The Post Correspondence Problem and equalisers for certain free group and monoid morphisms
aa r X i v : . [ m a t h . G R ] S e p THE POST CORRESPONDENCE PROBLEM AND EQUALISERS FORCERTAIN FREE GROUP AND MONOID MORPHISMS
LAURA CIOBANU AND ALAN D. LOGAN
Abstract.
A marked free monoid morphism is a morphism for which the image of eachgenerator starts with a different letter, and immersions are the analogous maps in free groups.We show that the (simultaneous) PCP is decidable for immersions of free groups, and providean algorithm to compute bases for the sets, called equalisers, on which the immersions takethe same values. We also answer a question of Stallings about the rank of the equaliser.Analogous results are proven for marked morphisms of free monoids. Introduction
In this paper we prove results about the classical Post Correspondence Problem (PCP FM ),which we state in terms of equalisers of free monoid morphisms, and the analogue problemPCP FG for free groups ([CMV08], [MNU14]), and we describe the solutions to PCP FM andPCP FG for certain classes of morphisms. While the classical PCP FM is famously undecidablefor arbitrary maps of free monoids [Pos46] (see also the survey [HK97] and the recent resultof Neary [Nea15]), PCP FG for free groups is an important open question [DKLM19, Problem5.1.4]. Additionally, for both free monoids and free groups there are only few results describingalgebraically the solutions to classes of instances known to have decidable PCP FM or PCP FG .Our results apply to marked morphisms in the monoid case, and to their counterparts in freegroups, called immersions . Marked morphisms are the key tool used in resolving the PCP FM for the free monoid of rank two [EKR82], and therefore understanding the solutions to thePCP FG for immersions is an important step towards resolving the PCP FG for the free groupof rank two. The density of marked morphisms and immersions among all the free monoid orgroup maps is strictly positive (Section 10), so our results concern a significant proportion ofinstances.An instance of the PCP FM is a tuple I = (Σ , ∆ , g, h ), where Σ , ∆ are finite alphabets,Σ ∗ , ∆ ∗ are the respective free monoids, and g, h : Σ ∗ → ∆ ∗ are morphisms. The equaliser of g, h is Eq( g, h ) = { x ∈ Σ ∗ | g ( x ) = h ( x ) } . The PCP FM is the decision problem:PCP FM PCP FM PCP FM : Given I = (Σ , ∆ , g, h ), is the equaliser Eq( g, h ) trivial?Analogously, an instance of the PCP FG is a four-tuple I = (Σ , ∆ , g, h ) with g, h : F (Σ) → F (∆) morphisms between the free groups F (Σ) and F (∆), and PCP FG is the decision problempertaining to the similarly defined Eq( g, h ) in free groups.Beyond PCP FM , in this paper we also consider the Algorithmic Equaliser Problem , denotedAEP FM (or AEP FG in the group case), which for an instance I = (Σ , ∆ , g, h ) with g, h freemonoid morphisms (or free group morphisms for AEP FG ), says: Mathematics Subject Classification.
Key words and phrases.
Post Correspondence Problem, free group, free monoid, marked map, immersion.
AEP FM AEP FM AEP FM : Given I = (Σ , ∆ , g, h ), output(a) a finite basis for Eq( g, h ), or(b) a finite automaton recognising the set Eq( g, h ).If a finite basis or finite automaton for Eq( g, h ) does not exist then Part (a) or (b), respec-tively, of the problem is insoluble. Note that (a) and (b) are connected: for free groups thesetwo problems are in fact the same when Eq( g, h ) is finitely generated, while for free monoids((a)) implies ((b)). Part ((a)) of the AEP FM is known to be soluble when | Σ | = 2 and one of g or h is non-periodic, and insoluble otherwise [Hol03] [HK97, Corollary 6]. Sets of morphisms.
We are particularly interested in sets S of morphisms (not just twomorphisms f , g ) and their equalisers Eq( S ) = T g,h ∈ S Eq( g, h ), and we prove structural resultsfor arbitrary sets and algorithmic results for finite sets. Our results resolve the simultaneous
PCP FG and PCP FM for immersions and marked morphisms; these problems take as input afinite set S of maps and ask the same questions about equalisers as in the classical setting.Analogously, one could further define the “simultaneous AEP FG and AEP FM ”. However, thesimultaneous AEP FG is equivalent to the AEP FG , and Part ((b)) of the simultaneous AEP FM is equivalent to Part ((b)) of the AEP FM , as follows. As bases of intersections of finitelygenerated subgroups of free groups are computable (and as Parts ((a)) and ((b)) of the AEP FG are equivalent), if the AEP FG is soluble for a class C of maps then there exists an algorithmwith input a finite set S of morphisms from F (Σ) to F (∆), S ⊆ C , and output a basis forEq( S ). Similarly, automata accepting intersections of regular languages are computable, andso if Part ((b)) of the AEP FM is soluble for a class C of maps then there exists an algorithmwith input a finite set S of morphisms from Σ ∗ to ∆ ∗ , S ⊆ C , and output a finite automatonwhose language is Eq( S ) = T g,h ∈ S Eq( g, h ). Main results.
A set of words s ⊆ ∆ ∗ is marked if any two distinct u, v ∈ s start with adifferent letter of ∆, which implies | s | ≤ | ∆ | . A free monoid morphism f : Σ ∗ → ∆ ∗ is marked if the set f (Σ) is marked. An immersion of free groups is a morphism f : F (Σ) → F (∆) wherethe set f (Σ ∪ Σ − ) is marked (see Section 3 for equivalent formulations). Halava, Hirvensaloand de Wolf [HHdW01] showed that PCP FM is decidable for marked morphisms; inspired bytheir methods we were able to obtain stronger results (Theorem A) for this kind of map,as well as expand to the world of free groups (Theorem C), where we employ ‘finite stateautomata’-like objects called Stallings graphs . Theorem A. If S is a set of marked morphisms from Σ ∗ to ∆ ∗ , then there exists a finitealphabet Σ S and a marked morphism ψ S : Σ ∗ S → Σ ∗ such that Image( ψ S ) = Eq( S ) . Moreover,for S finite, there exists an algorithm with input S and output the marked morphism ψ S . Corollary B.
The simultaneous
PCP FM is decidable for marked morphisms of free monoids. Theorem C. If S is a set of immersions from F (Σ) to F (∆) , then there exists a finitealphabet Σ S and an immersion ψ S : F (Σ S ) → F (Σ) such that Image( ψ S ) = Eq( S ) . Moreover,when S is finite, there exists an algorithm with input S and output the immersion ψ S . Corollary D.
The simultaneous
PCP FG is decidable for immersions of free groups. The Equaliser Conjecture.
Our work was partially motivated by Stallings’ Equaliser
CP AND EQUALISERS FOR CERTAIN MORPHISMS 3
Conjecture for free groups, which dates from 1984 [Sta87, Problems P1 & 5] (also [DV96,Problem 6] [Ven02, Conjecture 8.3] [BMS02, Problem F31]). Here rk( H ) stands for the rank ,or minimum number of generators, of a subgroup H : Conjecture 1.1 (The Equalizer Conjecture, 1984) . If g, h : F (Σ) → F (∆) are injectivemorphisms then rk(Eq( g, h )) ≤ | Σ | . This conjecture has its roots in “fixed subgroups” Fix( φ ) of free group endomorphisms φ : F (Σ) → F (Σ) (if Σ = ∆ then Fix( φ ) = Eq( φ, id)), where Bestvina and Handel provedthat rk(Fix( φ )) ≤ | Σ | for φ an automorphism [BH92], and Imrich and Turner extended thisbound to all endomorphisms [IT89]. Bergman further extended this bound to all sets ofendomorphisms [Ber99]. Like Bergman’s result, our first corollary of Theorem C considerssets of immersions, which are injective, and answers Conjecture 1.1 for immersions. Corollary E. If S is a set of immersions from F (Σ) to F (∆) then rk(Eq( S )) ≤ | Σ | . In free monoids, equalisers of injections are free [HK97, Corollary 4] but they are notnecessarily regular languages (and hence not necessarily finitely generated) [HK97, Example6]. In order to understand equalisers Eq( S ) of sets of maps we need to understand intersectionsin free monoids. Recall that the intersection A ∗ ∩ B ∗ of two finitely generated free submonoidsof a free monoid Σ ∗ is free [Til72] and one can find a regular expression that represents a basisof A ∗ ∩ B ∗ [BH77]. However, the intersection is not necessarily finitely generated [Kar84].The following result is surprising because we have finite generation, even for the intersectionEq( S ) = T g,h ∈ S Eq( g, h ). Corollary F. If S is a set of marked morphisms from Σ ∗ to ∆ ∗ then Eq( S ) is a free monoidwith rk(Eq( S )) ≤ | Σ | . The Algorithmic Equaliser Problem.
