ∪ t − d W ′ d i is in fact equal to (cid:0) Mon h W − ∪ ( W ′ ) − t ∪ ( W ′ ) − t ∪ · · · ∪ ( W ′ d ) − t d ∪ tW − ∪ · · · ∪ t d W − d i (cid:1) − , which enables us to invoke Theorem D.By standard results on HNN extensions (see e.g. [27, Ch. IV]) every element g ∈ G ∗ t,φ : A → B can be written written as g = g t ε g t ε . . . t ε n g n , where g , g , . . . , g n ∈ G and ε i ∈ { , − } for all 1 ≤ i ≤ n . This expression issaid to be reduced if for all 1 ≤ i ≤ n − g i A whenever ε i = − ε i +1 = 1, and g i B whenever ε i = 1 and ε i +1 = −
1. Similarly, for words w , w , . . . , w n ∈ X ∗ and ε i ∈ { , − } (1 ≤ i ≤ n ) the word w t ε w t ε . . . t ε n w n , is said to be a word in reduced form if and only if ( w π ) t ε ( w π ) t ε . . . t ε n ( w n π ) isa reduced expression. The following result is standard and can be proved e.g. byapplying [27, Lemma IV.2.3] and Britton’s Lemma. Lemma 6.2.
An equality of two reduced forms g t ε g t ε . . . t ε n g n = h t δ h t δ . . . t δ m h m HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 33 holds in the HNN extension G ∗ t,φ : A → B if and only if n = m , ε i = δ i for all ≤ i ≤ n , and there exist α , α , . . . , α n , α n +1 = 1 ∈ A ∪ B such that for all ≤ i ≤ n we have α i ∈ A if ε i = − , α i ∈ B if ε i = 1 , and h i = α − i g i ( t ε i +1 α i +1 t − ε i +1 ) . Lemma 6.3.
Let G ∗ = G ∗ t,φ : A → B and let M be a submonoid of G ∗ such that M = Mon ( M ∩ G ) ∪ { t, t − } . Then every element g ∈ M can be written in reduced form g = g t ε g t ε . . . t ε n g n such that g i ∈ M ∩ G for all ≤ i ≤ n .Proof. Assume that g ∈ M . Then by assumption g = h t δ h t δ . . . t δ k h k for some h i ∈ M ∩ G , 0 ≤ i ≤ k , and δ i ∈ { , − } , 1 ≤ i ≤ k . By [27, page 184], theabove product, which itself is not necessarily reduced, can be transformed into areduced form by applying a finite number of t -reductions. Recall that t -reductionsthe following operations • replace t − gt , where g ∈ A , by gφ , or • replace tgt − , where g ∈ B , by gφ − .Hence to prove the lemma it suffices to show that, under our assumptions, applica-tions of t -reductions to a product of the above form preserves the property of the G -terms belonging to M .Consider a product p t γ p t γ . . . t γ m p m , such that p i ∈ M for 1 ≤ i ≤ m , towhich a t -reduction can be applied. Suppose without loss of generality this is a t -reduction of the first kind listed above. Applying this t -reduction will result in aproduct of the form p t γ p t γ . . . p i − ( p i φ ) p i +1 . . . t γ m p m , where γ i = − p i ∈ A and γ i +1 = −
1. Since t, t − ∈ M by assumption, it follows that p i φ = t − p i t ∈ M and hence p i − p i p i +1 ∈ M . Hence the G -terms in the above product still all belongto M .The argument when a t -reduction of the second kind is applied is analogous. (cid:3) In the following lemma we identify conditions under which the process in Lemma6.3 can be performed algorithmically.
Lemma 6.4.
Let G ∗ = G ∗ t,φ : A → B such that G , A and B are all finitely generated, G has recursively enumerable word problem, and the membership problems for A and B within G are both decidable. Then there an algorithm which takes any word w ∈ ( X ∪ { t } ) ∗ as input and returns a word in reduced form w = w t ε w t ε . . . t ε n w n , with w i ∈ X ∗ for ≤ i ≤ n .Proof. This may be proved by combining the comments from [27, pages 184–185]discussing conditions under which the process of conducting t -reductions is effective,together with Lemma 4.4. (cid:3) Proof of Theorem C.
A key ingredient in our proof is given by the followingauxiliary result, which strengthens Lemma 6.3 under the stronger conditions ofTheorem C.
Proposition 6.5.
Assuming all the notation and conditions from Theorem C, let g = g t ε g t ε . . . t ε n g n be an element of G ∗ = G ∗ t,φ : A → B written in reduced form. Then g ∈ M if and onlyif g i ∈ M ∩ G for all ≤ i ≤ n .Proof. ( ⇒ ) By Lemma 6.3, since g ∈ M there exists a reduced form g = m t µ m t µ . . . t µ k m k such that m i ∈ M for all 0 ≤ i ≤ k . By Lemma 6.2, we must have k = n , µ i = ε i and g i = α − i m i ( t ε i +1 α i +1 t − ε i +1 )for all 0 ≤ i ≤ n and some 1 = α , α . . . , α n , α n +1 = 1 ∈ A ∪ B (such that α i ∈ A whenever ε i = − α i ∈ B otherwise). It follows that g i ∈ ( A ∪ B )( M ∩ G )( A ∪ B ) ⊆ M ∩ G. ( ⇐ ) is trivial, since t, t − ∈ M . (cid:3) Proof of Theorem C.
To prove the theorem we must show that there is an algorithmwhich takes any word w from ( X ∪ { t } ) ∗ as input and decides whether or not theword represents an element of the submonoid M . The hypotheses of Lemma 6.4are clearly satisfied Applying this lemma we conclude that there is an algorithmthat given any such word w returns a word w = w t ε w t ε . . . t ε n w n , with w i ∈ X ∗ for 1 ≤ i ≤ n , that is in reduced form. It follows from Proposition 6.5that w represents an element of M if and only if w i ∈ M ∩ G for all 0 ≤ i ≤ n .However, this is decidable by assumption (iii), and this completes the proof. (cid:3) Combining Benois’ Theorem and Britton’s Lemma with Theorem C gives thefollowing result.
