aa r X i v : . [ m a t h . G R ] S e p The equation w ( x, y ) = u over free groups: analgebraic approach Nicholas W.M. TouikanDepartment of Mathematics and Statistics, McGill UniversityMontr´eal, Qu´ebec, Canada [email protected]
October 24, 2018
Abstract
Using the theory developed by Olga Kharlampovich, Alexei Miasnikov,and, independently, by Zlil Sela to describe the set of homomorphisms ofa f.g. group G into a free group F , we describe the solutions to equationswith coefficients from F and unknowns x, y of the form w ( x, y ) = u , where u lies in F and w ( x, y ) is a word in { x, y } ± . We also give an example of asingle equation whose solutions cannot be described with only one “level” ofautomorphisms. Solving systems of equations over free groups has been a very important topicin group theory. A major achievement was the algorithm due to Makaninand Razborov [13, 15] which produces a complete description of the solutionset of an arbitrary finite system of equations over a free group. In practice,however, the algorithm is quite complicated and does not readily imply theresults of this paper.Much has already been said about solutions to certain types of systemsof equations. Solutions of systems of equations in one unknown over a freegroup were described in 1960 by Lyndon [11]. In 1971, Hmelevski˘ı gave in[6] an algorithm to decide solvability as well as a description of the solutionsof equations in unknowns x, y with coefficients in a free group F of the form w ( x, y ) = u , and t ( x, F ) = u ( y, F ). In 1972 Wicks [21] also described amethod for find all the the solutions of the equation w ( x, y ) = u . In hispaper, Wicks gives a way to find a finite set of solutions to an equation andshows how to generate all the possible solutions from this finite set usingautomorphisms. It has also been shown by Laura Ciabanu in [4] that thereis a polynomial time algorithm to determine if w ( x, y ) = u has a solution.So far all the approaches have been combinatorial.In this paper we tackle the equation w ( x, y ) = u from a different point ofview. We will use the theory developed by Olga Kharlampovich, Alexei Mias-nikov, and, independently, by Zlil Sela to describe the set of homomorphisms f a f.g. group G into a free group F . We start by considering the fully resid-ually F groups (also called the Limit groups relative to F ) corresponding tothe equation w ( x, y ) = u . These groups were shown by Remeslennikov in[16] to be key in the study of systems of equations. We then systematicallydescribe the possible canonical F -automorphisms of these groups and givethe possible Hom (also called
Makanin-Razborov ) diagrams that arise.In so doing we get an algebraic proof that solutions to equations of theform w ( x, y ) = u , can be parametrized by a finite set of minimal solutions and a group of canonical automorphisms , which gives us a very explicit de-scription of the arising algebraic varieties (see Theorem 2.28). We also ex-hibit an equation E ( F, x, y ) = 1 whose solutions cannot be described thisway (see Theorem 3.1). In particular, we recover some of the aforementionedresults of Hmelevski˘ı and Wicks, but our description of the solutions is by farthe most transparent. In our opinion this paper also serves as an illustrationof some of the very important ideas and techniques that have recently beenapplied fully residually free (or limit) groups. F -groups and Algebraic Geometry A complete account of the material in this section can be found in [2]. Fixa free group F . An equation in variables x, y over F is an expression of theform E ( x, y ) = 1where E ( x, y ) = f z m . . . z nm n f n +1 ; f i ∈ F, z j ∈ { x, y } and m k ∈ Z . Byan equation of the form w ( x, y ) = u we mean an equation z m . . . z nm n u − = 1where u ∈ F, z j ∈ { x, y } .We view an equation as an element of the group F [ x, y ] = F ∗ F ( x, y ). A solution of an equation is a substitution x g , y g ; g i ∈ F (1)so that in F the product E ( g , g ) = F
1. A system of equations in vari-ables x, y ; S ( x, y ) = S ; is a subset of F [ x, y ] and a solution of S ( x, y ) is asubstitution as in (1) so that all the elements of S ( x, y ) vanish in F . Definition 1.1.
A group G equipped with a distinguished monomorphism i : F ֒ → G is called an F -group we denote this ( G, i ). Given F -groups ( G , i ) and( G , i ), we define an F − homomorphism to be a homomorphism of groups f such that the following diagram commutes: G f / / G F i O O i = = {{{{{{{{ We denote by Hom F ( G , G ) the set of F -homomorphisms from ( G , i ) to( G , i ). n the remainder the distinguished monomorphisms will in general beobvious and not explicitly mentioned. It is clear that every mapping of theform (1) induces an F -homomorphism φ ( g , g ) : F [ x, y ] → F , it is also clearthat every f ∈ Hom F ( F [ x, y ] , F ) is induced from such a mapping. It followsthat we have a natural bijective correspondenceHom F ( F [ x, y ] , F ) ↔ F × F = { ( g , g ) | g i ∈ F } Definition 1.2.
Let S = S ( x, y ) be a system of equations. The subset V ( S ) = { ( g , g ) ∈ F × F | x g , y g is a solution of S } is called the algebraic variety of S .We have a natural bijective correspondenceHom F ( F [ x, y ] /ncl ( S ) , F ) ↔ V ( S ) Definition 1.3.
The radical of S is the normal subgroup Rad ( S ) = \ f ∈ Hom F ( F [ x,y ] /ncl ( S ) ,F ) ker( f )and we denote the coordinate group of SF R ( S ) = F [ x, y ] /Rad ( S )It follows that there is a natural bijective correspondenceHom F ( F [ x, y ] /ncl ( S ) , F ) ↔ Hom F ( F R ( S ) , F )so that V ( S ) = V ( Rad ( S )). We say that V ( S ) or S is reducible if it is aunion V ( S ) = V ( S ) ∪ V ( S ); V ( S ) ( ∪ V ( S ) ) V ( S )of algebraic varieties. An F -group G is said to be fully residually F if forevery finite subset P ⊂ G there is some f P ∈ Hom F ( G, F ) such that therestriction of f P to P is injective. Theorem 1.4 ([2]) . S is irreducible if and only if F R ( S ) is fully residually F . Theorem 1.5 ([2]) . Either F R ( S ) is fully residually F or V ( S ) = V ( S ) ∪ . . . V ( S n ) where the V ( S i ) are irreducible and there are canonical epimorphisms π i : F R ( S ) → F R ( S i ) such that each f ∈ Hom F ( F R ( S ) , F ) factors through some π i . Corollary 1.6. If F [ x, y ] /ncl ( S ) is fully residually F then F R ( S ) = F [ x, y ] /ncl ( S ) . .2 Rational Equivalence Definition 1.7. An F -automorphism of F [ x, y ] is an automorphism φ : F [ x, y ] → F [ x, y ]such that the restriction φ | F is the identity. Two systems of equations S, T are said to be rationally equivalent if φ ( S ) = T , for some φ ∈ Aut F ( F [ x, y ]). Proposition 1.8. (i)
Aut F ( F [ x, y ]) is generated by the elementary Nielsentransformations on the basis { F, x, y } that fix F elementwise.(ii) If S, T are rationally equivalent via φ ∈ Aut F ( F [ x, y ]) , then the naturalmap e φ in the commutative diagram below is an isomorphism. F [ x, y ] φ / / π (cid:15) (cid:15) F [ x, y ] π (cid:15) (cid:15) F R ( S ) e φ / / F R ( T ) Proposition 1.9.
