A graph-theoretic proof for Whitehead's second free-group algorithm
aa r X i v : . [ m a t h . G R ] J un A GRAPH-THEORETIC PROOF FORWHITEHEAD’S SECOND FREE-GROUP ALGORITHM
WARREN DICKS*
Abstract.
J. H. C. Whitehead’s second free-group algorithm determines whether or nottwo given elements of a free group lie in the same orbit of the automorphism group ofthe free group. The algorithm involves certain connected graphs, and Whitehead usedthree-manifold models to prove their connectedness; later, Rapaport and Higgins & Lyndongave group-theoretic proofs.Combined work of Gersten, Stallings, and Hoare showed that the three-manifold mod-els may be viewed as graphs. We give the direct translation of Whitehead’s topologicalargument into the language of graph theory. Minimal background
Whitehead(1936b) gave an algorithm which, with input two finite sequences S , S ofelements (or conjugacy classes of elements) of a finite-rank free group F , outputs eitheran F -automorphism ϕ such that ϕ ( S ) = S or an assurance that no such ϕ exists. Moreimportantly, he introduced certain connected graphs that have been of great interest togroup theorists. His nine-page proof of connectedness used a three-manifold model for each F -automorphism. Rapaport(1958) gave a twenty-page group-theoretic proof of connected-ness, and Higgins & Lyndon(1962, 1974) gave one of five pages; these proofs led the way toan even deeper understanding of F -automorphisms.Gersten(1987) constructed a graph model for each F -automorphism, and Stallings(1983)pointed out a connection between Gersten’s model and Whitehead’s. Krsti´c(1989) usedCayley trees to simplify Gersten’s construction. Hoare(1990) gave an explicit descriptionof Whitehead’s model in terms of Gersten’s. Below, we give the resulting translation ofWhitehead’s topological argument into the language of graph theory. This argument con-cerns changes of bases (free-generating sets) rather than automorphisms, and ours may bethe first treatment of Gersten’s graphs that does not mention group morphisms.All of the following will apply throughout.1.1.
Notation.
Set N := { , , , . . . } .Let F be a finite-rank free group. By a straight word in F , we mean an element of F ;by a cyclic word in F , we mean the F -conjugacy class of an element of F ; and, by a wordin F , we mean a straight-or-cyclic word in F . Let R be a finite set of words in F . Let Mathematics Subject Classification.
Primary: 20E05; Secondary:20F10, 20E08.
Key words and phrases.
Free group, Whitehead algorithm, Gersten graph.*Research supported by MINECO (Spain) through project number MTM2014-53644. For his earlier free-group algorithm, Whitehead(1936a) also used three-manifold models, to prove hiscelebrated cutvertex lemma. Hoare(1988) gave the second proof, using Gersten’s graphs in place of manifolds.Dicks(2014, 2017), refining work of Stong(1997), proved a more general result by tricolouring a Cayley tree.The elegant folding theorem of Heusener & Weidmann(2014) leads to a yet more general result. and Y be F -bases. In Section 2, we shall recall the value h ( X ) := P r ∈ R X -length( r ) ∈ N .We write X ± := X ∪ X − . We say that Y is a Whitehead transform of X if there existssome x ∈ X ± such that Y ⊆ { , x }· X ·{ , x − } . We say that X is a local-minimum pointfor h if h ( X ) h ( X ′ ) for each Whitehead transform X ′ of X . (cid:3) In Section 3, we shall use Gersten’s graphs to define a value d(
X, Y ) ∈ N that Whiteheadused tacitly. What the topological portion of Whitehead’s argument shows is preciselyif X and Y are local-minimum points for h , then either X ± = Y ± or some(1.1) Whitehead transform Y ′ of Y satisfies h ( Y ′ ) = h ( Y ) and d( X, Y ′ ) < d( X, Y ).This will be stated as Theorem 3.3 below, and our sole objective is to give a self-containedgraph-theoretic proof that copies Whitehead’s. All the other parts of his article are graphtheoretic or group theoretic, and we shall not discuss them. However, Whitehead leaves themain consequence of (1.1) unsaid, and it is as follows.Let us say that Y is an F -neighbour of X if either Y ± = X ± (whence h ( Y ) = h ( X )) or Y is a Whitehead transform of X . Let Γ( F ) denote the graph with vertices the F -basesand with edges joining F -neighbours. Let Γ( h ) denote the subgraph of Γ( F ) with verticesthe local-minimum points for h and with edges joining F -neighbours. It is obvious, butimportant, that h is constant on each connected subgraph of Γ( h ), and that a simple al-gorithm outputs a strictly h -decreasing Γ( F )-path starting at any given Γ( F )-vertex andstopping when Γ( h ) is reached. Now suppose that X is a local-minimum point for h andthat h ( Y ) h ( Z ) for each F -basis Z . By induction on d( X, Y ), it follows from (1.1) thatthere exists some ( h -constant) Γ( h )-path from Y to X . On varying X , we find that Γ( h ) isconnected, which may be considered to be the main result of Whitehead(1936b); it greatlygeneralizes the result of Nielsen(1919) that Γ( F ) itself is connected.The connectedness of Γ( F ) was used in the arguments of Whitehead, Rapaport, Higgins & Lyndon, and Gersten. However, Krsti´c did not use it, and this will permit us to prove (1.1)without using it. 2.