The AEP FG is insoluble in general, as equalisers infree groups are not necessarily finitely generated [Ven02, Section 3], and is an open problem ofStallings’ if both maps are injective [Sta87, Problems P3 & 5]. Our next corollary of TheoremC resolves this open problem for immersions. Corollary G.
The
AEP FG is soluble for immersions of free groups. The AEP FM is insoluble in general, primarily as equalisers are not necessarily regular lan-guages [ER78, Example 4.6]. Even for maps whose equalisers form regular languages, theproblem remains insoluble [KS10]. Another corollary of Theorem A is the following. Corollary H.
The
AEP FM is soluble for marked morphisms of free monoids. Outline of the article.
In Section 2 we prove Theorem A and its corollaries. The remainderof the paper focuses on free groups, where the central result is Theorem 6.2, which is TheoremC for | S | = 2. In Section 3 we reformulate immersions in terms of Stallings’ graphs. In Section4 we define the “reduction” I ′ = (Σ ′ , ∆ ′ , g ′ , h ′ ) of an instance I = (Σ , ∆ , g, h ) of the AEP FG for immersions. Repeatedly computing reductions is the key process in our algorithm. InSection 5 we prove the process of reduction reduces the “prefix complexity” of an instance (sothe word “reduction” makes sense). In Section 6 we prove Theorem 6.2, mentioned above. InSection 7 we prove Theorem C and its corollaries. In Section 9 we give a complexity analysisfor both our free monoid and free group algorithms, and in Section 10 we show that the densityof marked morphisms and immersions among all the free monoid or group maps is strictlypositive. LAURA CIOBANU AND ALAN D. LOGAN
Acknowledgements
The authors were supported by EPSRC Standard Grant EP/R035814/1. The first-namedauthor would like to thank the organisers of the Dagstuhl seminar 19131
Algorithmic Prob-lems in Group Theory , where the topics addressed in this paper were discussed and listed asimportant open questions in the theory of free groups [DKLM19, Direction 5.1.4].2.
Marked morphisms in free monoids
In this section we prove Theorem A and its corollaries. We use the following immediatefact.
Lemma 2.1.
Marked morphisms of free monoids are injective.Proof.
Let f : Σ ∗ → ∆ ∗ be marked and let x = y be nontrivial. One can write x = zax ′ and y = zby ′ , where a, b ∈ Σ are the first letter where x and y differ. As f is marked, f ( a ) = f ( b ),hence f ( x ) = f ( z ) f ( a ) f ( x ′ ) = f ( z ) f ( b ) f ( y ′ ) = f ( y ), so f is injective. (cid:3) We may assume Σ ⊆ ∆, as | Σ | ≤ | ∆ | holds whenever f : Σ ∗ → ∆ ∗ is marked.Consider morphisms g : Σ ∗ → ∆ ∗ and h : Σ ∗ → ∆ ∗ . The set of non-empty words over analphabet Σ is denoted Σ + . For a ∈ ∆, a pair ( u, v ) ∈ Σ +1 × Σ +2 is an a -block if (i) g ( u ) = h ( v )starts with a , and (ii) u and v are minimal, that is, the length | g ( u ) | = | h ( v ) | is minimal amongall such pairs. If the pair ( g, h ) has blocks a i = ( u i , v i ), 1 ≤ i ≤ m , then let Σ ′ be the alphabetconsisting of these blocks and define g ′ : (Σ ′ ) ∗ Σ ∗ by g ′ ( a i ) = u i and h ′ : (Σ ′ ) ∗ Σ ∗ by h ′ ( a i ) = v i . These maps are computable and, by an identical logic to [HHdW01, Section 2],are seen to be marked. Then gg ′ = hh ′ , and we let k = gg ′ = hh ′ (so k : (Σ ′ ) ∗ → ∆ ∗ ). Since k is the composition of marked morphisms, it is itself marked. We therefore have the following. Lemma 2.2. If g : Σ ∗ → ∆ ∗ and h : Σ ∗ → ∆ ∗ are marked morphisms then the correspondingmaps g ′ : Σ ′∗ → Σ ∗ , h ′ : Σ ′∗ → Σ ∗ and k : Σ ∗ → ∆ ∗ , k = gg ′ = hh ′ , are marked and arecomputable. The reduction of an instance I = (Σ , ∆ , g, h ) of the marked PCP FM , as defined in [HHdW01],is the instance I ′ := (Σ ′ , ∆ , g ′ , h ′ ) where Σ ′ is defined as above, and where g ′ and h ′ are asabove, but with codomain ∆ (which we may do as Σ ⊆ ∆). We additionally assume thatΣ ′ ⊆ Σ; we can do this as | Σ ′ | ≤ | Σ | by Lemma 2.2.The following relies on [HHdW01, Lemma 1], which we strengthen by replacing the notionof “equivalence” with that of “strong equivalence”: Two instances I and I of the PCP FM are strongly equivalent if their equalisers are isomorphic, which we write as Eq( I ) ∼ = Eq( I ). Lemma 2.3.
Let I ′ = (Σ ′ , ∆ ′ , g ′ , h ′ ) be the reduction of I = (Σ , ∆ , g, h ) where g and h aremarked. Then I and I ′ are strongly equivalent, and g ′ (Eq( I ′ )) = Eq( I ) = h ′ (Eq( I ′ )) .Proof. Firstly, note that g ′ (Eq( I ′ )) ≤ Eq( I ) [HHdW01, Lemma 1, paragraph 2]. From[HHdW01, Lemma 1, paragraph 1] it follows that g ′ (Eq( I ′ )) ≥ Eq( I ), so g ′ (Eq( I ′ )) = Eq( I ) .As g ′ is injective, the map g ′ | Eq( I ′ ) is an isomorphism. Hence, I and I ′ are strongly equivalent,and, by symmetry for the h ′ map, g ′ (Eq( I ′ )) = Eq( I ) = h ′ (Eq( I ′ )) as required. (cid:3) We can now improve the existing result on the marked PCP FM . We store a morphism f : Σ ∗ → ∆ ∗ as a list ( f ( a )) a ∈ Σ . Theorem 2.4. If I = (Σ , ∆ , g, h ) is an instance of the marked PCP FM then there existsan alphabet Σ g,h and a marked morphism ψ g,h : Σ ∗ g,h → Σ ∗ such that Image( ψ g,h ) = Eq( I ) .Moreover, there exists an algorithm with input I and output the marked morphism ψ g,h . CP AND EQUALISERS FOR CERTAIN MORPHISMS 5
Proof.