Suppose w ( x, y ) is a primitive (by primitive we mean anelement that belongs to some basis) element of F ( x, y ) , then there exist words X ( u, z ) , Y ( u, z ) such that the set of solutions of w ( x, y ) = u correspond tothe set of pairs ( x, y ) = ( X ( u, z ) , Y ( u, z )) where z takes arbitrary values in F .Proof. Let S = { w ( x, y ) u } . By assumption there is φ ∈ Aut F ( F [ x, y ]) thatsends w ( x, y ) to x and φ extends to an F automorphism of F [ x, y ]. Thismeans that S is rationally equivalent to T = { xu − } . The first thing to noteis that F R ( T ) is a free group, hence so is F R ( S ) . Hom F ( F R ( T ) , F ) is given by V ( T ) = { ( x, y ) ∈ F × F | x = u, y ∈ F } the result now follows by precomposing with e φ − , as defined in Proposition1.8. Lemma 1.10.
Suppose the free group F ( x, y ) on generators { x, y } admitsa presentation F ( x, y ) = h ξ, ζ, p | [ ξ, ζ ] p − i where ξ, ζ, p ∈ F ( x, y ) . Then the mapping φ ( ξ ) = x, φ ( ζ ) = y, φ ( p ) = [ x, y ] ,extends to an automorphism φ : F ( x, y ) → F ( x, y ) .Proof. Notice that the basis elements x, y of [ x, y ] obviously satisfy the iden-tity [ x, y ][ x, y ] − = 1, so the mapping φ gives an automorphism. We assume the reader is familiar with Bass-Serre theory, so we only describeenough to explain our notation. efinition 1.11. A graph of groups G ( A ) consists of a connected directedgraph A with vertex set V A and edges EA . A is directed in the sense that toeach e ∈ EA there are functions i : EA → V A, t : EA → V A correspondingto the initial and terminal vertices of edges. To A we associate the following: • To each v ∈ V A we assign a vertex group G v . • To each e ∈ EA we assign an edge group G e . • For each edge e ∈ EA we have monomorphisms σ e : G e → G i ( e ) , τ e : G e → G t ( e ) we call the maps σ e , τ e boundary monomorphisms and the images ofthese maps boundary subgroups .A graph of groups has a fundamental group denoted π ( G ( A )). We saythat a group splits as the fundamental group as a graph of groups if G = π ( G ( A )) and refer to the data D = ( G, G ( A )) as a splitting . Definition 1.12 (Moves on G ( A )) . We have the following moves on G ( A )that do not change the fundamental group. • Change the orientation of edges in G ( A ), and relabel the boundarymonomorphisms. • Conjugate boundary monomorphisms , i.e. replace σ e by γ g ◦ σ e where γ g denotes conjugation by g and g ∈ G i ( e ) . • Slide , i.e. if there are edges e, f such that σ e ( G e ) = σ f ( G f ) then wechange X by setting i ( f ) = t ( e ) and replacing σ f by τ e ◦ σ − e ◦ σ f . • Folding , i.e. if σ e ( G e ) ≤ A ≤ G i ( e ) , then replace G t ( e ) by G t ( e ) ∗ τ e ( G e ) A ,replace G e by a copy of A and change the boundary monomorphismaccordingly. • Collapse an edge e , i.e. for some edge e ∈ EA , take the subgraph star ( e ) = { i ( e ) , e, t ( e ) } and consider the quotient of the graph A , sub-ject to the relation ∼ that collapses star ( e ) to a point. The resultinggraph A ′ = A/ ∼ is again a directed graph. Denote the equivalenceclass v ′ = [ star ( e )] ∈ A ′ , then we have G v ′ = G i ( e ) ∗ G e G t ( e ) or G i ( e ) ∗ G e depending whether i ( e ) = t ( e ) or not. For each edge f of A incidentto either i ( e ) or t ( e ), we have boundary monomorphisms G f → G v ′ given by σ ′ f = i ◦ σ f or τ ′ f = i ◦ τ f , where i is the one of the inclusion G t ( e ) ⊂ G v ′ or G i ( e ) ⊂ G v ′ . • Conjugation , i.e. for some g ∈ G replace all the vertex groups by G gv and postcompose boundary monomorphisms with γ g (which denotesconjugation by g ). Definition 1.13. An elementary cyclic splitting D of G is a splitting of G as either a free product with amalgamation or an HNN extension over acyclic subgroup. We define the Dehn twist along D , δ D , as follows. • If G = A ∗ h γ i B then δ D ( x ) = x if x ∈ Ax γ if x ∈ B If G = h A, t | t − γt = β i , γ, β ∈ A then δ D ( x ) = x if x ∈ Atβ if x = t A Dehn twist generates a cyclic subgroup of
Aut ( G ). A splitting such thatall the edge groups are nontrivial and cyclic is called a cyclic splitting .We can generalize the notion of a Dehn twist to arbitrary cyclic splittings. Definition 1.14. let D be a cyclic splitting of G with underlying graph A and let e be an edge of of A . Then a Dehn twist along e is an automorphismthat can be obtained by collapsing all the other edges in A to get a splitting D ′ of G with only the edge e and applying one the applicable automorphismsof Definition 1.13 Definition 1.15. (i) A subgroup H ≤ G is elliptic in a splitting D if H isconjugable into a vertex group of D , otherwise we say it is hyperbolic.(ii) Let D and D ′ be two elementary cyclic splittings of a group G withboundary subgroups C and C ′ , respectively. We say that D ′ is ellipticin D if C ′ is elliptic in D . Otherwise D ′ is hyperbolic in D A splitting D of an F -group is said to be modulo F if the subgroup F iscontained in a vertex group.The following is proved in [17]: Theorem 1.16. (i) Let G be freely indecomposable (modulo F ) and let D ′ , D be two elementary cyclic splittings of G (modulo F ). D ′ is ellipticin D if and only if D is elliptic in D ′ .(ii) Moreover if D ′ is hyperbolic in D then G admits a splitting E suchthat one of its vertex groups is the fundamental group Q = π ( S ) of apunctured surface S such that the boundary subgroups of Q are punc-ture subgroups. Moreover the cyclic subgroups h d i , h d ′ i correspondingto D, D ′ respectively are both conjugate into Q . Definition 1.17.