Review of Cayley trees
Definitions.
By a graph , we mean a quintuple ( Γ , VΓ , EΓ , ι, τ ) such that Γ is a set,VΓ and EΓ are disjoint subsets of Γ whose union is Γ, and ι and τ are maps from EΓto VΓ. We use the same symbol Γ to denote both the graph and the set. We call VΓand EΓ the vertex-set and edge-set of Γ respectively, and call their elements Γ -vertices andΓ -edges respectively. The maps ι and τ are called the initial and terminal incidence functionsrespectively.Each e ∈ EΓ has an inverse in the free group h EΓ | ∅i , and we set ι ( e − ) := τ ( e ) and τ ( e − ) := ι ( e ). For each v ∈ VΓ, by the Γ -valence of v , we mean (cid:12)(cid:12) { e ∈ (EΓ) ± : ιe = v } (cid:12)(cid:12) .By a Γ -path , we mean a sequence of the form p = ( v , e , v , e , v , . . . , v ℓ − , e ℓ , v ℓ ), where ℓ ∈ N and, for each i ∈ { , , . . . , ℓ } , e i ∈ (EΓ) ± , v i − = ιe i , and v i = τ e i . We sometimesabbreviate p to ( e , e , . . . , e ℓ ), even if ℓ = 0 when v is specified. The path p is said to be from v to v ℓ , and to have length ℓ . For each e ∈ EΓ, by the number of times p traverses e ,we mean (cid:12)(cid:12)(cid:8) i ∈ { , , . . . , ℓ } : e i ∈ { e } ± (cid:9)(cid:12)(cid:12) . We call the element e e · · · e ℓ of h EΓ | ∅i theΓ -label of p . If v ℓ = v , then we say that p is a closed path based at v . If e i = e − i − for each i ∈ { , , . . . , ℓ } , then we say that p is a reduced path. or v, w ∈ VΓ, let Γ[ v, w ] denote the set of all Γ-paths from v to w ; we then have the inversion map Γ[ v, w ] → Γ[ w, v ], p p − , where ( e , e , . . . , e ℓ ) − := ( e − ℓ , . . . , e − , e − ). For u, v, w ∈ VΓ, we have the concatenation map Γ[ u, v ] × Γ[ v, w ] → Γ[ u, w ], ( p , p ) p p , where ( e , e , . . . , e ℓ ) e ′ , e ′ , . . . , e ′ m ) := ( e , e , . . . , e ℓ , e ′ , e ′ , . . . , e ′ m ) . If a Γ-path p is closedand p p is reduced, we say that p is cyclically reduced .We say that Γ is a tree if VΓ = ∅ and, for all v, w ∈ VΓ, there exists a unique reducedΓ-path from v to w . We say that Γ is connected if, for all v, w ∈ VΓ, there exists a Γ-pathfrom v to w . By a component of Γ, we mean a maximal nonempty connected subgraphof Γ. Thus, Γ equals the disjoint union of its components. We say that Γ is a forest if eachcomponent of Γ is a tree. Thus, Γ is not a forest if and only if some closed Γ-path traversessome Γ-edge exactly once.For any group G , we say that Γ is a left G -graph if VΓ and EΓ are left G -sets, and ι and τ are left- G -set morphisms; right G -graphs are defined similarly. (cid:3) Recall that F is a finite-rank free group, and that X and Y are F -bases. The finite-rankhypothesis will not be used in this section.2.2. Definitions.