We explain the algorithm, and note at the end that the output is a marked morphism ψ g,h : F (Σ g,h ) → F (Σ) with the required properties, and so the result follows.Begin by making reductions I , I , I , . . . , starting with I = I = (Σ , ∆ , g, h ), the inputinstance. Then by [HHdW01, Section 5, paragraph 1] we will obtain an instance I j =(Σ j , ∆ , g j , h j ) such that one of the following will occur:(1) | Σ j | = 1.(2) | g j ( a ) | = 1 = | h j ( a ) | for all a ∈ Σ j .(3) There exists some i < j with I i = I j (sequence starts cycling).Keeping in mind the fact that reductions preserve equalisers (Lemma 2.3), we obtain in eachcase a subset Σ g,h (possibly empty) which forms a basis for Eq( I j ): For Case (1), writingΣ j = { a } , the result holds as if g ( a i ) = h ( a i ) then g ( a ) i = h ( a ) i and so g ( a ) = h ( a ) as rootsare unique in a free monoid. For Case (2), suppose g j ( x ) = h j ( x ). Then g j and h j agree onthe first letter of x ∈ Σ ∗ j because the image of each letter has length one, and inductively wesee that they agree on every letter of x . Hence, a subset Σ g,h of Σ j forms a basis for Eq( I j ).For Case (3), suppose there is a sequence of reductions beginning and ending at I j : I j → I j +1 → · · · → I j +( i − → I j + i = I j and write r := j + i . By Lemma 2.3, Eq( I j ) = g j +1 g j +2 . . . g r (Eq( I r )) = Eq( I r ); thus g r := g j +1 g j +2 . . . g r restricts to an automorphism of Eq( I j ), so g r | Eq( I j ) ∈ Aut(Eq( I j )). Theautomorphism g r is necessarily length-preserving ( | g r ( w ) | = | w | for all w ∈ Eq( I j )). Consider x ∈ Eq( I j ) = Eq( I r ). Then g r maps the letters occurring in x r to letters and so g j (= g r ) and h j (= h r ) map the letters occuring in x to letters, and it follows that every letter occuring in x is a solution to I r = I j . Hence, a subset Σ g,h of Σ j forms a basis for Eq( I j ) as required.Therefore, in all three cases a subset Σ g,h of Σ j forms a basis for Eq( I j ), and since Σ j iscomputable, this basis is as well. In order to prove the theorem, it is sufficient to prove thatthere is a computable immersion ψ g,h : Σ ∗ g,h → Σ ∗ . Consider the map ˜ g = g g · · · g j : Σ ∗ j → Σ ∗ (and the analogous ˜ h ). Now, each g i is marked, by Lemma 2.2, and so ˜ g is the composition ofmarked morphisms and hence is marked itself. Define ψ g,h := ˜ g | Σ ∗ g,h . This map is computablefrom ˜ g , and as Σ g,h ⊆ Σ j , the map ψ g,h is marked. As Image( ψ g,h ) = g g . . . g j (Eq( I j )) =Eq( I ), by Lemma 2.3 and the above, the result follows. (cid:3) Theorem 2.4 combines with the following general result to give the non-algorithmic part ofTheorem A. A subsemigroup M of a free monoid Σ ∗ is marked if it is the image of a markedmorphism. Lemma 2.5. If { M j } j ∈ J is a set of marked subsemigroups of Σ ∗ then the intersection T j ∈ J M j is marked.Proof. Firstly, suppose x, y ∈ M j for some j ∈ J . Then there exist two words x . . . x l and y . . . y k , with x i , y i ∈ Σ, such that φ ( x . . . x l ) = x and φ ( y . . . y k ) = y , where φ is a markedmorphism. If x and y have a nontrivial common prefix, then because φ is marked we get x = y , and φ ( x ) is a prefix of both x and y , and in particular φ ( x ) ∈ M j . By continuingthis argument, if z is a maximal common prefix of x and y , then z ∈ M j .Now, suppose x, y ∈ T j ∈ J M j , and suppose they both begin with some letter a ∈ Σ ∪ Σ − .By the above, their maximal common prefix z a is contained in each M j and so is contained in T j ∈ J M j . Therefore, z a is a prefix of every element of T j ∈ J M j beginning with an a . It followsthat T j ∈ J M j is immersed, as required. (cid:3) We now prove the algorithmic part of Theorem A (this is independent of Lemma 2.5).
LAURA CIOBANU AND ALAN D. LOGAN
Lemma 2.6.
There exists an algorithm with input a finite set of marked morphisms S from Σ ∗ to ∆ ∗ and output a marked morphism ψ S : Σ ∗ S → Σ ∗ such that Image( ψ S ) = Eq( S ) .Proof. We use induction on | S | . By Theorem 2.4, the result holds if | S | = 2. Suppose the resultholds for all sets of n marked morphisms, n ≥
2, and let S be a set of n + 1 marked morphisms.Take elements g, h ∈ S , and write S g = S \ { g } . By hypothesis, we can algorithmically obtainmarked morphisms ψ S g : Σ ∗ S g → Σ ∗ and ψ g,h : Σ ∗ g,h → Σ ∗ such that Image( ψ S g ) = Eq( S g ) andImage( ψ g,h ) = Eq( g, h ).By Lemma 2.2, there exists a (computable) marked morphism ψ S : Σ ∗ S → Σ ∗ such thatImage( ψ S ) = Image( ψ S g ) ∩ Image( ψ g,h ) (the map ψ S corresponds to the map k in Lemma2.2, and Σ S to Σ ′ ). Then, as required: Image( ψ S ) = Image( ψ S g ) ∩ Image( ψ g,h ) = Eq( S g ) ∩ Eq( g, h ) = Eq( S ) . (cid:3) We now prove Theorem A, which states that the equaliser is the image of a marked map.
Proof of Theorem A.
By applying Lemma 2.5 to Theorem 2.4, there exists an alphabet Σ S and a marked morphism ψ S : Σ ∗ S → Σ ∗ such that Image( ψ S ) = Eq( S ), while by Lemma 2.6 if S is finite then such a marked morphism can be algorithmically found. (cid:3) We now prove Corollary F, which says that Eq( S ) is free of rank ≤ | Σ | . Proof of Corollary F.
Consider the marked morphism ψ S : Σ ∗ S → Σ ∗ given by Theorem A. ByLemma 2.1, ψ S is injective so Image( ψ S ) is free. As ψ S is marked the map Σ S → Σ takingeach a ∈ Σ S to the initial letter of ψ S ( a ) is an injection, so | Σ S | ≤ | Σ | as required. (cid:3) We now prove a strong form of the AEP FM for marked morphisms. Corollary 2.7.
There exists an algorithm with input a finite set S of marked morphisms from Σ ∗ to ∆ ∗ and output a basis for Eq( S ) .Proof. To algorithmically obtain a basis for Eq ( S ), first use the algorithm of Theorem A toobtain the marked morphism ψ S : Σ ∗ S → Σ ∗ such that Image( ψ S ) = Eq( S ). Then, recallingthat we store ψ S as a list ( ψ S ( a )) a ∈ Σ , the required basis is the set of elements in this list, sothe set { ψ S ( a ) } a ∈ Σ . (cid:3) Corollary H, the AEP FM for marked morphisms, follows from Corollary 2.7 by taking | S | =2, while Corollary B, the simultaneous PCP FM , also follows as Eq ( S ) is trivial if and only ifits basis is empty. 3. Immersions of free groups
We denote the free group with finite generating set Σ by F (Σ), and view it as the set ofall freely reduced words over Σ ± = Σ ∪ Σ − , that is, words not containing xx − as subwords, x ∈ Σ ± , together with the operations of concatenation and free reduction (that is, the removalof any xx − that might occur when concatenating two words).We now begin our study of immersions of free groups, as defined in the introduction. Wefirst state the characterising lemma, then explain the terms involved before giving the proof. Lemma 3.1.