A subgroup Q ≤ G is a quadratically hanging (QH) sub-group if for some cyclic splitting D of G , Q is a vertex group that arises asin item (ii) of Theorem 1.16.Not every surface with punctures can yield a QH subgroup. By Theorem3 of [8], the projective plane with puncture(s) and the Klein bottle withpuncture(s) cannot give QH subgroups. It has also been noted that surfacesthat can give QH subgroups must admit pseudo-Anosov homeomorphisms. Definition 1.18. (i) A QH subgroup Q of G is a maximal QH (MQH)subgroup if for any other QH subgroup Q ′ of G , if Q ≤ Q ′ then Q = Q ′ .(ii) Let D be a splitting of G with Q be a QH vertex subgroup and let C be a splitting of Q with boundary subgroup h c i then there is a splitting D ′ of G called a refinement of D along C such that D is obtained froma collapse of D ′ along an edge whose corresponding group is h c i . Definition 1.19. (i) A splitting D is almost reduced if vertices of valencyone and two properly contain the images of edge groups, except verticesbetween two MQH subgroups that may coincide with one of the edgegroups. ii) A splitting D of G is unfolded if D can not be obtained from anothersplitting D ′ via a folding move (See Definition 1.12). Theorem 1.20 (Proposition 2.15 of [10]) . Let H be a freely indecomposablemodulo F f.g. fully residually F group. Then there exists an almost reducedunfolded cyclic splitting D called the cyclic JSJ splitting of H modulo F withthe following properties:(1) Every MQH subgroup of H can be conjugated into a vertex group in D ; every QH subgroup of H can be conjugated into one of the MQHsubgroups of H ; non-MQH [vertex] subgroups in D are of two types:maximal abelian and non-abelian [rigid], every non-MQH vertex groupin D is elliptic in every cyclic splitting of H modulo F .(2) If an elementary cyclic splitting H = A ∗ C B or H = A ∗ C is hyperbolicin another elementary cyclic splitting, then C can be conjugated intosome MQH subgroup.(3) Every elementary cyclic splitting H = A ∗ C B or H = A ∗ C modulo F which is elliptic with respect to any other elementary cyclic splittingmodulo F of H can be obtained from D by a sequence of moves givenin Definition 1.12.(4) If D is another cyclic splitting of H modulo F that has properties(1)-(2) then D can be obtained from D by a sequence of slidings, con-jugations, and modifying boundary monomorphisms by conjugation (seeDefinition 1.12.) Definition 1.21. (For simplicity we consider only the case where F R ( S ) isfreely indecomposable modulo F .) Given D , a cyclic JSJ decomposition of F R ( S ) modulo F , we define the group ∆ of canonical F − automorphisms of F R ( S ) to be generated by the following: • Dehn twists along edges of D ; or by Dehn twists along edges e ′ obtainedfrom refinements of D along cyclic splittings of MQH subgroups; thatfix F ≤ F R ( S ) . • Automorphisms of the abelian vertex groups that fix edge groups.The following Theorem is proved in [9, 18].
Theorem 1.22. If F R ( S ) = F and is freely indecomposable (modulo F ) thenit admits a non trivial cyclic JSJ decomposition modulo F . F ( F R ( S ) , F ) Definition 1.23. A Hom diagram for Hom F ( G, F ), denoted Diag F ( G, F ),consists of a finite directed rooted tree T such that the root, v , has noincoming edges and otherwise every vertex has at most one incoming edgealong with the following data: • To each vertex, except the root, v of T we associate a fully residually F group F R ( S v ) . • The group associated to each leaf of T is a free product F ∗ H ∗ . . . ∗ H n ,where the H i are isomorphic to subgroups of F . (The H i can be thoughtas free variables) F F FFFFF FF R ( S ) F R ( S ) F R ( S ) F V − V ( S ) V − V ( S ) π π π Figure 1: Hom diagrams corresponding to cases 1 ., . , and 3 . of Corollary 2.12, π , π , π are given in Proposition 2.14. • To each edge e with initial vertex v i and terminal vertex v t we have aproper F -epimorphism π e : F R ( S v ) → F R ( S v ) We point out that in the work of Sela, the
Hom diagram is called aMakanin-Razborov diagram (relative to F) and that our fully residually F groups are limit groups (relative to F). The following theorem gives a finiteparametrization of the solutions of systems of equations over a free group. Theorem 1.24 ([9, 18]) . For any system of equations S ( x , . . . , x n ) there ex-ists a Hom diagram Diag F ( F R ( S ) , F ) such that for every f ∈ Hom F ( F R ( S ) , F ) there is a path v , e , v , e , . . . , e m +1 , v m +1 from the root v to a leaf v m +1 such that f = ρ ◦ π v m +1 ◦ σ v m ◦ . . . ◦ σ v ◦ π e where the σ v j are canonical F -automorphisms of F R ( S vj ) , the π j are epi-morphisms π j : F R ( S vj ) → F R ( S vj +1 ) inside Diag F ( F R ( S ) , F ) , and ρ is any F -homomorphism ρ : F R ( S vm +1 ) → F from the free group F R ( S vm +1 ) to F . S = { w ( x, y ) u − } Definition 2.1.
Let φ be a solution of S , then the rank of φ is the rank ofthe subgroup h φ ( x ) , φ ( y ) i ≤ F .If all solutions of S are of rank 1, then V ( S ) is easy to describe and isgiven in Section 2.1. If S has solutions of rank 2, then there will be infinitelymany such solutions. For this case we will prove that Diag F ( F R ( S ) , F ) cor-respond to one of three cases (see Figure 1.) We will moreover describethe possible splittings of F R ( S ) and the associated canonical automorphisms.This description along with Theorem 1.24, will enable us to describe V ( S )as a set of pairs of words in F (see Theorem 2.28). .1 Easy Cases and Reductions By Proposition 1.9 we need only concern ourselves with the case where w ( x, y ) is not primitive. We state some results that enable us to simplifymatters: Lemma 2.2.
The equation w ( x, y ) = 1 doesn’t admit any rank 2 solutions. Let σ x ( w ) and σ y ( w ) be the exponents sums of x and y respectively inthe word w ( x, y ). Then it is easy to see that V ( S ) = { ( r n , r n ) ∈ F × F | r ∈ F ; n σ x ( w ) + n σ y ( w ) = 0 } (2)In this case we have that F R ( S ) ≈ F ∗ < t > and the mapping F [ x, y ] / ncl( S ) → F R ( S ) is given by the mapping f f, f ∈ Fx t r x y t r y where ( r x , r y ) is a generator of the subgroup { ( a, b ) ∈ Z ⊕ Z | aσ x ( w ) + bσ y ( w ) = 0 } . Lemma 2.3. If w ( x, y ) = v ( x, y ) n , n > then either the variety V ( { w ( x, y ) u − } ) is empty or u = r n for some r ∈ F and we have the equality V ( { w ( x, y ) u − } ) = V ( { v ( x, y ) r − } ) . We will always assume that w ( x, y ) is not a proper power. Although thismay seem somewhat contrived, our reason for doing so is twofold: firstly,requiring that an element is primitive is not enough; in our theorems we wantto exclude the case where w ( x, y ) is a proper power of a primitive elementas, again, solutions are easy to describe. Secondly, if w ( x, y ) = v ( x, y ) n with n maximal, then in the cyclic JSJ splitting of F R ( S ) modulo F , the edgegroup will be generated by v ( x, y ) and not w ( x, y ). For the next result weneed the following theorem: Theorem 2.4 (Main Theorem of [1]) . Let w = w ( x , x , . . . , x n ) be anelement of a free group F freely generated by x , x , . . . , x n which is neithera proper power nor a primitive. If g , g , . . . , g n , g are elements of a freegroup connected by the relation w ( g , g , . . . , g n ) = g m ( m > then the rank of the group generated by g , g , . . . , g n is at most n − . Corollary 2.5.