For any g ∈ F , we let · g and g · denote the permutations F → F givenby v vg and v gv respectively. For any subset S of F , we write · S := {· g : g ∈ S } and S · := { g · : g ∈ S } .We let F y Y denote the (Cayley) graph with vertex-set F and edge-set F × · Y , for whicheach edge ( v, · y ) has initial vertex v and terminal vertex vy . The ( F y Y )-paths ( v, ( v, · y ) , vy )and ( vy, ( v, · y ) − , v ) are depicted as v · y −→− vy and vy · y − −−→− v respectively. An ( F y Y )-path p will sometimes be depicted in the form v · y −→− vy · y −→− vy y →− · · · →− vy y · · · y ℓ − · y ℓ −→− vy y · · · y ℓ − y ℓ , for a unique Y ± -sequence σ = ( y , y , . . . , y ℓ ), that is, an ℓ -tuple of elements of Y ± for some ℓ ∈ N . We call σ the right Y ± -label of p . We say that σ is reduced if y i = y − i − for each i ∈ { , , . . . , ℓ } , and that σ is cyclically reduced if ( y , y , . . . , y ℓ , y , y , . . . , y ℓ ) is reduced.Thus, p is a reduced ( F y Y )-path if and only its right Y ± -label is a reduced Y ± -sequence.We let X y F denote the graph with vertex-set F and edge-set X · × F , for which eachedge ( x · , v ) has initial vertex v and terminal vertex xv . The ( X y F )-paths ( v, ( x · , v ) , xv ) and( xv, ( x · , v ) − , v ) are depicted as v x · −→− xv and xv x − · −−→− v respectively. An ( X y F )-path p willsometimes be depicted in the form v x · −→− x v x · −→− x x v →− · · · →− x ℓ − · · · x x v x ℓ · −→− x ℓ x ℓ − · · · x x v, for a unique X ± -sequence σ = ( x ℓ , . . . , x , x ), called the left X ± -label of p . Again, p is areduced ( X y F )-path if and only its left X ± -label is a reduced X ± -sequence.We let X y F y Y denote the graph with vertex-set F and edge-set the (disjoint) union of X · × F and F × · Y , with initial and terminal vertices as before. Thus, X y F and F y Y aresubgraphs of X y F y Y which are being amalgamated over their common vertex-set F . (cid:3) Dehn(1910) initiated the study of Cayley graphs of infinite groups, particularly surfacegroups, and he must have known the following at the start.2.3.
Theorem.
The left F -graph F y Y is a tree.Proof (Fox(1953), streamlined by Dicks(1980)). Set T := F y Y . For each ( v, y ) ∈ F × Y , set v ⊗ y := ( v, · y ) ∈ F × · Y = E T ; thus, ι ( v ⊗ y ) = v and τ ( v ⊗ y ) = vy . learly, T is nonempty.Let ∼ denote the inclusion-smallest equivalence relation on V T such that ι ( v ⊗ y ) ∼ τ ( v ⊗ y )for each T -edge v ⊗ y . There exists a left- F -set isomorphism between the set of componentsof T and the set of equivalence classes of ∼ . Also, ∼ is the inclusion-smallest equivalencerelation on F such that v ∼ vy for each ( v, y ) ∈ F × Y . In particular, the equivalence class[1] of 1 satisfies [1] = [ y ] = y · [1] for each y ∈ Y . Hence, the subgroup { f ∈ F : f · [1] = [1] } of F includes Y . Thus, for all f ∈ h Y i = F , [1] = f · [1] = [ f ] . Hence, [1] = F . Thus, T isconnected.For each set S , we let Z [ S ] denote the free Z -module on S . The maps ι, τ : E T → V T induce Z -module morphisms ˆ ι, ˆ τ : Z [ E T ] → Z [ V T ]. For each closed T -path p which tra-verses some T -edge exactly once, the abelianization map h E T | ∅i → Z [ E T ] carries the T -label of p to a nonzero element of the kernel of ˆ τ − ˆ ι . Thus, to show that T is a tree, itsuffices to show that ˆ τ − ˆ ι is injective. Using the natural left F -action on Z [ E T ], we mayform the semi-direct-product group ( F Z [ E T ] { } { } ) with matrix-style multiplication, whereineach element ( a b ) is denoted ⌈ a, b ⌉ . Since Y is an F -basis, there exists a unique group mor-phism F → ( F Z [ E T ] { } { } ), f
7→ ⌈ ϕf, αf ⌉ , such that ⌈ ϕy, αy ⌉ = ⌈ y, ⊗ y ⌉ for each y ∈ Y . Forall f, g ∈ F , ⌈ ϕ ( f g ) , α ( f g ) ⌉ = ⌈ ϕf, αf ⌉⌈ ϕg, αg ⌉ = ⌈ ( ϕf )( ϕg ) , ( ϕf )( αg ) + αf ⌉ . Then ϕ : F → F is the identity map, since ϕy = y and ϕ ( f g ) = ( ϕf )( ϕg ). The map α : F → Z [ E T ] satisfies αy = 1 ⊗ y and α ( f g ) = ( ϕf )( αg ) + αf . Thus, we have a map α : V T → Z [ E T ] such that, for each v ⊗ y ∈ E T , α (cid:0) τ ( v ⊗ y ) (cid:1) − α (cid:0) ι ( v ⊗ y ) (cid:1) = α ( vy ) − α ( v ) = ( ϕv )( αy ) = ( v )(1 ⊗ y ) = v ⊗ y. Now α induces a Z -module morphism ˆ α : Z [ V T ] → Z [ E T ], and the composite Z [ E T ] ˆ τ − ˆ ι −−→ Z [ V T ] ˆ α −→ Z [ E T ]is the identity map on Z [ E T ], since it carries each v ⊗ y ∈ E T to itself. Hence, ˆ τ − ˆ ι isinjective, as desired. (cid:3) Definitions.
For each straight word r in F , there exists some reduced Y ± -sequence( y , y , . . . , y ℓ ) such that y y · · · y ℓ = r . Here,1 · y −→− y · y −→− y y →− · · · →− y y · · · y ℓ − · y ℓ −→− y y · · · y ℓ = r is a reduced ( F y Y )-path from 1 to r , which is unique by Theorem 2.3. Thus, ( y , y , . . . , y ℓ )is unique, and we call it the reduced Y ± -sequence for r . We set Y -length( r ) := ℓ and Y | y -length( r ) := (cid:12)(cid:12)(cid:8) i ∈ { , , . . . , ℓ } : y i ∈ { y } ± (cid:9)(cid:12)(cid:12) , for each y ∈ Y ± .For each cyclic word r in F , there exists some cyclically reduced Y ± -sequence ( y , . . . , y ℓ )such that y y · · · y ℓ ∈ r . Here, ( y , . . . , y ℓ ) is unique up to cyclic permutation, as may be seenby considering the · Y -labelled quotient graphs h g i\ ( F y Y ), g ∈ r , which are all isomorphic.We set Y -length( r ) := ℓ and Y | y -length( r ) := (cid:12)(cid:12)(cid:8) i ∈ { , , . . . , ℓ } : y i ∈ { y } ± (cid:9)(cid:12)(cid:12) for y ∈ Y ± .Recall that R is a finite set of words in F . We set h ( Y ) := P r ∈ R Y -length( r ) and h ( Y | y ) := P r ∈ R Y | y -length( r ). It is clear that h ( Y | y − ) = h ( Y | y ) and h ( Y ) = P y ∈ Y h ( Y | y ). (cid:3) Gersten’s graphs and Whitehead’s proof
Definitions.