Let g : F (Σ) → F (∆) be a free group morphism. The following are equivalent.(1) The map g is an immersion of free groups.(2) Every word in the language L (Γ g , v g ) is freely reduced.(3) For all x, y ∈ Σ ∪ Σ − such that xy = 1 , the length identity | g ( xy ) | = | g ( x ) | + | g ( y ) | holds. CP AND EQUALISERS FOR CERTAIN MORPHISMS 7
Characterisation (3) is the established definition of Kapovich [Kap00]. Characterisation (2)is the one we shall work with in this article. It uses “Stallings graphs”, which are essentiallyfinite state automata that recognise the elements of finitely generated subgroups of free groups.We define these now, and refer the reader to [KM02] for background on Stallings graphs.The (unfolded) Stallings graph Γ g of the free group morphism g is the directed graph formedby taking a bouquet with | Σ | petals attached at a central vertex we call v g , where each petalconsists of a path labeled by g ( x ) ∈ (∆ ∪ ∆ − ) ∗ ; the elements of ∆ − occur as edges traversedbackwards and we denote by e − the edge e in opposite direction, and by E Γ ± g the sets ofedges in both directions. A path q = ( e , . . . , e n ), e i ∈ E Γ ± g edges, is reduced if it has nobacktracking, that is, e − i = e i +1 for all 1 ≤ i < n . We denote by ι ( p ) the initial vertex ofa path p and τ ( p ) for the terminal vertex, and call a reduced path p with ι ( p ) = u = τ ( p ) a closed reduced circuit .We shall view Γ g as a finite state automaton (Γ g , v g ) with start and accept states both equalto v g . Then the extended language accepted by (Γ g , v g ) is the set of words labelling reducedclosed circuits at v g in Γ g : L (Γ g , v g ) = { label ( p ) | p is a reduced path with ι ( p ) = u = τ ( p ) } . Immersions are precisely those maps g such that every element of L (Γ g , v g ) is freely reduced;this corresponds to the automaton (Γ g , v g ) and the “reversed” automaton (Γ g , v g ) − , whereedge directions are reversed, both being deterministic (map g in Figure 1 is not an immersion;although the automaton (Γ g , v g ) is deterministic, (Γ g , v g ) − is not). For such maps, L (Γ g , v g )is precisely the image of the map g [KM02, Proposition 3.8]. Proof of Lemma 3.1. (1) ⇔ (2). Every element of L (Γ g , v g ) is freely reduced if and only if thegraph Γ g , with base vertex v g , is such that for all e , e ∈ ( E Γ g ) ± such that both edes startat v g or both edges end at v g , then e and e have different labels (so γ g ( e ) = γ g ( e )). Thiscondition on labels is equivalent to g (Σ ∪ Σ − ) being marked, as required.(1) ⇔ (3). Condition (3) is equivalent to the condition that for all x, y ∈ Σ ∪ Σ − such that xy = 1, free cancellation does not happen between g ( x ) and g ( y ), which in turn is equivalentto the condition that for all such x, y the elements g ( x − ) and g ( y ) start with different lettersof ∆ ∪ ∆ − . This is equivalent to g (Σ ∪ Σ − ) being marked, as required. (cid:3) Example 3.2.
Let g : F ( a, b ) → F ( x, y ) be the map defined by g ( a ) = x − y and g ( b ) = y x .Then the graph Γ g , where the double arrow represents x and the single arrow y , is depictedin Figure 1. The map g is not an immersion since there are two edges labeled x entering v g (violating Characterisation (2)). Similarly, g ( a ) and g ( b − ) both start with x − (violatingCharacterisation (1)) and | g ( ba ) | = 4 < | g ( a ) | + | g ( b ) | (violating Characterisation (3)). Γ g = v g g ( a ) g ( b ) Figure 1.
The graph Γ g for the map g : F ( a, b ) → F ( x, y ) defined by g ( a ) = x − y , g ( b ) = y x − .Using Characterisation (2), we see that immersions are injective [KM02, Proposition 3.8]: LAURA CIOBANU AND ALAN D. LOGAN
Lemma 3.3. If g : F (Σ) → F (∆) is an immersion then it is injective. The reduction of an instance in free groups
By an immersed instance of the PCP FG we mean an instance I = (Σ , ∆ , g, h ) where both g and h are immersions. In this section we define the “reduction” of an immersed instance ofthe PCP FG , which is similar to the reduction in the free monoid case.Let Γ be a directed, labeled graph and u ∈ V Γ a vertex of Γ. The core graph of Γ at u , written Core u (Γ), is the maximal subgraph of Γ containing u but no vertices of degree1, except possibly u itself. Note that L (Core u (Γ) , u ) = L (Γ , u ). For Γ , Γ directed, labeledgraphs, the product graph of Γ and Γ , denoted Γ ⊗ Γ , is the subgraph of Γ × Γ with vertexset V Γ × V Γ and edge set { ( e , e ) | e i ∈ E Γ ± i , label ( e ) = label ( e ) } . One may think of thestandard construction of an automaton recognising the intersection of two regular languages,each given by a finite state automaton Γ i with start state s i , where the core of Γ ⊗ Γ at( s , s ) is the automaton recognising this intersection. Core graph of a pair of morphisms.
Let g : F (Σ ) → F (∆), h : F (Σ ) → F (∆) bemorphisms. The core graph of the pair ( g, h ), denoted Core( g, h ), is the core graph of Γ g ⊗ Γ h at the vertex v g,h = ( v g , v h ), so Core( g, h ) = Core v g,h (Γ g ⊗ Γ h ). We shall refer to v g,h as the central vertex of Core( g, h ). Note that Core( g, h ) represents the intersection of the two images[KM02, Lemma 9.3], in the sense that L (Core( g, h ) , v g,h ) = Image( g ) ∩ Image( h ) . Write δ g : Core( g, h ) → Γ g and δ h : Core( g, h ) → Γ h for the restriction of Core( g, h ) to the g and h components, respectively, so δ g ( e , e ) = e , etc.Now, let g, h be immersions. The graph Core( g, h ) is a bouquet and every element of L (Core( g, h ) , v g,h ) is freely reduced [KM02, Lemma 9.2]. We therefore have free group mor-phisms g ′ : L (Core( g, h ) , v g,h ) → L (Γ g , v g ) and h ′ : L (Core( g, h ) , v g,h ) → L (Γ h , v h ) inducedby the maps δ g , δ h , where L (Γ g , v g ) = F (Σ ) and L (Γ h , v h ) = F (Σ ). These maps are com-putable [KM02, Corollary 9.5]. Let Σ ′ be the alphabet whose elements consist of the petals ofCore( g, h ). Then Σ ′ generates the free group L (Core( g, h ) , v g,h ), so F (Σ ′ ) = L (Core( g, h ) , v g,h ),and we see that both g ′ and h ′ are immersions with g ′ : F (Σ ′ ) → F (Σ ), h ′ : F (Σ ′ ) → F (Σ ).The map gg ′ = hh ′ , which we shall call k (so k : F (Σ ′ ) → F (∆)) is the composition ofimmersions and hence is itself an immersion. We therefore have the following. Lemma 4.1. If g : F (Σ ) → F (∆) and h : F (Σ ) → F (∆) are immersions then the cor-responding maps g ′ : F (Σ ′ ) → F (Σ ) , h ′ : F (Σ ′ ) → F (Σ ) and k : F (Σ) → F (∆) , where k = gg ′ = hh ′ , are immersions and are computable. Reduction.
The reduction of an immersed instance I = (Σ , ∆ , g, h ) of the PCP FG is theinstance I ′ = (Σ ′ , ∆ , g ′ , h ′ ) where Σ ′ is defined as above, and where g ′ and h ′ are as above,but with codomain ∆ (which we may do as Σ ⊆ ∆). We additionally assume that Σ ′ ⊆ Σ; wecan do this as | Σ ′ | ≤ | Σ | by Lemma 4.1. As I is immersed, it follows from Lemma 4.1 that I ′ is also immersed. In the next section we show that the name “reduction” makes sense, as itreduces the “prefix complexity” of instances. Example 4.2.