Suppose that w ( x, y ) is neither primitive nor a proper power.If u = r n , n > is a proper power then the equation w ( x, y ) = u doesn’t haveany rank 2 solutions.Proof. Suppose not then there is a solution φ : F R ( S ) → F such that x = φx, y = φy and [ x, y ] = 1 which means that h x, y i is free group of rank two.But we have the identity w ( x, y ) = r n , which by Theorem 2.4 implies thatrank of h x, y i is at most one –contradiction. .2 Possible cyclic JSJ splittings of F R ( S ) and canon-ical automorphisms Lemma 2.6.
Suppose that w ( x, y ) is neither primitive nor a proper power.If w ( x, y ) = u has a rank 2 solution then the group F [ x, y ] / ncl ( S ) ≈ F ∗ u = w ( x,y ) h x, y i is fully residually F and, in particular, we have that F R ( S ) = F ∗ u = w ( x,y ) h x, y i Proof.
Let ( x, y ) be a rank 2 solution. Let F = h F, t | t − ut = u i , F is a rankone free extension of a centralizer of F , and therefore is fully residually F .By definition F − subgroups are also fully residually F . Let H = h x, y i ≤ F and let H ′ = t − Ht . By Britton’s Lemma we see that H ′ ∩ F = h u i andthat h F, H i ≈ F u = w ( x t ,y t ) H ′ ≈ F ∗ u = w ( x,y ) h x, y i so this gives an F − embedding F ∗ u = w ( x,y ) h x, y i ֒ → F so F ∗ u = w ( x,y ) h x, y i is fully residually F . By Corollary 1.6 we obtain the equality F R ( S ) = F [ x, y ] / ncl( S ) Lemma 2.7. If w ( x, y ) is not primitive nor a proper power then F R ( S ) = F ∗ u = w ( x,y ) h x, y i is freely indecomposable modulo F .Proof. Suppose not. Since h x, y i is a free group of rank 2, if it splits freelywith nontrivial factors, then it must split as a free product of two cyclicgroups. Since any splitting of F R ( S ) modulo F must also be modulo w ( x, y )we have that w ( x, y ) must lie in one of these free cyclic factors, contradictingthe hypotheses of the lemma.Given this first decomposition as an amalgam, we wish to see how it canbe refined to a cyclic JSJ decomposition modulo F . By the Freiheitssatz,the subgroup h x, y i ≤ F R ( S ) is free of rank 2. So to investigate cyclic JSJdecomposition modulo F , we must first look at the possible cyclic splittingof h x, y i . Our main tool will be the following theorem of Swarup: Theorem 2.8 (Theorem 1 of [20]) . (A) Let G = G ∗ H G be an amalga-mated free product decomposition of a free group G with H finitely generated.Then, there is a non-trivial free factor H ′ of H such that H ′ is a free factorof either G or G .(B) Let G = J ∗ H,t be an HNN decomposition of a free group G with H finitely generated. Then there are decompositions H = H ∗ H , J = J ∗ J with H non trivial such that H is a free factor of J and t − H t is conjugatein J to a subgroup of J . Corollary 2.9. If G = G ∗ h γ i G is an amalgamated free product decom-position of a free group over a nontrivial cyclic subgroup, then rank ( G ) = rank ( G ) + rank ( G ) − . emma 2.10. Let G be a free group of rank 2 and let w ∈ G be nonprimitive, and not a proper power. Then the only possible almost reduced(see Definition 1.19) nontrivial cyclic splittings of G as the fundamentalgroup of a graph of groups with w elliptic are as(i) a star of groups, specifically a graph of groups whose underlying graphis simply connected, consisting of a center vertex v c and a collection ofperipheral vertices v , . . . , v m connected to v c by an edge. The groupassociated to v c , called the central group , is free of rank 2 and eachedge group is nontrivial, cyclic and is a proper finite index subgroup ofthe associated “peripheral” vertex group; or(ii) as an HNN extension G = h H, t | t − pt = q i ; p, q ∈ H − { } where w ∈ H and H is another free group of rank 2. Moreover we havethat H = h p i ∗ h q i i.e. G = h p, t i .Proof. Let D be a splitting of G . If G splits as a free product with amalga-mation G = G ∗ h γ i G then if γ is not trivial, Corollary 2.9 forces one of thefactors to be cyclic. Since we are assuming almost reducedness we must havethat the edge group is a finite index subgroup of one of the cyclic factors.Suppose G is a cyclic factor and let z be a generator of G . Then the freegroup G is obtained by adjoining the n th root z of the element γ ∈ G , whichis a free group of rank 2. It is however impossible to have a further splitting G ∗ h γ i ∗ G ∗ h γ ′ i ∗ G with G and G cyclic and with h γ i , h γ ′ i proper finiteindex subgroups of G , G (resp.) since then, by an easy computation usingnormal forms, it would be possible to get a counter example to commutationtransitivity, which must hold in a free group. The general star case follows.If the underlying graph of D is simply connected and one of the edgegroups is trivial, then we can collapse D to a free product G ∗ G withnontrivial factors, and with w lying in one of the vertex groups, by Grushko’sTheorem we must have rank( G ) = rank( G ) = 1 and our assumption that w is elliptic in D and not a proper power forces w to be primitive –contradiction.We have therefore covered the case where the underlying graph is simplyconnected.If the underlying graphs has two cycles (and a nontrivial vertex group),then we would have a proper epimorphism G → F ( a, b ) which contradictsthe Hopf property. Claim: If G = h H, t | t − pt = q i , then H is a free group of rank 2. By Theo-rem 2.8 ( B ) and conjugating boundary monomorphisms we can arrange sothat H = H ∗ H with p ∈ H and q ∈ H (3)Theorem 2.8 ( B ) moreover gives us that without loss of generality we canassume that h q i is a free factor of H . This means that H = H ′ ∗ h q i (4)Letting H ′ = H ∗ H ′ we get that H = H ′ ∗h q i so combining (3) and (4) givesus a presentation G = h H ′ , t, q | t − pt = q i which via a Tietze transformationgives us G = h H ′ , t |∅i (5) hich forces H ′ to be cyclic which means that H has rank 2. Moreover, wesee immediately that H = h p i ∗ h q i .We denote by ∆ the group of canonical F − automorphisms of F R ( S ) (seeDefinition 1.21.) Convention 2.11.