Corollary 6.6.
Let G ∗ = G ∗ t,φ : A → B where G = F G ( X ) is a free group of finiterank, and A and B are finitely generated subgroups of G . Let T be a finitely gen-erated submonoid of G containing A , and let M be the submonoid of G ∗ generatedby T ∪ { t, t − } . Then the membership problem for M in G ∗ is decidable. This can be applied to show that membership is decidable in certain finitelygenerated submonoids of surface groups. It is an open problem whether in generalthe submonoid membership problem is decidable for surface groups.6.2.
Proof of Theorem D.
The following lemma of combinatorial nature willturn out to be crucial in what follows.
Lemma 6.7.
Let G ∗ = G ∗ t,φ : A → B be an HNN extension of a finitely generatedgroup G with finitely generated associated subgroups A, B . Let G be finitely gener-ated by X with canonical homomorphism π : X ∗ → G . Then there is an algorithm HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 35 which takes any finite list of finite subsets W , W , . . . , W d , W ′ , . . . , W ′ d of G andany integer m ≥ and returns an integer C m ≥ and a finite set of FSA {N ( j ) m,i : 0 ≤ i ≤ m, ≤ j ≤ C m } over X such that with N ( j ) m,i = [ L ( N ( j ) m,i )] π and D m = h th t . . . th m : ( h , . . . , h m ) ∈ [ ≤ j ≤ C m (cid:16) N ( j ) m, × · · · × N ( j ) m,m (cid:17) , we have Mon h W ∪ W t ∪ W t ∪ · · · ∪ W d t d ∪ tW ′ ∪ · · · ∪ t d W ′ d i = [ m ≥ D m . Proof.
We will first describe the algorithm, and then verify that the sets D m doindeed satisfy the property claimed in the statement of the lemma.The algorithm we describe will in fact output rational expressions for the sets N ( j ) m,i . By the comments in Section 2, this suffices to conclude that there is analgorithm computing the corresponding FSA N ( j ) m,i .The algorithm is recursive. When m = 0 the algorithm returns C = 1 and therational expression W ∗ so that N (1)0 , = Mon h W i . Now consider a typical stage m of the algorithm with m >
0, assuming that the algorithm already constructed theintegers C p and rational expressions for sets N ( j ) p,i , 1 ≤ i ≤ p , 1 ≤ j ≤ C p , for allvalues p < m .The algorithm then proceeds as follows. For any 1 ≤ µ ≤ min( d, m ) constructthe following two collections of sequences of subsets of G (all of length m + 1): • Mon h W i W µ , { } , . . . , { } | {z } µ − , N ( j ) m − µ, , . . . , N ( j ) m − µ,m − µ , • Mon h W i , { } , . . . , { } | {z } µ − , W ′ µ N ( j ) m − µ, , . . . , N ( j ) m − µ,m − µ ,where in both cases 1 ≤ j ≤ C m − µ . Set C m = min( d,m ) X µ =1 C m − µ . That is, C m is the total number of sequences constructed above. Then set N ( j ) m,i tobe the set appearing in position i of the j th sequence in the above list, 1 ≤ j ≤ C m .The algorithm computes the above sequences, and the number C m . It is clearfrom the definition of these sequences all the sets appearing in these sequences arerational subsets of G , and the algorithm can be instructed to output the appropriatecorresponding rational expressions.To complete the proof of the lemma we must now verify thatMon h W ∪ W t ∪ W t ∪ · · · ∪ W d t d ∪ tW ′ ∪ · · · ∪ t d W ′ d i = [ m ≥ D m holds.Set K = Mon h W ∪ W t ∪ W t ∪ · · · ∪ W d t d ∪ tW ′ ∪ · · · ∪ t d W ′ d i . For all m ≥ K m be the subset of K of all elements that when written in reduced form have precisely m occurrences of t . This is well-defined by Britton’s Lemma and K is adisjoint union of the sets K m with m ≥
0. Note that it also follows from Britton’sLemma that S m ≥ D m is a union of pairwise disjoint sets.Hence to finish the proof of the lemma it will suffice to prove that K m = D m forall m ≥
0, which we shall prove by induction on m .The base case m = 0 holds because K = Mon h W i = D where the secondequality follows by the construction of the algorithm.For the induction step, consider K m and D m and suppose by induction that K m ′ = D m ′ for all 0 ≤ m ′ < m .To see that K m ⊆ D m , let g ∈ K m be arbitrary. By the definition of Kg = u . . . u k , where each term u r , 1 ≤ r ≤ k , is either of the form w r t δ r for some w r ∈ W δ r , orof the form t δ r w r for some w r ∈ W ′ δ r , where 0 ≤ δ r ≤ d .Since there is no occurrence of t − in u . . . u k it follows that this is in reducedform thus, as g ∈ K m , it follows from Britton’s Lemma that δ + · · · + δ k = m .Let s be the smallest index such that in the above decomposition of g we have δ s = µ >
0. This implies that u . . . u s − ∈ Mon h W i and µ ≤ min( d, m ) because δ + · · · + δ k = m . As for u s , we have either u s ∈ W µ t µ , or u s ∈ t µ W ′ µ . Finally, u s +1 . . . u k is an element of the monoid K with the property that any of its reducedforms has precisely m − µ occurrences of t . Therefore, u s +1 . . . u k ∈ K m − µ , whichby induction hypothesis implies that u s +1 . . . u k ∈ D m − µ . We conclude that either g ∈ Mon h W i W µ t µ D m − µ , or g ∈ Mon h W i t µ W ′ µ D m − µ . In both cases, by the description of our algorithm, we have g ∈ N ( j ) m, tN ( j ) m, t . . . tN ( j ) m,m for some 1 ≤ j ≤ C m , yielding g ∈ D m .Conversely, to see that D m ⊆ K m , let g ∈ D m . Then there exists an index j , 1 ≤ j ≤ C m , such that g ∈ N ( j ) m, tN ( j ) m, t . . . tN ( j ) m,m . Now, the sequence of sets N ( j ) m, , N ( j ) m, , . . . , N ( j ) m,m is obtained in one of the two ways as described in the defini-tion of our algorithm. Therefore, there is an integer µ , 1 ≤ µ ≤ min( d, m ), such thateither g ∈ Mon h W i W µ t µ D m − µ , or g ∈ Mon h W i t µ W ′ µ D m − µ . However, by the in-duction hypothesis, D m − µ = K m − µ , so we have either g ∈ Mon h W i W µ t µ K m − µ ⊆ K m , or g ∈ Mon h W i t µ W ′ µ K m − µ ⊆ K m . In summary, in either case we concludethat g ∈ K m , which completes our proof. (cid:3) The following result which gives necessary and sufficient conditions for an elementin reduced form to belong to M , will be essential for the proof of Theorem D. Proposition 6.8.