Consider the maps g, h : F ( a, b, c ) → F ( x, y, z ) given by g ( x ) = aba , g ( b ) = y − , g ( c ) = zxz and h ( a ) = x, h ( b ) = yx y, h ( c ) = z . CP AND EQUALISERS FOR CERTAIN MORPHISMS 9
Then the graph
Core( g, h ) is a bouquet with two petals labelled xyx y and zxz , and Image( g ) ∩ Image( h ) = h xyx y, zxz i . Moreover, g ( ab − ) = h ( ab ) = xyx y and g ( z ) = h ( zxz ) = zxz. Then we can take Σ ′ = { a ′ , b ′ } , and the maps given by g ′ ( a ′ ) = ab − , g ( b ′ ) = c and h ′ ( a ′ ) = ab, h ′ ( b ′ ) = cac are the reduction of ( g, h ) . We now prove that reduction preserves equalisers. Two instances I and I of the PCP FG are strongly equivalent if the equalisers are isomorphic, which we write as Eq( I ) ∼ = Eq( I ). Lemma 4.3.
Let I ′ = (Σ ′ , ∆ ′ , g ′ , h ′ ) be the reduction of I = (Σ , ∆ , g, h ) where g and h areimmersions. Then I and I ′ are strongly equivalent, and g ′ (Eq( I ′ )) = Eq( I ) = h ′ (Eq( I ′ )) .Proof. It is sufficient to prove that g ′ | Eq( I ′ ) is injective and g ′ (Eq( I ′ )) = Eq( I ); that h ′ (Eq( I ′ )) =Eq( I ) follows as g ′ | Eq( I ′ ) = h ′ | Eq( I ′ ) .As g ′ is an immersion it is injective, by Lemma 3.3. Therefore, g ′ | Eq( I ′ ) is injective. Tosee that Image( g ′ | Eq( I ′ ) ) ≤ Eq( I ), suppose x ′ ∈ Eq( I ′ ). Writing x = g ′ ( x ′ ) = h ′ ( x ′ ), we have g ( x ) = gg ′ ( x ′ ) = hh ′ ( x ′ ) = h ( x ) and so x = g ′ ( x ′ ) ∈ Eq( I ), as required.To see that Image( g ′ | Eq( I ′ ) ) ≥ Eq( I ), suppose x ∈ Eq( I ). Then there exists a path p x ∈ Core( g, h ), ι ( p x ) = v g,h = τ ( p x ), such that γ g δ g ( p x ) = g ( x ) = h ( x ) = γ h δ h ( p x ) [KM02,Proposition 9.4], where γ g : Γ g → Γ ∆ is the canonical morphism of directed, labeled graphsfrom Γ g to the bouquet Γ ∆ with ∆ petals. Hence, writing x ′ for the element of F (Σ ′ ) corre-sponding to p x ∈ L (Core( g, h ) , v g,h ), we have that gg ′ ( x ′ ) = g ( x ) = h ( x ) = hh ′ ( x ). As h and g are injective, by Lemma 3.3, we have that g ′ ( x ′ ) = x = h ′ ( x ′ ) as required. (cid:3) Prefix complexity of immersions in free groups
In this section we associate to an instance I of the PCP FG a certain complexity, called the“prefix complexity”. We prove that the process of reduction does not increase this complexity,and that for all n ∈ N there are only finitely many instances with complexity ≤ n .Let I = (Σ , ∆ , g, h ) be an immersive instance of the PCP FG . We define, analogously to[HHdW01, Section 4] (see also [EKR82]), the prefix complexity σ ( I ) as: σ ( I ) = |∪ a ∈ Σ ± { x ∈ F (∆) | x is a proper prefix of g ( a ) }| + |∪ a ∈ Σ ± { x ∈ F (∆) | x is a proper prefix of h ( a ) }| . In the maps in Example 4.2, σ ( I ) = 10 + 6 = 16, and σ ( I ′ ) = 2 + 4 = 6 . The process of reduction does not increase the prefix complexity, and we prove this by usingthe fact that, for any a ∈ Σ ± , the proper prefixes of g ( a ) and h ( a ) are in bijection with theproper initial subpaths of the petals of Γ g and Γ h , respectively. Lemma 5.1.
Let I = (Σ , ∆ , g, h ) be an instance of the PCP FG with g and h immersions, andlet I ′ be the reduction of I . Then σ ( I ′ ) ≤ σ ( I ) .Proof. We write V g Core( g, h ) = { ( v g , v ) ∈ V Core( g, h ) | v ∈ Γ h } = δ − g ( v g ) for the set ofvertices in the Core( g, h ) whose first component is the central vertex v g of Γ g , and similarlyfor V h Core( g, h ). Note that V g Core( g, h ) ∩ V h Core( g, h ) = { v g,h } .By construction, each petal of Γ g and Γ h corresponds to a letter a ∈ Σ ± , and we shalldenote the petal also by a . Write P Γ for the set of reduced paths in a graph Γ. Similarly to a ∈ P Γ g and a ∈ P Γ h , we map write a ∈ P Core( g, h ) for the petal in Core( g, h ) corresponding to a ∈ (Σ ′ ) ± . From now on, all paths are assumed to be reduced. Define G = ∪ a ∈ Σ ± { p ∈ Γ g | p is a proper initial subpath of petal a ∈ Γ g } ,G ′ = ∪ a ∈ (Σ ′ ) ± { p ∈ Core( g, h ) | p is a proper initial subpathof a ∈ Core( g, h ) s.t. its end vertex τ ( p ) ∈ V g Core( g, h ) } , and define H and H ′ analogously. Hence, σ ( I ) = | G | + | H | and analogously σ ( I ′ ) = | G ′ | + | H ′ | .For a ∈ (Σ ′ ) ± let q ∈ G ′ be a subpath of a ∈ P Core( g, h ). Denote by r q the shortestsubpath of a intersecting q at only one point, their common end vertex (that is, τ ( q ) = τ ( r q ) = q ∩ r q ), such that ι ( r q ) ∈ V h Core( g, h ); the paths q and r q can be seen as “facing”one another on a . As V g Core( g, h ) ∩ V h Core( g, h ) = { v g,h } , and as q is a proper initialsubpath of a , the projection δ h ( r q ) is a non-trivial path in Γ h . Note also that ι ( δ h ( r q )) = v h ,as ι ( r q ) ∈ V h Core( g, h ), hence there exists some b ∈ Σ ± such that δ h ( r q ) is a proper initialsubpath of the petal b ∈ Γ h . Therefore, δ h ( r q ) ∈ H . Let ξ H : G ′ → H be the map given by ξ H ( q ) = δ h ( r q ).We now prove that ξ H is injective. Suppose p, q ∈ G ′ are such that ξ H ( p ) = ξ H ( q ), andlet r p and r q be the paths obtained from p and q , respectively, such that ξ H ( p ) = δ h ( r p ) and ξ H ( q ) = δ h ( r q ). Write e p for the terminal edge of r p , and e q for the terminal edge of r q ,and note that these two edges have the same label and direction as δ h ( e p ) = δ h ( e q ). Now, δ g ( τ ( e p )) = v g = δ g ( τ ( e q )) as τ ( r p ) , τ ( r q ) ∈ V g Core( g, h ), and as e p and e q have the same labeland direction we have that δ g ( e p ) = δ g ( e q ). Therefore, both δ g and δ h agree on e p and e q , andso as Core( g, h ) is a subgraph of Γ g × Γ h we have that e p = e q . As Core( g, h ) is a bouquet,there exists a unique shortest reduced path s such that ι ( s ) = v g,h and τ ( s ) = s ∩ e p = τ ( e p ).Hence, p = s = q as required.Thus ξ H is injective, and so | G ′ | ≤ | H | . The same will hold for an analogously definedfunction ξ H from H ′ to G , so | H ′ | ≤ | G | . Therefore, σ ( I ′ ) = | G ′ | + | H ′ | ≤ | G | + | H | = σ ( I ). (cid:3) For a fixed number n ≥ ≤ n proper prefixes, and so the following is clear: Lemma 5.2.