Whenever we have a “star” splitting of the subgroup h x, y i , as given in item (i) in the statement of Lemma 2.10, we will collapsethe whole splitting to a single vertex group. The first reason being that theDehn twists around the edge groups fixing the central group act trivially.Secondly, by uniqueness of n th roots in a free group, we see that any mappingof the central group into a free group has at most a unique extension tothe whole group. It follows that to describe solutions to the equation, thecollapsed splitting is sufficient. Corollary 2.12.
There are three possible classes of cyclic JSJ decompositionmodulo F of F R ( S ) :1. F R ( S ) ≈ F ∗ u = w ( x,y ) h x, y i and ∆ = h γ w i , where γ w is the automorphismthat extend the mapping: γ w : f f ; f ∈ Fz w − zw ; z ∈ h x, y i
2. The subgroup h x, y i splits as a cyclic HNN-extension: h x, y i = h H, t | t − pt = q i with w ( x, y ) ∈ H so that F R ( S ) ≈ F ∗ u = w ( x,y ) h H, t | t − pt = q i and ∆ = h γ w , τ i where these are the automorphisms that extend the mappings: γ w : f f ; f ∈ Fz w − zw ; z ∈ h x, y i ; τ : z z ; z ∈ h F, H i t tq F R ( S ) ≈ F ∗ u = w ( x,y ) Q where Q is a QH subgroup and, up to rationalequivalence, Q = h x, y, w | [ x, y ] w − i . ∆ is generated by the automor-phisms extending the mappings: γ w ; δ x : x yx identity on F ∪ { y } ; δ y : y xy identity on F ∪ { x } Proof.
Suppose first that the cyclic JSJ decomposition of F R ( S ) modulo F has a QH subgroup Q . Then Q must be a subgroup of h x, y i , in particularthere must be a splitting of h x, y i modulo w such that Q is one of its vertexgroups. By Lemma 2.10 we must either have that Q = h x, y i , or h x, y i isan HNN extension of Q . Either way we must have that Q is a free group ofrank 2. The possible punctured surfaces S such that π ( S ) is a free group ofrank 2 are the once punctured torus or the once punctured Klein bottle, thelatter is not allowed (see Theorem 3 of [8].) Moreover, we see that if h x, y i isan HNN extension of Q then the associated subgroups must be conjugate in Q , which would imply that h x, y i contains an abelian free group of rank 2 –contradiction. It follows from Corollary 1.10 that, up to rational equivalence,the only possibility is as in case 3. of the statement.The rest of the statement follows immediately from Lemma 2.10 andDefinition 1.21. .3 Solutions of rank 1 We now consider solutions of rank 1. Although everything can easily bedescribed in terms of linear algebra, it is instructive to explain this in termsof Hom diagrams and canonical automorphisms, because as we shall see theseprovide examples of canonical epimorphisms that are not strict (see [18] fora definition.)As we saw earlier, rank 1 solutions occur when we are solving w ( x, y ) = 1.More generally a rank 1 solutions occurs if and only if w ( x, y ) = u = v d where d = gcd ( σ x ( w ) , σ y ( w )); σ x ( w ) , σ y ( w ) denote the exponent sums of x, y in w ( x, y ). Corollary 2.5 states that if d >
1, but w ( x, y ) not primitiveand not a proper power, then all solution of w ( x, y ) = u have rank 1. If d = 1 then w ( x, y ) = u may have both rank 1 and rank 2 solutions.Let S = { w ( x, y ) u − , [ x, y ] } , then all rank 1 solutions must factor through F R ( S ) . If d > Rad ( { w ( x, y ) u − } ) = ncl ( { w ( x, y ) u − , [ x, y ] } ). As a set, thesesolutions are easy to describe: V ( S ) = { ( u n , u n ) ∈ F × F | n σ x ( w ) + n σ y ( w ) = d } (6)Let p, q be integers such that pσ x ( w ) + qσ y ( w ) = d (7)then doing some linear algebra we have that n , n in (6) are given by( n , n ) = ( p, q ) + m ( σ y ( w ) , − σ x ( w )); m ∈ Z (8)We now investigate the situation where w ( x, y ) = u has rank 1 and rank2 solutions, i.e V ( S ) ) V ( S ). We first want to understand F R ( S ) . Lemma 2.13.
Suppose that w ( x, y ) is not primitive nor a proper powerand suppose moreover that w ( x, y ) = u admits rank 1 and rank 2 solutions.Then there F R ( S ) is isomorphic to h F, s | [ u, s ] = 1 i = F . The F − morphism π : F R ( S ) → F given by π ( x ) = u p s σ y ( w ) = x ; π ( y ) = u q s − σ x ( w ) = y (9) where p, q are as in equation (7), realizes this isomorphism.Proof. Consider the F − epimorphism π : F R ( S ) → h F, s | [ u, s ] = 1 i = F given by (9) On one hand we see that π is surjective which gives an injectionHom F ( F , F ) ֒ → Hom F ( F R ( S ) , F ) (10)via pullbacks f f ◦ π . On the other hand F , a free rank 1 extensionof a centralizer, is fully residually free. On the third hand the group ∆ ofcanonical F automorphisms of F is generated by the automorphism givenby: δ : s suf f f ∈ F and if we consider the F − epimorphism π : F → F given by π ( s ) = u thenwe immediately see that the set V = { ( π ( σ m ( x )) , π ( σ m ( y )) ∈ F × F | σ ∈ ∆ } f images of ( x, y ) via the mappings π ◦ σ ◦ π , σ ∈ ∆ coincides with V ( S ).And since Hom F ( F , F ) = { π ◦ σ | σ ∈ ∆ } we get that the correspondence(10) is in fact a bijective correspondence. It follows that F R ( S ) ≈ F F . Proposition 2.14.