Let G ∗ = G ∗ t,φ : A → B be an HNN extension of a finitely generatedgroup G with finitely generated associated subgroups A, B . Let M = Mon h W ∪ W t ∪ W t ∪ · · · ∪ W d t d ∪ tW ′ ∪ · · · ∪ t d W ′ d i for some finite W , W , . . . , W d , W ′ , . . . , W ′ d ⊆ G . In addition, for n ≥ , let C n ≥ and N ( j ) n,i , ≤ i ≤ n , ≤ j ≤ C n , be the integers and rational subsets of G givenby Lemma 6.7. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 37
Let g = g t ε g t ε . . . t ε n g n , be an element of G ∗ in reduced form where g i ∈ G for ≤ i ≤ n and ε j ∈ { , − } for ≤ j ≤ n . For i ∈ { , . . . , n } and ≤ j ≤ C n define subsets Q ( j ) i = Q ( j ) i ( g , . . . , g n ) of G in the following way: Q ( j )0 = { } ,Q ( j ) i +1 = (( N ( j ) n,i ) − Q ( j ) i g i ∩ A ) φ, for ≤ i ≤ n − .Then g ∈ M if and only if ε = · · · = ε n = 1 and g n ∈ [ ≤ j ≤ C n ( Q ( j ) n ) − N ( j ) n,n . Proof. ( ⇒ ) Assume that g ∈ M . By Lemma 6.7 there exist m ≥ ≤ j ≤ C m such that g = h th t . . . h m − th m holds for some h i ∈ N ( j ) m,i , 0 ≤ i ≤ m . Since the right-hand side of the pre-vious equality is in reduced form and g = g t ε g t ε . . . t ε n g n is a reduced form,by Lemma 6.2, it follows that m = n , and ε = · · · = ε n = 1, and there exist1 = α , α , . . . , α n , α n +1 = 1 ∈ B such that g i = α − i h i ( tα i +1 t − )holds for all 0 ≤ i ≤ n . In particular, we have g = h ( tα t − ), so α φ − = tα t − = h − g ∈ ( N ( j ) n, ) − g ∩ A = ( N ( j ) n, ) − Q ( j )0 g ∩ A, implying α ∈ (( N ( j ) n, ) − Q ( j )0 g ∩ A ) φ = Q ( j )1 .Now we proceed by induction to prove that for all 0 ≤ i ≤ n , α i ∈ Q ( j ) i . Supposethat α k ∈ Q ( j ) k for some 0 ≤ k ≤ n and consider α k +1 . We have α k +1 = ( tα k +1 t − ) φ = ( h − k α k g k ) φ ∈ (( N ( j ) n,k ) − Q ( j ) k g k ∩ A ) φ = Q ( j ) k +1 , completing the induction step. It follows that g n = α − n h n ∈ ( Q ( j ) n ) − N ( j ) n,n , asrequired.( ⇐ ) Let g = g t ε g t ε . . . t ε n g n be an element of G ∗ in reduced form such that ε i = 1 for all 1 ≤ i ≤ n and g n ∈ ( Q ( j ) n ) − N ( j ) n,n for some 1 ≤ j ≤ C n . Then we canwrite g n = β − n k n for some β n ∈ Q ( j ) n and k n ∈ N ( j ) n,n . Therefore tβ n t − = β n φ − ∈ Q ( j ) n φ − = ( N ( j ) n,n − ) − Q ( j ) n − g n − ∩ A, by definition of Q ( j ) n . So there exist k n − ∈ N ( j ) n,n − and β n − ∈ Q ( j ) n − such that tβ n t − = k − n − β n − g n − . Rearranging this yields g n − = β − n − k n − ( tβ n t − ) . Continuing in this way we obtain for i = n − , n − , . . . , k i ∈ N ( j ) n,i and β i ∈ Q ( j ) i (0 ≤ i < n ) such that g i = β − i k i ( tβ i +1 t − ) . Note that β = 1 because Q ( j )0 = { } . Substituting into the reduced form of g , andcancelling gives adjacent inverse pairs, we obtain g = g tg t . . . tg n = k tk t . . . tk n ∈ M by Lemma 6.7. (cid:3) Lemma 6.9.