Consider any subset V of F . We let X y V , V y Y , and X y V y Y denote thefull subgraphs of X y F , F y Y , and X y F y Y with vertex-set V respectively, where a subgraph Γ of a graph Γ is said to be full if Γ contains every Γ-edge whose initial and terminal vertices ie in Γ . By Theorem 2.3, X y F and F y Y are trees; thus, X y V and V y Y are forests. A subsetof X y F y Y is said to be 1 -containing it it contains 1. We say that V is an ( X, Y ) -translator if V is a 1-containing F -generating set such that X y V and V y Y are trees. In this event, welet ( X y V y Y ) > denote the set of elements of V −{ } which have ( X y V y Y )-valence at least 3.Notice that | V −{ }| > rank( F ), since V −{ } generates F .Clearly, F itself is an ( X, Y )-translator. Let κ denote the minimum value for | V −{ }| as V ranges over the set of all ( X, Y )-translators. If κ > rank( F ), we define d( X, Y ) := κ . Oth-erwise, κ = rank( F ), and we then define d( X, Y ) to be the minimum value for (cid:12)(cid:12) ( X y V y Y ) > (cid:12)(cid:12) as V ranges over the set of all ( X, Y )-translators of cardinal 1+ rank( F ). (cid:3) Lemma (Gersten(1987)) . d( X, Y ) ∈ N .Proof (Krsti´c(1989), here streamlined). For each finite 1-containing subset W of F , we let˘ X W and ˘ Y W denote the vertex-sets of the 1-containing components of the forests X y W and W y Y respectively; also, we let X W and Y W denote the vertex-sets of the tree-closuresof W in the trees X y F and F y Y respectively, where the tree-closure of W in a tree is theinclusion-smallest subtree which includes W . We have now defined four self-maps of the setof finite 1-containing subsets of F .Set ˜ Y := { } ∪ Y ± and V := ˘ YX Y X ˜ Y .Then V is a finite 1-containing subset of F , V y Y is a tree, and(3.1) V = ˘ Y (cid:0) X ( Y X ˜ Y ) (cid:1) ⊇ ˘ Y (cid:0) ( Y X ˜ Y ) (cid:1) = Y X ˜ Y ⊇ X ˜ Y ⊇ ˜ Y .
In particular, V is an F -generating set.We now prove that ( ˘ X V ) · ˜ Y ⊆ ˘ X ( V · ˜ Y ). Let y ∈ ˜ Y and v ∈ ˘ X V ; thus, V ⊇ X { v, } . Then V ⊇ X { v, , y − } , since V ⊇ X ˜ Y , by (3.1). Now V · ˜ Y ⊇ ( X { v, , y − } ) · y = X { v · y, y, } ,since X y F is a right F -tree. Thus, v · y ∈ ˘ X ( V · ˜ Y ), as desired.It follows from the definition of ˘ Y that V is the inclusion-smallest 1-containing subset of F such that X Y X ˜ Y ∩ V · ˜ Y ⊆ V . Now ˘ X V is a 1-containing subset of V , and X Y X ˜ Y ∩ ( ˘ X V ) · ˜ Y ⊆ ˘ X ( X Y X ˜ Y ) ∩ ˘ X ( V · ˜ Y ) ⊆ ˘ X ( X Y X ˜ Y ∩ V · ˜ Y ) ⊆ ˘ X ( V ) . It follows from the minimality property of V that ˘ X V = V . Thus, X y V is a tree.Hence, V is a finite ( X, Y )-translator. (cid:3)
Theorem (Whitehead(1936b)) . With
Notation 1.1 , if X and Y are local-minimumpoints for h , then either X ± = Y ± or some Whitehead transform Y ′ of Y satisfies h ( Y ′ ) = h ( Y ) and d( X, Y ′ ) < d( X, Y ) .Proof (Whitehead(1936b), here translated). For all v, g ∈ F , we let v X : g · – · · · →− g · v denote theunique reduced ( X y F )-path from v to g · v , and v · g : Y – · · · →− v · g denote the unique reduced( F y Y )-path from v to v · g . If g = 1, then these paths have length zero.We shall obtain information about Whitehead transforms of Y that are constructed usinga procedure that depends on d( X, Y ). We begin by describing features that apply wheneverwe have an (