There exist only finitely many distinct instances I = (Σ , ∆ , g, h ) of the PCP FG that satisfy σ ( I ) ≤ n . As the reduction I ′ of an instance I gives σ ( I ′ ) ≤ σ ( I ), and as | Σ ′ | ⊆ | Σ | , this means thatthe process of iteratively computing reductions will eventually cycle.6. Solving the Algorithmic Equaliser Problem in free groups (
AEP FG ) The algorithm for solving the AEP FG for immersions is analogous to the algorithm formarked free monoid morphisms in Section 2. Our algorithm starts by making reductions I , I , I , . . . , beginning with I = I , the input instance. By Lemma 5.2, we will obtain aninstance I j = (Σ j , ∆ , g j , h j ) such that one of the following will occur:(1) | Σ j | = 1.(2) σ ( I j ) = 0.(3) there exists some i < j with I i = I j (sequence starts cycling).Keeping in mind the fact that reductions preserve equalisers (Lemma 4.3), we obtain in eachcase a subset Σ g,h (possibly empty) which forms a basis for Eq( I j ): For Case (1), writingΣ j = { a } , the result holds as if g ( a i ) = h ( a i ) then g ( a ) i = h ( a ) i and so g ( a ) = h ( a ) as roots areunique in a free group. For Case (2), σ ( I j ) = 0 is equivalent to | g ( a ) | = | h ( a ) | = 1 for all a ∈ Σ.Suppose there exists some non-trivial reduced word x = a ǫ i · · · a ǫ n i n such that g ( x ) = h ( x ). CP AND EQUALISERS FOR CERTAIN MORPHISMS 11
Then as g and h are injective, the words g ( a i ) ǫ · · · g ( a i n ) ǫ n and h ( a i ) ǫ · · · h ( a i n ) ǫ n are freelyreduced and hence are the same word, and so g ( a i j ) = h ( a i j ). The result then follows for Case(2). Case (3) has a more involved proof. Lemma 6.1.
Let I = (Σ , ∆ , g, h ) be an immersive instance of the PCP FG that starts a cycle(i.e. starting the reduction process with I eventually gives I again). If Eq( I ) is non-trivialthen a subset of Σ forms a basis for Eq( I ) .Proof. There is a sequence of reductions beginning and ending at I : I = I → I → · · · → I r − → I r = I where I i = (Σ i , ∆ , g i , h i ). By Lemma 4.3, Eq( I ) = g g . . . g r (Eq( I r )) = Eq( I r ) and so g r = g g . . . g r restricts to an automorphism of Eq( I ), that is, g r | Eq( I ) ∈ Aut(Eq( I )). For h r defined analogously, h r | Eq( I ) ∈ Aut(Eq( I )). Write Eq( I k ) ( n ) for the set of words in Eq( I k ) oflength precisely n , and Eq( I k ) ( ≤ n ) for the set of words in Eq( I k ) of length at most n . Considersome x ∈ Eq( I ) and write x r = g r − ( x ). Then x = g g . . . g r ( x r ) = g r ( x r ) ,x = h h . . . h r ( x r ) = h r ( x r ) . By Lemma 4.1, both g i and h i are immersions for each i , and so by Characterisation (3) ofLemma 3.1 we see that | g i ( w ) | ≥ | w | for all w ∈ F (Σ i ). Hence, | x | ≥ | x r | . Therefore, for all m ≥ g r induces a map g r ( m ) : Eq( I r ) ( m ) → Eq( I r ) ( ≤ m ) . Clearly g r (1) is a bijection,and so we inductively see that g r ( m ) has image Eq( I r ) ( m ) . Therefore, the automorphism g r ofEq( I ) is length-preserving ( | g r ( w ) | = | w | for all w ∈ Eq( I )), and so maps the letters occurringin x r to letters. Hence, g (= g r ) and h (= h r ) map the letters occuring in x r to letters, andit follows that every letter occuring in x r is a solution to I . Hence, a subset Σ g,h of Σ r formsa basis for Eq( I r ). (cid:3) We now prove the central theorem of this article, which gives an algorithm to describe Eq( I )as the image of an immersion. Note that not every subgroup of a free group is the image ofan immersion: for example, if | Σ | = n , then no subgroup of F (Σ) of rank > n is the image ofan immersion. We store a morphism f : F (Σ) → F (∆) as a list ( f ( a )) a ∈ Σ . Theorem 6.2.
There exist an algorithm with input an immersive instance I = (Σ , ∆ , g, h ) ofthe PCP FG and output an immersion ψ g,h : F (Σ g,h ) → F (Σ) such that Image( ψ g,h ) = Eq( I ) .Proof. Start by making reductions I = I → I → · · · . By Lemma 5.2 we will obtain aninstance I j = (Σ j , ∆ , g j , h j ) satisfying one of the Cases (1)–(3) above, and in each case asubset Σ g,h of Σ j forms a basis for Eq( I j ). Since Σ j is computable, this basis is as well.In order to prove the theorem, it is sufficient to prove that there is a computable immersion ψ g,h : F (Σ g,h ) → F (Σ). Consider the map ˜ g = g g · · · g j : F (Σ j ) → F (Σ) (and the analogous˜ h ). Now, each g i is an immersion, so ˜ g is the composition of immersions and hence is animmersion. Define ψ g,h = ˜ g | F (Σ g,h ) . This map is computable from ˜ g , and as Σ g,h ⊆ Σ j , themap ψ g,h is an immersion. As Image( ψ g,h ) = g g . . . g j (Eq( I j )) = Eq( I ), by Lemma 4.3 andthe above, the result follows. (cid:3) We now prove Corollary G, which solves the AEP FG for immersions of free groups. Proof of Corollary G.
To algorithmically obtain a basis for Eq( I ), first obtain the immersion ψ g,h : F (Σ g,h ) → F (Σ) given by Theorem 6.2. Then, recalling that we store ψ g,h as a list( ψ g,h ( a )) a ∈ Σ , the required basis is the set of elements in this list, so the set { ψ g,h ( a ) } a ∈ Σ . (cid:3) Sets of immersions
We now prove Theorem C and its corollaries. We first give a general result, from whichthe non-algorithmic part of Theorem C follows quickly. An immersed subgroup H of a freegroup F (Σ) is a subgroup which is the image of an immersion. The proof of Lemma 7.1 isfundamentally identical to the proof of Lemma 2.5, via Characterisation 1 of Lemma 3.1. Lemma 7.1. If { H j } j ∈ J is a set of immersed subgroups of F (Σ) then the intersection T j ∈ J H j is immersed.Proof. Firstly, suppose x, y ∈ H j for some j ∈ J , and let z be their maximal common prefix.Then z decomposes uniquely as z z · · · z n z ′ n +1 such that each z k ∈ H j . As H j is immersed,and as z is a maximal common prefix of x and y , we have that z ∈ H j .Now, suppose x, y ∈ T j ∈ J H j , and suppose they both begin with some letter a ∈ Σ ∪ Σ − .By the above, their maximal common prefix z a is contained in each H j and so is contained in T j ∈ J H j . Therefore, z a is a prefix of every element of T j ∈ J H j beginning with an a . It followsthat T j ∈ J H j is immersed, as required. (cid:3) The following lemma corresponds to the algorithmic part of Theorem C. Similar to theabove, the proof of the lemma is fundamentally identical to the proof of Lemma 2.6.