Under the assumptions of Theorem D, in the statement of Proposi-tion 6.8 every set Q ( j ) i ( ≤ i ≤ n , ≤ j ≤ C n ) is a rational subset of B .Proof. Clearly Q ( j )0 is a rational subset of B . We prove by induction on i (with j fixed) that Q ( j ) i is rational for 0 ≤ i ≤ n . For the induction step suppose that Q ( j ) k is rational and consider Q ( j ) k +1 . By definition Q ( j ) k +1 = (( N ( j ) n,k ) − Q ( j ) k g k ∩ A ) φ. By Lemma 6.7 the set N ( j ) n,k is a rational subset of G . It then follows from condition(ii) in Theorem D and the induction hypothesis that R = ( N ( j ) n,k ) − Q ( j ) k g k ∩ A is a rational subset of G . Since R ⊆ A it then follows from Theorem 2.9 that R ∈ Rat( A ). Then since φ is an isomorphism it follows that Q ( j ) k +1 = Rφ ∈ Rat( B ). (cid:3) Proof of Theorem D.
Similarly to the proof of Theorem C, to prove the theorem wemust show that there is an algorithm which takes any word w from ( X ∪ { t } ) ∗ asinput and decides whether or not the word represents an element of the submonoid M . By assumption (i) it follows that the membership problems for A in G and B in G are both decidable, since G has decidable rational subset membership problemand A and B are both finitely generated. Also, condition (i) implies that G has de-cidable subgroup membership problem, and hence decidable word problem. Hence,the hypotheses of Lemma 6.4 are satisfied. Applying this lemma we conclude thatthere is an algorithm that given any such word w returns a word w = w t ε w t ε . . . t ε n w n , with w i ∈ X ∗ for 1 ≤ i ≤ n , and ε i ∈ { , − } for 1 ≤ i ≤ n , that is in reducedform.At this point, the algorithm calls as a subroutine the algorithm from Lemma 6.7which will return an integer C n ≥ {N ( j ) n,i : 0 ≤ i ≤ n, ≤ j ≤ C n } over X such that with N ( j ) n,i = [ L ( N ( j ) n,i )] π the conditions in the statement ofLemma 6.7 are satisfied.Set g i = w i π for 0 ≤ i ≤ n . For each i ∈ { , . . . , n } and j ∈ { , . . . , C n } let Q ( j ) i = Q ( j ) i ( g , . . . , g n ) be defined as in the statement of Proposition 6.8. Then byLemma 6.9 each of these sets Q ( j ) i is a rational subset of B , and therefore also arational subset of G . Claim.
There exists an algorithm which for each i ∈ { , . . . , n } and j ∈ { , . . . , C n } computes • a finite state automaton G ( j ) i over X with [ L ( G ( j ) i )] π = Q ( j ) i , and • a finite state automaton B ( j ) i over Z with [ L ( B ( j ) i )] ξ = Q ( j ) i . HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 39
Proof of claim.
For each 1 ≤ j ≤ C n , the algorithm iteratively constructs the pairs( G ( j ) i , B ( j ) i ) in the following way. When i = 0 we have Q ( j ) i = { } and it is clearthat an appropriate pair ( G ( j )0 , B ( j )0 ) can be computed e.g. by taking automata thataccept only the empty word in each case. Now consider a typical stage i with i > Q ( j ) i = (( N ( j ) n,i − ) − Q ( j ) i − g i − ∩ A ) φ. The algorithm constructs the automaton M ( j ) n,i − over X such that [ L ( M ( j ) n,i − )] π =( N ( j ) n,i − ) − from the automaton N ( j ) n,i − . Using M ( j ) n,i − and G ( j ) i − the algorithmthen produces, in the obvious way, a FSA D ( j ) i over X such that [ L ( D ( j ) i )] π =( N ( j ) n,i − ) − Q ( j ) i − g i − . The algorithm given by assumption (ii), in the statement ofthe theorem, is then applied to the automaton D ( j ) i which yields an automaton C ( j ) i over X satisfying [ L ( C ( j ) i )] π = ( N ( j ) n,i − ) − Q ( j ) i − g i − ∩ A. The algorithm then calls as a subroutine the algorithm from Theorem 2.9 whichreturns a FSA A ( j ) i over Y satisfying[ L ( A ( j ) i )] θ = ( N ( j ) n,i − ) − Q ( j ) i − g i − ∩ A. By replacing each letter y l ∈ Y , 1 ≤ l ≤ k , in the transitions of the automaton A ( j ) i by the corresponding letter z l ∈ Z , the algorithm computes the automaton B ( j ) i over Z such that [ L ( B ( j ) i )] ξ = (( N ( j ) n,i − ) − Q ( j ) i − g i − ∩ A ) φ = Q ( j ) i . Finally, the algorithm calls Theorem 2.9 as a subroutine which returns the automa-ton G ( j ) i over X satisfying[ L ( G ( j ) i )] π = (( N ( j ) n,i − ) − Q ( j ) i − g i − ∩ A ) φ = Q ( j ) i . This completes the proof of the claim. (cid:3)
To complete the proof, by Proposition 6.8, we have g ∈ M if and only if ε = · · · = ε n = 1 and g n ∈ ( Q ( j ) n ) − N ( j ) n,n . Using the automata G ( j ) n and N ( j ) n,n , thealgorithm produces, in the obvious way, a FSA A ( w ) over X such that [ L ( A ( w ))] π =( Q ( j ) n ) − N ( j ) n,n .Therefore, in summary we have shown that there is an algorithm which givenany word w ∈ ( X ∪ { t } ) ∗ computes a word w t ε w t ε . . . t ε n w n in reduced form,equal to w in G , and also computes a FSA A ( w ) over X such that w representsan element of M if and only if ε = · · · = ε n = 1 and w n π ∈ [ L ( A ( w ))] π . This isdecidable by condition (i) of the theorem. (cid:3) Corollary 6.10.
For a finite alphabet X let G = F G ( X ) ∗ t,φ : A → B be an HNN extension such that A, B are finitely generated. Then for any finitesubsets W , W , . . . , W d , W ′ , . . . , W ′ d of F G ( X ) the membership problem for M = Mon h W ∪ W t ∪ W t ∪ · · · ∪ W d t d ∪ tW ′ ∪ · · · ∪ t d W ′ d i is decidable. Proof.