Let w ( x, y ) be non primitive and not a proper power.Suppose moreover that w ( x, y ) = u has rank 1 and rank 2 solutions. Then(i) if F R ( S ) is as in . in Corollary 2.12 , then V ( S ) is represented by thefollowing branch in Diag F ( F R ( S ) , F ) : F R ( S ) π / / F σ (cid:25) (cid:25) π / / F (11) where σ ∈ ∆ .(ii) If F R ( S ) is as in . in Corollary 2.12 , then V ( S ) is represented by thefollowing branch in Diag F ( F R ( S ) , F ) : F R ( S ) σ (cid:23) (cid:23) π / / F (12) where σ ∈ ∆ and π = π ◦ π Where π , π and ∆ were defined in the previous proof.Proof. We first note that if F R ( S ) corresponds to case 3 . of Corollary 2.12,then the equality (7) is impossible. In both possible cases we have epimor-phisms F R ( S ) π / / F π / / F (13)We saw that all solutions rank 1 solutions factor through π . If F R ( S ) is asin 1 . in Corollary 2.12 then ∆ is generated by γ w , now since π ◦ γ w = π wehave that solutions in V ( S ) must factor through F and are parametrizedby ∆ .If F R ( S ) is as in 2 . in Corollary 2.12, then h x, y i splits as h H, t | t − pt = q i ; p, q ∈ H , moreover by Lemma 2.10 we have that h x, y i = h p, t i . We consider thisbasis of h x, y i . Let π ( t ) = t, π ( p ) = p , then the subgroup Z ⊕ Z ≈ A = h u, s i ≤ F is generated by p, t . We note that in F R ( S ) , as written as aword in { p, t } ± , w ( x, y ) = w ′ ( p, t ) = u has exponent sum zero in the letter t . Since A is the abelianization of h x, y i , we have that in A, u = 0 t + np and since u lies in a minimal generating set of A we must have n = ±
1. Ittherefore follows that for the Dehn twist τ , which sends t tq , we have π ◦ τ = δ ◦ π , where δ is the generator of ∆ . It follows that the canonical F -automorphisms of F in (13) can be “lifted” to F R ( S ) and the branch (12)gives us a parametrization of V ( S ). .4 Solutions of rank 2 Before being able to make our finiteness arguments we need some preliminarysetup. We will study more closely mappings F ( x, y ) → F . Definition 2.15. (i) Let ( f , f ) be a pair of words in a free group, thenan elementary Nielsen move (e.N.m.) is a mapping of the form( f , f ) ( f , ( f ǫ f ǫ ) ǫ ) or ( f , f ) (( f ǫ f ǫ ) ǫ ) , f )with ǫ , ǫ ∈ {− , } and ǫ ∈ {− , , } .(ii) For F ( x, y ), the free group on the basis { x, y } , an elementary Nielsentransformation (e.N.t.) is an element of Aut ( F ( x, y )) that is definedby the mappings: x ( x ǫ y ǫ ) ǫ y y or x xy ( y ǫ x ǫ ) ǫ with ǫ , ǫ ∈ {− , } and ǫ ∈ {− , , } . Lemma 2.16.
Suppose φ , given by ( x , y ) ∈ F × F , is a rank 2 solution of w ( x, y ) = u , let ( x , y ) m / / . . . m n / / ( x n , y n ) be a sequence of e.N.m. then(i) there is a corresponding sequence of e.N.t t , . . . , t n such that letting w ( x, y ) = w ( x, y ) and w j +1 ( x, y ) = t j +1 ( w j ( x, y )) we have the equali-ties u = w ( x , y ) = . . . = w n ( x n , y n ) (14) (ii) Let α = t n ◦ . . . ◦ t ∈ Aut ( F ( x, y )) (15) then the mapping φ ′ = φ ◦ α − : F ( x, y ) → F is given by the pair ( x n , y n ) sketch of proof. Noting that a rank 2 solution isomorphically identifies thesubgroup h x, y i ≤ F R ( S ) with a rank 2 subgroup of a free group, the proofis essentially the same as the proof that elementary Nielsen transformationsgenerate the automorphisms of a f.g. free group (See Proposition I.4.1. of[12]).The reader can look at Section I.2 of [12] for the necessary backgroundfor the next lemma. Lemma 2.17.
Fix a basis X of F , then to any subgroup H ≤ F of rank n we can canonically associate an ordered set of Nielsen reduced generators ( j , . . . , j n ) , moreover this ordered set can be obtained from any ordered n − tuple of generators ( h , . . . , h n ) via a sequence of e.N.m. We now give names to all of these:
Definition 2.18.
Let φ , given by ( x , y ), be a solution of w ( x, y ) = u . Let( x , y ) m / / . . . m n / / ( x n , y n )be the sequence of e.N.m. that brings the pair ( x , y ) to the canonical pair( x n , y n ) of generators of h x , y i guaranteed by Lemma 2.17. Then we have: The pair ( x n , y n ) is called the terminal pair of φ (denoted tp ( φ ).) • The word w n ( x, y ) ∈ h x, y i in (14) is called the terminal word of φ (denoted tw ( φ ).) • The automorphism α ∈ Aut ( F ( x, y )), is the automorphism associated to φ (denoted α φ .) Proposition 2.19.
Let S = { w ( x, y ) = u } and let U ⊂ V ( S ) be the opensubvariety of rank 2 solutions, then there are only finitely many possibleterminal pairs and terminal words that can be associated to solutions φ ∈ U .Proof. Fix a basis X of F , we first show finiteness of possible terminal pairs.Let φ be a solution, given by ( x , y ) and let H = h x , y i ≤ F and let Γbe the Stallings graph for H (See, for instance, [19].) Then there is a pathin Γ with label u . We also have that Nielsen generators can be read directlyoff Γ (see [7]) as labels of simple closed paths. If we define the radius ofΓ to be the distance between the basepoint of Γ and the “farthest” vertex,then we see that the length of Nielsen the generators ( x m , y m ) is boundedby two times the radius. Moreover since w ( x, y ) is neither primitive nor aproper power in F ( x, y ) ≈ H , u is not primitive nor a proper power in H .It follows that the reduced path in Γ labeled u must cover the whole graphwhich means | u | is at least twice the radius, hence | x m | , | y m | ≤ | u | so the number of possible terminal pairs is bounded.Consider now the terminal word w n ( x, y ). Since ( x m , y m ) ∈ F × F is aNielsen reduced pair we have that | w n ( x, y ) | { x,y } ≤ | w n ( x n , y n ) | X = | u | X which bounds the number of terminal words.We now connect all these ideas to solutions of equations. The next ob-servation is obvious but critical. Lemma 2.20.
Let F R ( S ) be the coordinate group of w ( x, y ) = u , with w ( x, y ) not primitive, not a proper power and such that w ( x, y ) has a rank 2 solution.Then the group of F -automorphisms of F R ( S ) are induced by the automor-phisms of the free subgroup h x, y i that fix w ( x, y ) . Proposition 2.21.
Suppose that φ and φ ′ are solutions F R ( S ) → F of w ( x, y ) = u . And suppose moreover that tp ( φ ) = tp ( φ ′ ) and tw ( φ ) = tw ( φ ′ ) ,then there is an automorphism β ∈ Aut F ( F R ( S ) ) such that φ ′ = φ ◦ β .Proof. Let φ be given by ( x , y ) and let φ ′ be given by ( x ′ , y ′ ). Then wehave a sequence of e.N.m.( x , y ) m / / . . . m n / / tp ( φ ) = tp ( φ ′ ) . . . m ′ r o o ( x ′ , y ′ ) m ′ o o And we have automorphisms α φ , α φ ′ such that α φ ( w ( x, y )) = α φ ′ ( w ( x, y )) = tw ( φ ). On one hand we have that β = α − φ ◦ α φ ′ ∈ stab( w ), so, by Lemma2.20, β ∈ Aut ( F ( x, y )) extends to an automorphism of F R ( S ) . We moreover ave by Lemma 2.16 we have that the mappings F ( x, y ) → F , φ ′ ◦ α − φ ′ = φ ◦ α − φ which means that φ ′ = φ ◦ α − φ ◦ α φ ′ = φ ◦ β So we have proved that all rank 2 solutions are obtained from a finitefamily φ , . . . φ N of solutions and precomposition with F − automorphism of F R ( S ) . Nothing so far has been said about canonical automorphisms. Definition 2.22.