X, Y )-translator V .For each x ∈ X ± , we set ˆ ι X x := x − · V ∩ V and ˆ τ X x := x · V ∩ V = x · ˆ ι X . For each y ∈ Y ± , we set ˆ ι Y y := V · y − ∩ V and ˆ τ Y y := V · y ∩ V = ˆ ι Y · y .Consider any y ∈ Y ± . We shall now show that X y (ˆ ι Y y ) and X y (ˆ τ Y y ) are subtreesof the tree X y V , and that (cid:0) X y (ˆ ι Y y ) (cid:1) · y = X y (ˆ τ Y y ) . We first show that ˆ ι Y y = ∅ . Since V generates F , there exists some u ∈ V −h Y −{ y } ± i . Let ( y , y , . . . , y ℓ ) be the reduced ± -sequence for u ; thus, there exists some k ∈ { , , . . . , ℓ } such that { y k } ± = { y } ± . Thereduced ( F y Y )-path from 1 to u is then1 = u · y −→− u · y −→− u · · · · y ℓ −→− u ℓ = u ;this is a ( V y Y )-path, since the endpoints lie in V , and the subpath u k − · y k −→− u k meets ˆ ι Y y ,as desired. Now consider any v, w ∈ ˆ ι Y y . Then v · y, w · y ∈ ˆ τ Y y . Let ( x ℓ , x ℓ − , . . . , x ) be thereduced X ± -sequence for w · v − = ( w · y ) · ( v · y ) − . The reduced ( X y F )-paths v = v x · −→− v · · · x ℓ · −→− v ℓ = w and v · y = v · y x · −→− v · y · · · x ℓ · −→− v ℓ · y = w · y are ( X y V )-paths, since their endpoints lie in V . Thus, { v , v , . . . , v ℓ }·{ , y } ⊆ V. This provesthat X y (ˆ ι Y y ) is a subtree of the tree X y V . Also, (cid:0) X y (ˆ ι Y y ) (cid:1) · y = X y (ˆ τ Y y ) , and X y (ˆ τ Y y ) is asubtree of the tree X y V .Analogous assertions hold for (ˆ ι X x ) y Y and (ˆ τ X x ) y Y .Consider any v, w ∈ V and any ( X y V y Y )-path p from v to w . Let ( x · , x · , . . . , x ℓ · ) be thesequence of X ± · -labels encountered along p . We call the X ± -sequence ( x ℓ , . . . , x , x ) the left X ± -label of p , and call g := x ℓ · · · x x the left F -label of p . Let ( · y , · y , . . . , · y ℓ ′ ) be thesequence of · Y ± -labels encountered along p . We call the Y ± -sequence ( y , y , . . . , y ℓ ′ ) the right Y ± -label of p , and call g ′ := y y · · · y ℓ ′ the right F -label of p . It is not difficult to seethat gvg ′ = w in F . We may use ordinary path reductions and assume that p is a reduced( X y V y Y )-path without changing the left and right F -labels. If the right Y ± -label of p isstill not a reduced Y ± -sequence, then p has some subpath p ′ of the form u · y −→− u · y X : h · – · · · →− h · u · y · y − −−→− h · u, for some h ∈ F −{ } . Since X y V is a tree, we have the ( X y V )-path p ′′ which is u X : h · – · · · → h · u .The ( X y V y Y )-path obtained from p by replacing p ′ with p ′′ is said to be a right Y -reductionof p . This gives a shorter ( X y V y Y )-path from v to w with the same left and right F -labels,the same left X ± -label, and a shorter right Y ± -label. Similar considerations give left X -reductions of p . Any ( X y V y Y )-path yields an ( X y V y Y )-path with reduced left X ± - andright Y ± -labels after applying ordinary, left X -, and right Y -reductions sufficiently often.Similar considerations apply for cyclic ordinary, left X -, and right Y -reductions of closed( X y V y Y )-paths; these operations may change where the path is based.We write Paths( X y V y Y ) to denote the set of all ( X y V y Y )-paths. We construct a map F → Paths( X y V y Y ) which assigns to each g ∈ F a closed ( X y V y Y )-path based at 1 whoseleft X ± -label is the reduced X ± -sequence for g − , and whose right Y ± -label is the reduced Y ± -sequence for g . One way to do this is first to choose, for each x ∈ X , some v x ∈ ˆ ι X x ,and then the ( X y V y Y )-path 1 · x · v x : Y – · · · →− x · v x x − · −−→− v x · v − x : Y – · · · →− X ± -label ( x − ), which is the reduced X ± -sequence for x − . Using inversion andconcatenation of paths, we may now assign to each g ∈ F a closed ( X y V y Y )-path based at 1whose left X ± -label is the reduced X ± -sequence