It will suffice to prove that the hypotheses of Theorem D are satisfied. Since G is a free group, hypothesis (i) is satisfied by Benois’ Theorem (Theorem 2.7) andCorollary 2.8, while condition (ii) holds as a consequence of [3, Theorem 3.3 andCorollary 3.4]. (cid:3) Applications of HNN extension results to the prefix membershipproblem
In this section we present several applications of the results from the previoussection, mainly Theorem D and its Corollary 6.10. These applications include,among others, some presentations of Adjan type, conjugacy pinched presentationsand, in particular, Baumslag-Solitar groups.7.1.
Exponent sum zero theorem.
For a word w ∈ X ∗ and x ε ∈ X , where x ∈ X and ε ∈ { , − } , we write | w | x ε for the number of occurrences of x ε in w .For t ∈ X , the exponent sum of t in w is the number | w | t − | w | t − . We say that t has exponent sum zero in w if | w | t = | w | t − = 0.We now describe a well-known method due to McCool and Schupp [33] for ex-pressing certain one-relator groups as HNN extensions of one-relator groups with ashorter defining relator. See [27, Page 198].Let w ∈ X ∗ be a word in which the letter t ∈ X has exponent sum zero. Wedefine a word ρ t ( w ) over the infinite alphabetΞ = { x l : x ∈ X \ { t } , l ∈ Z } obtained from w by first replacing each occurrence of x ∈ X \ { t } by x − i where i is the exponent sum of t in the prefix of w preceding the considered occurrence of x , and then deleting every occurrence of t . For each x ∈ X \ { t } let µ x and m x berespectively the smallest and the greatest value of j such that x j actually appearsin ρ t ( w ). For example ρ t ( bt − at bt − a ) = b a b − a , and µ a = 0 and m a = 1.The following result is originally due to Moldavanski˘ı [35]. Its proof can beextracted from the proof of Freiheitssatz given in [27, Section IV.5] which followsthe approach in [33]. Proposition 7.1.
Let w ∈ X ∗ be a word in which t ∈ X has exponent sum zerosuch that ρ t ( w ) is cyclically reduced. Then the group G = Gp h X | w = 1 i is an HNNextension of the group H = Gp h Ξ w | ρ t ( w ) = 1 i where Ξ w = { x l : x ∈ X \ { t } , µ x ≤ l ≤ m x } . The associated subgroups A and B in this extension are free groups freely generated by Ξ w \ { x m x : x ∈ X \ { t }} and Ξ w \ { x µ x : x ∈ X \ { t }} , respectively, with the isomorphism φ : x i x i +1 for all x ∈ X \ { t } and µ x ≤ i < m x . We say that w is prefix t -positive if | u | t − | u | t − ≥ u of w .Analogously, w is said to be prefix t -negative if | u | t − | u | t − ≤ u of w . HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 41
Theorem 7.2.
Maintaining the notation above, let G = h X | w = 1 i be a one-relator group presentation such that some t ∈ X has exponent sum zero in w andthat ρ t ( w ) is cyclically reduced, and let H = Gp h Ξ w | ρ t ( w ) = 1 i . Let A be the subgroup of H generated by Ξ w \ { x m x : x ∈ X \ { t }} . Suppose that (i) the rational subset membership problem is decidable in H , and (ii) A ≤ H is effectively closed for rational intersections.(In particular conditions (i) and (ii) both hold in the case that H is a free group.) If w is either prefix t -positive or prefix t -negative, then the group G defined by the pre-sentation Gp h X | w = 1 i has decidable prefix membership problem. Consequently, ifthese conditions hold and the inverse monoid Inv h X | w = 1 i is E -unitary, then ithas decidable word problem.Proof. First of all, the group G = Gp h X | w = 1 i is an HNN extension of H byProposition 7.1. So, to prove the theorem, it suffices to show that the prefix monoid P w ⊆ G has a generating set of the form given in Theorem D.We consider only the prefix t -positive case, the prefix t -negative case being anal-ogous. Write w ≡ u τ u τ u . . . τ n u n , where τ i ∈ { t, t − } + for all 1 ≤ i ≤ n and u i ∈ X \ { t } ∗ , where the words u , . . . , u n − are all non-empty. Set ξ i to be the t -exponent sum of τ i for all1 ≤ i ≤ n .The exponent sum zero condition translates to ξ + ξ + · · · + ξ n = 0 , while it is easy to see that the prefix t -positive condition is implies that the inequal-ities σ r = ξ + · · · + ξ r ≥ ≤ r ≤ n . This implies that for any letter x = t appearing in w wehave µ x , m x ≤ µ x ≤ − σ r ≤ m x for all 1 ≤ r ≤ n . In particular, σ = ξ > ρ t ( w ) ≡ u ′ u ′ . . . u ′ n − u ′ n , where both u ′ , u ′ n are obtained from u , u n , respectively, by equipping each of theirletters by the subscript 0, and for all 1 ≤ i ≤ n −
1, the word u ′ i is obtained from u i by putting − σ i in the subscript of each letter in u i .The prefixes p of w can be classified into the following three types:(1) p is a prefix of u ;(2) p ≡ u τ . . . τ i q , where q is a prefix of u i for some 1 ≤ i ≤ n ;(3) p ≡ u τ . . . u i − θ i for some 1 ≤ i ≤ n and prefix θ i of τ i .Our aim is to rewrite the elements of G represented by these prefixes with respectto the generating set Ξ w ∪ { t } of the presentation of G given in Proposition 7.1 asan HNN-extension H ∗ t,φ : A → B .Indeed, upon recalling from [27, Page 198] that the generator x − s of H representsthe element t s xt − s in G = H ∗ t,φ : A → B , we immediately see that the prefixes of u translate into prefixes of u ′ , a finite subset of H . This deals with the prefixes oftype (1). For prefixes of type (2), in G we have the equality p = u ( t σ u t − σ ) . . . ( t σ i − u i − t − σ i − )( t σ i qt − σ i ) t σ i , which means that in the presentation for the HNN extension H ∗ t,φ : A → B we have p = u ′ . . . u ′ i − q ′ t σ i where q ′ is the word obtained by putting the subscript − σ i on every letter of q ,and thus q ′ is a prefix of u ′ i . By a very similar argument, prefixes of type (3) areexpressed as u ′ . . . u ′ i − t σ i − θ i . Let α i ∈ Z be the t -exponent sum of θ i . By the prefix t -positive condition, σ i − + α i ≥