Lemma 7.2.
There exists an algorithm with input a finite set of immersions S from F (Σ) to F (∆) and output an immersion ψ S : F (Σ S ) → F (Σ) such that Image( ψ S ) = Eq( S ) .Proof. We proceed by inducting on | S | . By Theorem 6.2, the result holds if | S | = 2. Supposethe result holds for all sets of n immersions, n ≥
2, and let S be a set of n + 1 immersions.Take elements g, h ∈ S , and write S g = S \ { g } . By hypothesis, we can algorithmically obtainimmersions ψ S g : F (Σ S g ) → F (Σ) and ψ g,h : F (Σ g,h ) → F (Σ) such that Image( ψ S g ) = Eq( S g )and Image( ψ g,h ) = Eq( g, h ).By Lemma 4.1, there exists a (computable) immersion ψ S : F (Σ S ) → F (Σ) such thatImage( ψ S ) = Image( ψ S g ) ∩ Image( ψ g,h ) (the map ψ S corresponds to the map k in the lemma,and Σ S to Σ ′ ). Then we have the required equality:Image( ψ S ) = Image( ψ S g ) ∩ Image( ψ g,h )= Eq( S g ) ∩ Eq( g, h )= Eq( S ) . (cid:3) We now prove Theorem C, which states that the equaliser is the image of a computableimmersion.
Proof of Theorem C.
By Lemma 7.1, there exists an alphabet Σ S and an immersion ψ S : F (Σ S ) → F (Σ) such that Image( ψ S ) = Eq( S ), while by Lemma 7.2 if S is finite then such animmersion can be algorithmically found. (cid:3) We now prove Corollary D, which solves the simultaneous PCP FG for immersions. Proof of Corollary D.
First find a basis for Eq( I ): obtain the immersion ψ g,h : F (Σ g,h ) → F (Σ) given by Theorem C. Then, recalling that we store ψ g,h as a list ( ψ g,h ( a )) a ∈ Σ , therequired basis is the set of elements in this list, so the set { ψ g,h ( a ) } a ∈ Σ . Then Eq( S ) is trivialif and only if this basis is empty. (cid:3) CP AND EQUALISERS FOR CERTAIN MORPHISMS 13
Finally, we prove Corollary E, which says that Eq( S ) is of rank ≤ | Σ | . Proof of Corollary E.
Consider the immersion ψ g,h : F (Σ g,h ) → F (Σ) given by Theorem C.As Image( ψ g,h ) = Eq( S ) we have that rk(Eq( S )) ≤ | Σ g,h | , while as ψ g,h is an immersion wehave that | Σ g,h | ≤ | Σ | , and the result follows. (cid:3) Algorithm to compute the equaliser
Theorems A and C produce the equaliser of a set S of morphisms as the image of a com-putable map ψ S . For S = { g, h } , the structure of the algorithm that gives ψ S (as a list ofelements representing the images of the generators) is given below. The values for M in step3 correspond to the number of instances of complexity ≤ σ ( I ), as explained in Section 9.(1) Input I = (Σ , ∆ , g, h ).(2) Set c =; 0, i := 0, I := I (3) Set M := ( | ∆ | + 1) | Σ | ( σ ( I )+1) (monoids) or M := (2 | ∆ | ) | Σ | ( σ ( I )+1) (groups)(4) i := i + 1(5) Reduce instance I i − to I i (as in Sections 2 and 4); store I i in memory(6) If I i has source alphabet of size 1 or σ = 0 then:(a) Compute a basis B for Eq( I i )(b) Print composition( B , i ) (see below) and terminate.(7) If I i is simpler than I i − (smaller source alphabet or σ ) then set c = 0 and goto (4)(8) If c > M then there exists a cycle which starts with I i .(a) Compute a basis B for Eq( I i )(b) Print composition( B , i ) and terminate.Procedure composition( B , i ) computes the composition of a map, stored as a list B , withthe maps obtained in the reduction process, indexed from i downwards. composition( B , i ) (1) Set B := g i ( B ), where g i is loaded from memory(2) i := i − i ≥
0, goto (1); else, output B .9. Complexity analysis
The size of an instance I = (Σ , ∆ , S ), S a set of morphisms, is | Σ | + | ∆ | + P g ∈ S P a ∈ Σ ± | g ( a ) | .The algorithm underlying Theorem A can be run with O (2 n ) space, where n is the size ofthe input instance I , which gives a time bound of O (2 n ). The space grows exponentially,unlike in [HHdW01], because the algorithm computes instances that must each be stored (asthe immersion ψ g,h is their composition; this corresponds to the function composition( B , i ) , above). To obtain this space complexity, first suppose | S | = 2 (so consider the function pairs( g , h ) , above). There are at most ( | ∆ | + 1) | Σ | ( σ ( I )+1) instances I j with σ ( I j ) ≤ σ ( I )[HHdW01, Proof of Lemma 3], which is O (2 n ). Every other procedure requires asymptoticallyless space, and hence if | S | = 2 we require O (2 n ) space. For S = { g , . . . , g k } , note that we onlyneed to compute the immersions corresponding to Eq( g i , g i +1 ) for 1 ≤ i < k (as these intersectto give Eq( S )), and these can all be stored in ( k − × O (2 n ) = O (2 n ) space. Intersectioncorresponds to reduction, and reduction can be done in PSPACE [HHdW01, Section 6]. Hence,the algorithm can be run in O (2 n ) space.Similarly, the algorithm underlying Theorem C runs in O (2 n ) space, where n is the inputsize. The main difference to the above is that there are O (2 n ) instances I j with σ ( I j ) ≤ σ ( I ). To see this, write m := σ ( I ) and d := | ∆ | . If I j = (Σ j , ∆ j , g j , h j ) is such that σ ( I j ) ≤ m then | g ( a ) | ≤ m + 1 for all a ∈ Σ ± j , as g ( a ) has at most m proper prefixes, and similarly | h ( a ) | ≤ m + 1. There are 2 d (2 d − m freely reduced words of length m + 1 in F (Σ j ), andso (by using the empty word) we see that there are at most (2 d ) m +1 freely reduced wordsof length at most m + 1. As each list of 2 | Σ j | words defines an instance, there are at most(2 d ) | Σ j | ( m +1) ≤ (2 d ) | Σ | ( m +1) instances that satisfy σ ( I ) ≤ m . This is O (2 n ) as required.10. The density of marked morphisms and immersions
Here we show that immersions and marked morphisms are not a negligible (i.e. density zero)subset of the entire set of free group and free monoid morphisms, respectively, but representa strictly positive proportion of those.Suppose Σ = { a , . . . , a k } , and k = | Σ | ≥ | ∆ | = m . A morphism in a free monoid or freegroup, φ : Σ ∗ → ∆ ∗ or φ : F (Σ) → F (∆), is uniquely determined by ( φ ( a ) , . . . , φ ( a k )).We start with the monoid case. There are m n words of length n in ∆ ∗ , and P ≤ i ≤ n m i ∼ cm n words of length ≤ n , where c = mm − and we write a n ∼ b n for lim n →∞ a n b n = 1. If α n is thenumber of morphisms from Σ ∗ to ∆ ∗ with images of length at most n , then α n ∼ ( cm n ) k . Nowlet β n be the number of marked morphisms from Σ ∗ to ∆ ∗ with images of length at most n .For a marked morphism φ , each word in the list ( φ ( a ) , . . . , φ ( a k )) must start with a differentletter, followed by any word of length ≤ n −
1. Since there are (cid:0) mk (cid:1) k ! options for the firstletters, β n ∼ (cid:0) mk (cid:1) k !( cm n − ) k and we get: Proposition 10.1. If α n and β n are the numbers of morphisms and marked morphisms,respectively, from Σ ∗ to ∆ ∗ , with images of length at most n , then the density of the markedmorphisms among all morphisms is a positive constant: lim n →∞ β n α n = lim n →∞ (cid:0) mk (cid:1) k !( cm n − ) k ( cm n ) k = m ! m k ( m − k )! . In the free group case the counting is similar, but there are more restrictions on the imagesof an immersion: first, all images need to be reduced words, and second, not just their firstletters are constrained, but also their last letters. For some φ , let the set of first letters of( φ ( a ) , . . . , φ ( a k )) be F ⊂ ∆ ± , and the set of inverses of the last letters be L ⊂ ∆ ± . Then φ : F (Σ) F (∆) is an immersion if all letters in F are distinct, all the letters in L aredistinct, which implies | F | = | L | = k , and furthermore F ∩ L = ∅ . An image φ ( a i ) of length n has the form φ ( a i ) = αx x . . . x n − β , where α ∈ F , β − ∈ L , x i ∈ ∆ ± , and φ ( a i ) is reduced,so x = α − and x n − = β − . Counting such words is more delicate than in the monoid case,but the asymptotics are similar, due to the following result ([CR06, Proposition 1]). Proposition 10.2.