Let ∆ ≤ Aut ( F R ( S ) ) be the group of canonical F − auto-morphisms of F R ( S ) associated to a cyclic JSJ decomposition modulo F . Let φ, φ ′ ∈ Hom F ( F R ( S ) , F ), we say φ ∼ ∆ φ ′ if there is a σ ∈ ∆ such that φ ◦ σ = φ ′ . φ ∈ Hom F ( F R ( S ) , F ) is minimal if after fixing a basis X of F the quantity l f = | φ ( x ) | + | φ ( y ) | is minimal among all F -morphisms in φ ’s ∼ ∆ equivalence class.We wish to show that there are only finitely many ∆- minimal rank 2solutions to w ( x, y ) = u . In light of Proposition 2.21, this is equivalent tothe statement [stab( w ) : ∆] < ∞ . In [3], it is proved that for freely indecomposable fully residually free groups,the subgroup canonical automorphism is of finite finite index in the groupof outer automorphisms. Unfortunately, the result as formulated does notcover the case involving only automorphisms modulo F . We therefore provethis fact directly. What we will essentially show is that the internal F-automorphisms are of finite index in the whole group of F-automorphisms .The main pillars of the argument are that the JSJ decomposition is canonical in the sense of (4) of Theorem 1.20 and the following Theorem: Theorem 2.23 (Corollary 15.2 of [10]) . Let G be a nonabelian fully residu-ally free group, and let A = { A , . . . , A n } be a finite set of maximal abeliansubgroups of G . Denote by Out ( G ; A ) the set of those outer automorphismsof G which map each A i ∈ A onto a conjugate of itself. If Out ( G ; A ) is in-finite, then G has a nontrivial abelian splitting, where each subgroup in A iselliptic. There is an algorithm to decide whether Out ( G ; A ) is finite or infi-nite. If Out ( G ; A ) is infinite, the algorithm finds the splitting. If Out ( G ; A ) is finite, the algorithm finds all its elements. This next lemma follows immediately from the fact that in free groups n th roots are unique and centralizers of elements are cyclic. Lemma 2.24.
Let h x, y i be a free group and suppose h x, y i = h H, t | t − pt = q i ; p, q ∈ H − { } Suppose that for some g ∈ h x, y i we have the equality g − pg = q then g = tq j for some j ∈ Z . Proposition 2.25. ∆ ≤ Aut ( F ( x, y )) is of finite index in stab ( w ) roof. If w is conjugate to either [ x, y ] or [ y, x ] then the result follows imme-diately since the stab( w ) coincides with the automorphisms given in Corol-lary 2.12. (See, for instance, [14].) We first concentrate on the case wherethe JSJ of F R ( S ) is as in case 2 . of Corollary 2.12.Suppose the induced splitting of h x, y i is of the form h x, y i = h H, t | t − pt = q i p, q ∈ H − { } Let α ∈ stab( w ) ≤ Aut ( h x, y i ), then we can extend α to b α : F R ( S ) → F R ( S ) .We wish to understand the action of b α on F R ( S ) . First note that b α restrictedto F is the identity and b α ( h x, y i ) = h x, y i On the other hand, b α gives anothercyclic JSJ decomposition D modulo F : F R ( S ) = F ∗ u = w ( x,y ) h b α ( H ) , b α ( t ) | b α ( t ) − b α ( p ) b α ( t ) = b α ( q ) i (16)with w ∈ b α ( H ). By Theorem 1.20 (4), D can be obtained from D by asequence of slidings, conjugations and modifying boundary monomorphisms. b α ( H ) ∩ F = h w i , and H must be obtained from b α ( H ) as in (4) of Theorem1.20, i.e. by slidings, conjugating boundary monomorphisms and conjuga-tions. The only inner automorphism of F R ( S ) that fixes w is conjugation by w k ; k ∈ Z ; (use Bass-Serre theory and properties of free groups) and since b α ( H ) and H are attached to F at h w i , slidings will have no effect. It followsthat b α ( H ) = H . Applying Theorem 1.20 again forces p, q to be conjugatein H to b α ( p ) , b α ( q ) [respectively or in the other order]. We now have stronginformation enough on the dynamics of stab( w ) to apply Theorem 2.23.Indeed since b α ( H ) = H , we have a natural homomorphism ρ : stab( w ) → ^ stab( w ) ≤ Aut ( H ) given by the restriction α α | H . Moreover we seethat any almost reduced cyclic splitting of H modulo {h w i , h p i , h q i} mustbe trivial, otherwise contradicting Lemma 2.10. Let π : Aut ( H ) → Out ( H )be the canonical map (i.e. quotient out by Inn ( G ), the subgroup of innerautomorphisms). It therefore follows from Theorem 2.23 that the image π ◦ ρ (stab( w )) = stab( w ) must be finite.First note that Inn ( H ) ∩ ^ stab( w ) = h γ w i which means thatstab( w ) ≈ ^ stab( w ) / h γ w i and this isomorphism is natural. Let α ∈ ker ρ then we must have that α | H = 1. In particular we have α ( t ) − pα ( t ) = q which by Lemma 2.24 implies that α ( t ) = tq j it follows that ker( ρ ) ≤ h τ i .The other inclusion is obvious soker( ρ ) = h τ i There is a bijective correspondence between subgroups K of ^ stab( w ) andsubgroups of stab( w ) that contain h τ i given by K ρ − ( K ). Moreover thiscorrespondence sends normal subgroups to normal subgroups. It follows that ker ( π ◦ ρ ) = h τ, γ w i and so we get:stab( w ) / h τ, γ w i ≈ stab( w ) hich is finite. It follows that [stab( w ) : h τ, γ w i ] < ∞ .In the case where D , the cyclic JSJ of F R ( S ) modulo F is as in case 1 . ofCorollary 2.12 then again elements of α ∈ stab( w ) will give new splittings F R ( S ) = F ∗ u = w ( x,y ) b α ( H ). Arguing as before, we get that b α ( H ) = H andwe can apply Theorem 2.23 with A = {h w i} . We get that Out ( H ; A ) ≈ stab( w ) / h γ w i must be finite, otherwise H could split further, contradictingthe fact that D was a JSJ splitting, and the result follows.By Lemma 2.20, Propositions 2.19, 2.21, and 2.25 we get the second halfof our main result: Proposition 2.26.
Suppose that w ( x, y ) is not a proper power, nor is itprimitive. Then there are finitely many ∆ − minimal rank 2 solutions to theequation w ( x, y ) = u . V ( { w ( x, y ) u − } ) These next two results now follows immediately from Proposition 2.26, 2.14,Corollary 2.12, Lemma 2.10 and Theorem 1.24.