0. This implies that t σ i − θ i ∈ { t, t − } ∗ has non-negative t -exponent sumand therefore is equal in G to a non-negative power of t . Hence, upon defining d = max ≤ i ≤ n σ i we conclude that the monoid P w has a generating set of the form W ∪ W t ∪ · · · ∪ W d t d for some finite W , W , . . . , W d ⊆ H . We now see that all the requirements ofTheorem D are satisfied, so we conclude that the membership problem of P w in G is decidable. (cid:3) Remark . It is natural to compare the prefix t -positive condition in Theorem 7.2with of w -strictly positive presentations considered in [19] where it was shown thatgroups defined by such presentations have decidable prefix membership problem;see [19, Theorem 5.1]. In [19, Corollary 5.2] it is shown that if w is a cyclicallyreduced word such that Gp h X | w = 1 i is a w -strictly positive presentation then thegroup of units of Inv h X | w = 1 i is trivial. In contrast, the t -positive condition in ourtheorem certainly does not imply that the group of units is trivial, and in this waywe see that the class of examples to which Theorem 7.2 applies is distinct from thosedealt with by [19, Theorem 5.1]. For example, the inverse monoid presentationInv h a, b, t | tat − btat − = 1 i is t -positive and it may be shown the the group of units of this inverse monoid isthe infinite cyclic group. Indeed it may be shown that tat − and b are the minimalinvertible pieces of this relator, since it is easily seen that this inverse monoid isnot a group. These pieces satisfy the unique marker letter property, and hence byresults proved in [11] the group of units of this inverse monoid isomorphic to thegroup defined by the presentation Gp h x, y | xyx = 1 i . Hence the group of units ofthis monoid is the infinite cyclic group. Example 7.4.
Let M = Inv h a, b, c, t | t − atcbt − at cbt − at c = 1 i . Then t has exponent sum zero in the relator word, and, furthermore, this word isprefix t -negative. The corresponding group G = Gp h a, b, c, t | ( t − at ) cb ( t − at ) cb ( t − at ) c = 1 i is an HNN extension of the group H = Gp h a , a , a , b , c | a c b a c b a c = 1 i with A = Gp h a , a i , B = Gp h a , a i , and a i φ = t − a i t = a i +1 for i ∈ { , } . Thedefining relator in the presentation of H is a positive word and hence is cyclicallyreduced. Also, since the generator a only appears once in that word, it follows from HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 43 the Freiheitssatz that that H is a free group of finite rank. Hence, by Corollary 6.10and Theorem 7.2 the group defined by G = Gp h a, b, c, t | ( t − at ) cb ( t − at ) cb ( t − at ) c = 1 i has decidable prefix membership problem. Since the monoid M is E -unitary, asthe defining relator word is cyclically reduced, it follows that M has decidable wordproblem. Example 7.5.
For a slightly more involved example, let G = Inv h a, b, c, t | w = 1 i ,where w ≡ tbcbt bbct − ct − at bt − at ct − ct − . Note that w is not cyclically reduced; however, t has exponent zero in w and it is t -positive. Furthermore, ρ t ( w ) ≡ b − c − b − b − c − c − a b − a c − c − is a cyclically reduced word, so G is an HNN extension of H = Gp h a , b − , . . . , b − , c − , . . . , c − | ρ t ( w ) = 1 i . Note that ρ t ( w ) is a cyclically reduced word. Since the generator b − occurs onlyonce in ρ t ( w ) it follows by the Freiheitssatz that H is a free group of finite rank.Therefore, Theorem 7.2 tells us that the membership problem for P w in G is decid-able.As in the previous section, we now exhibit an example to which the methods ofthis section do not apply. Example 7.6.
Consider the presentation h a, b, t | bt − at bt − a = 1 i . Note that the relator word in this presentation is cyclically reduced and has expo-nent sum zero for the letter t . However, it is neither prefix t -positive, nor prefix t -negative. The group G defined by this presentation is an HNN extension of H = Gp h a , a , b − , b | b a b − a = 1 i , which is clearly a free group of rank 3. Theassociated subgroups A = Gp h a , b − i and B = Gp h a , b i are free groups of rank2. However, upon identifying all the prefixes of bt − at bt − a and expressing themin terms of the generators of the described HNN extension of H , we see that P w = Mon h W ∪ W t ∪ W − t − i , where W = { a , b , ( b − a ) − } , W = { a , ( b − a ) − } , W − = { b , ( b − a ) − } . Now we cannot apply Corollary 6.10 because of the ‘mixed’ nature of the generatingset of P w which contains both elements with t and with t − . The underlyingproblem now is that when we form an arbitrary product of such elements (that is,a product representing an element of P w ), we cannot guarantee anymore that sucha product is already in reduced form, as we had in Lemma 6.7 and Proposition6.8. Also, keeping track of the rationality of subsets containing elements of H occurring between consecutive instances of t and t − in such products becomesincreasingly troublesome as we are forced to make more and more (potentiallynested) t -reductions. Example 7.7.