Let A and B be subsets of ∆ ± . The number of elements of length n in F (∆) that do not start with a letter in A and do not end with a letter in B is equal to f A,B ( n ) = (2 m − | A | )(2 m − | B | )(2 m − n − + xm + ( − n ( | A || B | − ym )2 m , where x = | A ∩ B | − | A − ∩ B | , y = | A ∩ B | + | A − ∩ B | , and m = | ∆ | . Let A = { α − } and B = { β − } ; then since the number of possible φ ( a i ) of length ≤ n is equal to the number of reduced words of length ≤ n − α − and ending with a letter different from β − , this number is P ≤ j ≤ n − f A,B ( j ). Since | A | = | B | = 1, f A,B ( j ) is asymptotically (2 m − j , and the number of possible φ ( a i ) is CP AND EQUALISERS FOR CERTAIN MORPHISMS 15 ∼ c (2 m − n − , where c is a constant depending on m . Thus for fixed sets F and L thenumber of immersions φ with images in the ball of radius n is ∼ ( c (2 m − n − ) k . Since thereare only finitely many choices for sets F and L of first and last letters, respectively, and thenumber of k -tuples of elements in F (∆) of length ≤ n is ∼ ( c (2 m − n ) k for some constant c , the number of immersions over the total number of maps F (Σ) F (∆) is ∼ ( c (2 m − n − ) k ( c (2 m − n ) k ;so as n
7→ ∞ , this ratio is a positive constant depending on k and m . References [Ber99] George M. Bergman,
Supports of derivations, free factorizations, and ranks of fixed subgroups infree groups , Trans. Amer. Math. Soc. (1999), no. 4, 1531–1550. MR 1458296[BH77] Meera Blattner and Tom Head,
Automata that recognize intersections of free submonoids , Infor-mation and Control (1977), no. 3, 173–176. MR 460510[BH92] Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups , Ann. ofMath. (2) (1992), no. 1, 1–51. MR 1147956[BMS02] Gilbert Baumslag, Alexei G. Myasnikov, and Vladimir Shpilrain,
Open problems in combinatorialgroup theory. Second edition , Combinatorial and geometric group theory (New York, 2000/Hobo-ken, NJ, 2001), Contemp. Math., vol. 296, Amer. Math. Soc., Providence, RI, 2002, online version: , pp. 1–38. MR 1921705[CMV08] Laura Ciobanu, Armando Martino, and Enric Ventura,
The genericHanna Neumann Conjecture and Post Correspondence Problem , .[CR06] Laura Ciobanu and Sasa Radomirovic, Restricted walks in regular trees , Electronic J. of Combina-torics (2006), no. R93.[DKLM19] Volker Diekert, Olga Kharlampovich, Markus Lohrey, and Alexei Myasnikov, Algorithmic problems in group theory , Dagstuhl seminar report 19131 (2019), http://drops.dagstuhl.de/opus/volltexte/2019/11293/pdf/dagrep_v009_i003_p083_19131.pdf .[DV96] Warren Dicks and Enric Ventura,
The group fixed by a family of injective endomorphisms of afree group , Contemporary Mathematics, vol. 195, American Mathematical Society, Providence,RI, 1996. MR 1385923[EKR82] A. Ehrenfeucht, J. Karhum¨aki, and G. Rozenberg,
The (generalized) Post correspondence problemwith lists consisting of two words is decidable , Theoret. Comput. Sci. (1982), no. 2, 119–144.MR 677104[ER78] A. Ehrenfeucht and G. Rozenberg, Elementary homomorphisms and a solution of the
D0L sequenceequivalence problem , Theoret. Comput. Sci. (1978), no. 2, 169–183. MR 509015[HHdW01] Vesa Halava, Mika Hirvensalo, and Ronald de Wolf, Marked PCP is decidable , Theoret. Comput.Sci. (2001), no. 1-2, 193–204. MR 1819073[HK97] Tero Harju and Juhani Karhum¨aki,
Morphisms , Handbook of formal languages, Vol. 1, Springer,Berlin, 1997, pp. 439–510. MR 1469999[Hol03] ˇStˇep´an Holub,
Binary equality sets are generated by two words , J. Algebra (2003), no. 1, 1–42.MR 1953706[IT89] W. Imrich and E. C. Turner,
Endomorphisms of free groups and their fixed points , Math. Proc.Cambridge Philos. Soc. (1989), no. 3, 421–422. MR 985677[Kap00] Ilya Kapovich,
Mapping tori of endomorphisms of free groups , Comm. Algebra (2000), no. 6,2895–2917. MR 1757436[Kar84] Juhani Karhum¨aki, A note on intersections of free submonoids of a free monoid , Semigroup Forum (1984), no. 1-2, 183–205. MR 742132[KM02] Ilya Kapovich and Alexei Myasnikov, Stallings foldings and subgroups of free groups , J. Algebra (2002), no. 2, 608–668. MR 1882114[KS10] Juhani Karhum¨aki and Aleksi Saarela,
Noneffective regularity of equality languages and boundeddelay morphisms , Discrete Math. Theor. Comput. Sci. (2010), no. 4, 9–17. MR 2760332[MNU14] Alexei Myasnikov, Andrey Nikolaev, and Alexander Ushakov, The Post correspondence problemin groups , J. Group Theory (2014), no. 6, 991–1008. MR 3276224 [Nea15] Turlough Neary, Undecidability in binary tag systems and the post correspondence problem forfive pairs of words , 32nd International Symposium on Theoretical Aspects of Computer Science,LIPIcs. Leibniz Int. Proc. Inform., vol. 30, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2015,pp. 649–661. MR 3356447[Pos46] Emil L. Post,
A variant of a recursively unsolvable problem , Bull. Amer. Math. Soc. (1946),264–268. MR 15343[Sta87] John R. Stallings, Graphical theory of automorphisms of free groups , Combinatorial group theoryand topology (Alta, Utah, 1984), Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton,NJ, 1987, pp. 79–105. MR 895610[Til72] Bret Tilson,
The intersection of free submonoids of a free monoid is free , Semigroup Forum (1972), 345–350. MR 311807[Ven02] E. Ventura, Fixed subgroups in free groups: a survey , Combinatorial and geometric group theory(New York, 2000/Hoboken, NJ, 2001), Contemp. Math., vol. 296, Amer. Math. Soc., Providence,RI, 2002, pp. 231–255. MR 1922276
Heriot-Watt University, Edinburgh EH14 4AS, Scotland
E-mail address : [email protected] Heriot-Watt University, Edinburgh EH14 4AS, Scotland
E-mail address ::