Theorem 2.27.
Suppose that w ( x, y ) = u has rank 2 solutions and that w ( x, y ) is not a power of a primitive element. Then the possible Hom dia-grams are given in Figure 1. Theorem 2.28.
Suppose that w ( x, y ) = u has rank 2 solutions and that w ( x, y ) is neither primitive nor a proper power. Let { φ i | i ∈ I } be the col-lection of ∆ − minimal solution. Then V ( S ) = V ( S ) ∪ V ′ , where V ′ = V ( S ) − V ( S ) ,is given by the following:1. F R ( S ) ≈ F ∗ u = w ( x,y ) h x, y i , let φ i ( x ) = x i , φ i ( y ) = y i then V ( S ) = V ( S ) ∪ V ′ where V ′ = { ( u − n x i u n , u − n y i u n ) | i ∈ I and n ∈ Z } and if the exponent sums σ x ( w ) , σ y ( w ) of x, y respectively in w arerelatively prime, then V ( S ) is non empty and is given by (6).2. F R ( S ) ≈ F ∗ u = w ( x,y ) h H, t | t − pt = q i , H = h p, q i and we can write x, y ∈h x, y i as words x = X ( p, q, t ) , y = Y ( p, q, t ) . Let φ i ( p ) = p i , φ i ( q ) = q i , φ i ( t ) = t i then we have that V ( S ) = V ( S ) ∪ V ′ where V ′ = { ( X ( u − n p i u n , u − n q i u n , u − n t i q mi u n ) ,Y ( u − n p i u n , u − n q i u n , u − n t i q mi u n )) | i ∈ I, n, m ∈ Z } and if the exponent sums σ x ( w ) , σ y ( w ) of x, y respectively in w arerelatively prime, then V ( S ) is non empty and is given by (6).3. F R ( S ) ≈ F ∗ u = w ( x,y ) Q where Q is a QH subgroup and, up to rationalequivalence, Q = h x, y, w | [ x, y ] w − i . Then V ( S ) is empty. Let φ i ( x ) = x i , φ i ( y ) = y i then V ( S ) = { ( X σ ( x i , y i ) , Y σ ( x i , y i )) | σ ∈ ∆ } where the words σ ( x ) = X σ ( x, y ) , σ ( y ) = Y σ ( x, y ) ∈ h x, y i . We finally note that unless w ( x, y ) = u is orientable quadratic, thensolutions are given by “one level parametric” words (see [11] for a definition.) An Interesting Example
The Hom diagrams given for w ( x, y ) = u were very simple. In particular,modulo the slight technicalities of Theorem 2.28 item 1, we can say that;unless w ( x, y ) is a power of a primitive element; there are only finitely manyminimal solutions to w ( x, y ) = u with respect to a group of canonical au-tomorphisms. This translates as the Hom diagram having only one “level”.This also means that all fundamental sequences or strict resolutions of F R ( S ) have length 1 (see [9] or [18], respectively for definitions.) It is natural toask this holds true for general equations in two variables. We answer thisnegatively: Theorem 3.1.
Let F = F ( a, b ) then the Hom diagram associated to theequation with variables x, y [ a − ba [ b, a ][ x, y ] x, a ] = 1 (17) has branches corresponding to rank 2 solutions that have length at least .Proof. First note that via Tietze transformations, we have the followingisomorphism: h F, x, y | [ a − ba [ b, a ][ x, y ] x, a ] = 1 i≈ h F, x, y, t | [ x, y ] x = [ a, b ] a − b − at ; [ t, a ] = 1 i Let w ( x, y ) = [ x, y ] x and let u = [ a, b ] a − b − at . We now embed G = h F, x, y, t | w ( x, y ) = u, [ t, a ] = 1 i into a chain of extensions of centralizers. Let F = h F, t | [ t, a ] = 1 i and let F = h F , s | [ u, s ] = 1 i . Let x = b − t and y = b − ab . First note that[ x, y ] x = (( t − b )( b − a − b )( b − t )( b − ab )) ( b − t ) = [ a, b ] a − b − at = u We now form a double, i.e. we set x = x s , y = y s and let H = h x, y i = h x, y i s .By Britton’s Lemma we have that H ∩ e F = h u i and it follows that h F, x, y i is isomorphic to the amalgam F ∗ h u i H = G . Since chains of extensions ofcentralizers of F are fully residually F . We have that our equation (17) isan irreducible system of equations, we write F R ( S ) = G . We note that wehave the nontrivial cyclic splitting D : F R ( S ) ≈ F ∗ h u = w ( x,y ) i h x, y i moreover since w ( x, y ) = [ x, y ] x cannot belong to a basis (see [5]) of h x, y i we have that F R ( S ) is freely indecomposable modulo F . On the other hand,if we take the Grushko decomposition of F R ( S ) modulo FF R ( S ) = e F ∗ K ∗ . . . K n ; F ≤ e F we see that we must have F ≤ e F since [ t, a ] = 1 ⇒ t ∈ e F . It follows that F R ( S ) is actually freely indecomposable modulo F . It follows that D can berefined to a cyclic JSJ decomposition modulo F .Suppose towards a contradiction that all branches of the Hom diagramfor Hom F ( F R ( S ) , F ) corresponding to rank 2 solutions had length 1. Thismeans that there are finitely many minimal rank 2 solutions φ : F R ( S ) → F . n one hand the element t must be sent to arbitrarily high powers of a , since F R ( S ) is fully residually F . On the other hand, for there to be a canonicalautomorphism of F R ( S ) that sends t ta n , there must be a splitting D ′ of F R ( S ) with some conjugate of h a i as a boundary subgroup, but u would haveto be hyperbolic in such a splitting, and since h a i is elliptic in D , we wouldhave an elliptic-hyperbolic splitting which by Theorem 1.16 would contradictfree indecomposability modulo F .We now provide some illustration. We determined that F R ( S ) = F ∗ h u = w ( x,y ) i h x, y i with u = [ a, b ] ab − a − t . Now the mapping x x u and y y u ex-tends to a canonical automorphism of F R ( S ) and along some branch theremust be another canonical automorphism that maps t ta r . By check-ing directly we see that φ : F R ( S ) F given by x = b − a, y = b − ab is asolution, so we can get the family of solutions: x = ([ a, b ] ab − a − a n ) m ( b − a )([ a, b ] ab − a − a n ) − m y = ([ a, b ] ab − a − a n ) m ( b − ab )([ a, b ] ab − a − a n ) − m with n, m in Z . Notice that no precomposition by a canonical automorphismof F R ( S ) can affect the n parameter. It follows that the set of solution of(17) can not be given by precomposing a finite collection of maps φ , . . . , φ n : F R ( S ) → F with canonical automorphisms. References [1] Gilbert Baumslag. Residual nilpotence and relations in free groups.
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