In Example 5.11 we have seen that the orientable surface group ofgenus n ≥
2, defined by its standard presentation G n = Gp h a , . . . , a n , b , . . . , b n | [ a , b ][ a , b ] . . . [ a n , b n ] = 1 i , has decidable prefix membership problem. Now, by using Theorem 7.2 we canapply out results to give a new proof of [19, Theorem 5.3(b)] showing that theprefix membership problem is decidable for all cyclic conjugates of the relator wordin the above presentation.Indeed, upon denoting u ≡ [ a , b ] . . . [ a n , b n ], we have four cases to consider:(i) w ≡ a − b − a b u ;(ii) w ≡ b − a b ua − ;(iii) w ≡ a b ua − b − ;(iv) w ≡ b ua − b − a .Case (i) is already resolved in Example 5.11; to illustrate how to deal with theremaining ones, we consider Case (iii) the other cases being similar. Take a to bethe stable letter. The word w is cyclically reduced, a is exponent sum zero in w ,and w is a -positive. We conclude that G n is an HNN extension of H = Gp h ( b ) − , ( b ) , ( a ) − , ( b ) − , . . . | ( b ) − v (( b ) ) − = 1 i , where v is obtained from u by replacing each a i , b i , . . . by ( a i ) − , ( b i ) − , . . . , re-spectively, for 2 ≤ i ≤ n . So, H is a free group of rank 2 n − b ) − and ( b ) = ( b ) − v , respectively. By Theorem 7.2we obtain that P w has decidable membership in G n .We finish this subsection by presenting yet another application of Theorem 7.2which concerns the prefix membership problem for one-relator groups defined byAdjan presentations [16, 32] over a two-letter alphabet. Recall that a one-relatorgroup, inverse monoid, or monoid presentation is an Adjan presentation if it is ofthe form h X | u = v i , where u, v ∈ X ∗ are positive words such that the first lettersof u, v are different, and also the last letters of u, v are different. For our purposes,group presentations of Adjan type will be written as h X | uv − = 1 i ; note that thegiven conditions ensure that the word uv − is cyclically reduced. Theorem 7.8.
Let G = Gp h a, b | uv − = 1 i be a group defined by an Adjan presen-tation such that | u | a = | v | a . Assume that at least one of the following conditionshold: (i) one of the words u or v begins with ba ; (ii) one of the words u or v end with ab ; (iii) there exists an integer k , ≤ k < | u | a , such that there is a single b betweenthe k th and the ( k + 1) th occurrence of a in one of the words u, v , while inthe other word the k th and the ( k + 1) th occurrence of a are adjacent.Then the prefix membership problem for G is decidable, as is the word problem forthe inverse monoid Inv h a, b | uv − = 1 i .Proof. For each of the assumptions (i)–(iii), there are four cases to consider de-pending upon the first and last letters of u, v . However, all these cases are verysimilar, so we consider only one of them in each instance. Let p = | u | a = | v | a . HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 45
We begin by assuming that u begins with ba and ends with a . Then v beginswith a and ends with b , so we may write u = bab α . . . b α p a,v = ab β . . . ab β p +1 , for some integers α i , β i ≥
0. The word uv − = bab α . . . b α p ab − ( β p +1) a − . . . b − β a − , has exponent sum zero for a and is prefix a -positive. By considering a as the stableletter, it follows that G is an HNN extension of the group H = Gp h b , b − , . . . , b − p | b b α − . . . b α p − ( p − b − ( β p +1) − p . . . b − β − = 1 i . The defining relator is a cyclically reduced word, and H is a free group of finiterank by the Freiheitssatz since the generator b arises exactly once in the definingrelator. Hence in this case the result follows by Theorem 7.2.Similarly, if e.g. v begins with b and ends with ab then u both begins and endswith a , and so we may write u = ab α . . . b α p − a,v = b β +1 ab β . . . ab, for some integers α i , β i ≥
0. In this case, we conclude that G is an HNN extensionof the groupGp h b , b − , . . . , b − p | b α − . . . b α p − − ( p − b − − p . . . b − β − b − ( β +1)0 = 1 i . The defining relator is a cyclically reduced word, and H is a free group of finiterank by the Freiheitssatz since the generator b − p arises exactly once in the definingrelator. Hence in this case the result follows by Theorem 7.2.Finally, upon assuming (iii), let us further assume that u begins and ends with a , while v begins and ends with b . Then, for example, u = ab α . . . b α k − abab α k +1 . . . b α p − a,v = b β +1 ab β . . . b β k − aab β k +1 . . . ab β p +1 , or some integers α i , β i ≥
0. This leads to the conclusion that G is an HNN extensionof the groupGp h b , b − , . . . , b − p | b α − . . . b − k . . . b − ( β p +1) − p . . . b − β k +1 − ( k +1) b − β k − − ( k − . . . b − ( β +1)0 = 1 i . The defining relator is a cyclically reduced word, and H is a free group of finiterank by the Freiheitssatz since the generator b − arises exactly once in the definingrelator. Hence in this case the result follows by Theorem 7.2. (cid:3) Remark . There are examples to which Theorem 7.8 applies, which are nothandled in [32, Corollary 2.6]. For example, it covers a part of Case 4 from thatcorollary for which the decidability of the prefix membership problem is not deducedthere (one of the simplest examples is u ≡ aba , v ≡ baab ). This shows that ourresults are not consequences of the approach via distortion functions pursued in[32]. Conjugacy pinched presentations.
The “HNN analogue” of the class ofcyclically pinched groups are the conjugacy pinched groups : these are one-relatorgroups defined by a presentation of the formGp h X ∪ { t } | t − ut = v i , where u, v ∈ X ∗ are nonempty reduced words. Again, for our purposes, conjugacypinched group presentations will be written in the formGp h X ∪ { t } | t − utv − = 1 i . Theorem 7.10.
The prefix membership problem is decidable for any group definedby a conjugacy pinched presentation Gp h X ∪ { t } | t − utv − = 1 i . Consequently, the word problem is decidable for all one-relator inverse monoids ofthe form
Inv h X ∪ { t } | t − utv − = 1 i with u and v both reduced reduced words from X ∗ .Proof. By the Freiheitssatz, any conjugacy pinched group is the HNN extension ofthe free group
F G ( X ) with associated cyclic subgroups A = Gp h u i and B = Gp h v i .Hence, to prove the theorem it suffices to compute the set of prefixes of the word w ≡ t − utv − (which generate the prefix monoid P w ) and see that it has the formrequired by Theorem D. Indeed, we havepref( w ) = t − · pref( u ) ∪ t − ut · pref( v − ) . Note that in G we have t − ut · pref( v − ) = pref( v ), so P w is generated by W ∪ t − W ′ for W = pref( v ) and W ′ = pref( u ), whence the required result follows (seeRemark 6.1). (cid:3) Example 7.11.
As an application of the previous theorem, groups defined bypresentations of the form B ( m, n ) = Gp h a, b | b − a m ba − n = 1 i have decidable prefix membership problems. These are so-called Baumslag-Solitarpresentations. Hence, the inverse monoidsInv h a, b | b − a m ba − n = 1 i have decidable word problems (cf. [16, Theorem 4.2] for a highly related result).8. An undecidability result in the non-cyclically reduced case
In this article the main applications of our results have been to show that the pre-fix membership problem is decidable for certain groups defined by one-relator pre-sentations. On the other hand, in the recent paper [10] a word w (over a 3-elementalphabet X ) is constructed such that the inverse monoid M = Inv h X | w = 1 i hasundecidable word problem. Furthermore, it was proved in Theorem 3.8 of the samepaper that M is actually E -unitary. Combining this fact with [19, Theorem 3.1](see Theorem 2.5 for the statement) it follows that there does exists a one-relatorgroup G = Gp h X | w = 1 i with undecidable prefix membership problem. Hence,the following open problem arises naturally. HE PREFIX MEMBERSHIP PROBLEM FOR ONE-RELATOR GROUPS 47
Problem . Characterise the words w ∈ X ∗ with the property that the prefixmembership problem for Gp h X | w = 1 i is decidable. In particular, is the prefixmembership problem decidable when w is a cyclically reduced word?The latter question was stated in [5, Question 13.10]. By modifying some ideasand results from [10], we shall now show that if one weakens the hypothesis ofthis problem to insisting only that w is a reduced word, then this question has anegative answer. Theorem 8.2.
There is a finite alphabet X and a reduced word w ∈ X ∗ such that Gp h X | w = 1 i has undecidable prefix membership problem.Proof. Let H = Gp h a, b | abab − a − ba − b − = 1 i . It follows from [10, Theorem 2.4]that there is a finite set of words u , u , . . . , u k ∈ { a, b } ∗ such that the membershipproblem for T = Mon h u , u , . . . , u k i in H is undecidable. Set r ≡ abab − a − ba − b − and s ≡ a − b − abab − a − b , andlet β ≡ ( ara − )( brb − )( a − sa )( b − sb )and γ ≡ ( tu t − ) r ( tu − t − ) r ( tu t − ) r ( tu − t − ) r . . . r ( tu k t − ) r ( tu − k t − ) , where t is a new letter not in { a, b } . Now define w ≡ βγrγ − β − . It is easy to see that w is a reduced word in X ∗ where X = { a, b, t } . Weclaim that Gp h X | w = 1 i has undecidable prefix membership problem. Let P =Mon h pref( w ) i ≤ G . From the definition of w it follows that r = 1 in the group G . Since r = 1 and s is a cyclic conjugate of r , it follows that s = 1 in G . Usingthe fact that r = 1 and s = 1, by considering the prefixes of β we see that all of a, a − , b and b − belong to P (meaning that the elements these words represent allbelong to P ). Since β = 1 in P , considering prefixes of γ and using the fact that r = 1 in G we see that t belongs to P , and tu i t − belongs to P for all 1 ≤ i ≤ k .Since every other prefix of w is clearly expressible as a product of these elementswe conclude that P is equal to the submonoid of G generated by { a, b } ∪ { t } ∪ { tu i t − : i ∈ { , . . . , k }} . It may be shown (see [10, Lemma 3.6]) that for any word v ∈ { a, b } ∗ we have that tvt − represents an element of P if and only if in H the word v represents an elementin the submonoid T ≤ H . By assumption the submonoid membership problem for T in H is undecidable, and hence it follows that the membership problem for P within G is undecidable. Hence Gp h X | w = 1 i has undecidable prefix membershipproblem, where w ∈ X ∗ is a reduced word. (cid:3) Remark . Note that in the proof of Theorem 8.2 the initial presentationGp h a, b | abab − a − ba − b − = 1 i for the group H does have decidable prefix membership problem, and this followsas a consequence of Theorem 7.2. To see this, note that the letter a has exponentsum zero in the word r ≡ abab − a − ba − b − . Furthermore, r is prefix a -positive.Now following the method described in Proposition 7.1, working with respect to the exponent sum zero letter a , the group H arises as an HNN extension of thegroup H = Gp h b − , b − , b | b − b − − b − b − = 1 i , which is just the free group of rank 2 generated by b − and b − . Since H is a freegroup it follows that the hypotheses (i) and (ii) of Theorem 7.2 are both satisfied.Hence, Theorem 7.2 can be applied and it follows that the above presentation for H has decidable prefix membership problem.We conclude that the question of decidability of the prefix membership prob-lem depends on the presentation of the considered group; in this remark and inthe previous theorem we have just seen two presentations of the same group H ,one yielding undecidable prefix membership problem, whereas the same problem isdecidable with respect to the other presentation. Acknowledgements.
The authors are grateful to an anonymous referee for anumber of helpful comments. In particular, one of their suggestions led to theexample included in Remark 8.3.
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Department of Mathematics and Informatics, University of Novi Sad, Trg DositejaObradovi´ca 4, 21101 Novi Sad, Serbia
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