for g − . The left F -label is then g − ,and the right F -label must then be g . By applying right Y -reductions, we obtain aclosed ( X y V y Y )-path based at 1 whose left X ± -label is still the reduced X ± -sequencefor g − , whose right Y ± -label is a reduced Y ± -sequence, and whose right F -label isstill g . We call this the chosen ( X y V y Y ) -path representing g . The reduced Y ± -sequencefor g and the reverse of the reduced X ± -sequence for g − have been interlaced to form aclosed ( X y V y Y )-path based at 1. For our counting purposes, the reverse of the reduced ± -sequence for g − contains the same information as the reduced X ± -sequence for g ;previous authors amalgamated F y ( X − ) and F y Y over their vertex-sets via the inversionmap on F .We now construct a map R → Paths( X y V y Y ). We map each straight word r containedin R to the chosen ( X y V y Y )-path representing r . For each cyclic word r contained in R , wechoose an element g of r , and consider the chosen ( X y V y Y )-path representing g , and applycyclic ordinary, left X -, and right Y -reductions, until we get a closed ( X y V y Y )-path whoseright Y ± -label is a cyclically reduced Y ± -sequence and whose left X ± -label is a cyclicallyreduced X ± -sequence; then the right F -label is a conjugate of g , and the left F -label is aconjugate of g − . We call this the chosen ( X y V y Y ) -path representing r .Our map R → Paths( X y V y Y ) gives R the structure of a set of closed ( X y V y Y )-paths,with the straight words being based at 1. We may now speak of the number of times anelement of R traverses a given ( X y V y Y )-edge e , and by summing over all elements of R , wemay speak of the number of times R traverses e , and denote the number by e h ( e ).For each length-one ( X y V )-path v x · −→− w , we let v x · ⇋ w denote the ( X y V )-edge it tra-verses, and set e h ( v x · −→− w ) := e h ( v x · ⇋ w ); thus, w x − · ⇋ v equals v x · ⇋ w , and e h ( w x − · −−→− v )equals e h ( v x · −→− w ). For any element x of X ± , and subsets V and V of V , we set e h ( V x · −→− V ) := P v ∈ V ∩ ( x − · V ) e h ( v x · −→− x · v ) . Notice that e h ( V x · −→− V ) = e h (ˆ ι X x x · −→− ˆ τ X x ) = h ( X | x ), since the left X ± -labels of the chosen( X y V y Y )-paths are reduced, and cyclically reduced for cyclic words.Analogous notation applies with Y in place of X .For any x ∗ ∈ X ± , y ∗ ∈ Y ± , and v ∗ ∈ ˆ ι X x ∗ , we say that ( v ∗ , x ∗ , y ∗ ) is a first-stage triple ,and associate to it all of the following data.The ( X y V )-edge v ∗ x ∗ · ⇋ x ∗ · v ∗ is called the disconnecting edge. Let V denote the vertex-setof the 1-containing component of the forest ( X y V ) − { v ∗ x ∗ · ⇋ x ∗ · v ∗ } , and set V := V − V , thevertex-set of the other component. We let χ : V → { , } denote the characteristic functionof V ; thus, v ∈ V χ ( v ) for each v ∈ V . We define a map ˆ χ : ˆ ι Y ( Y ± ) → { , } as follows. For j ∈ { , } , let Y ± j -part denote the set of those y ∈ Y ± −{ y ∗ } such that χ restricted to ˆ ι Y y takes exactly j values. For each y ∈ Y ± , χ restricted to ˆ ι Y y takes exactly one value, andwe define ˆ χ (ˆ ι Y y ) to be that value. Let χ F : F → { , } denote the characteristic functionof the vertex-set of that component of ( X y F ) − { v ∗ x ∗ · ⇋ x ∗ · v ∗ } which does not contain 1; therestriction of χ F to V is then χ . For each y ∈ Y ± , we define ˆ χ (ˆ ι Y y ) := χ F ( v ∗ · y − ). Tocomplete the definition of the map ˆ χ : ˆ ι Y ( Y ± ) → { , } , we set ˆ χ (ˆ ι Y y ∗ ) := 1 − ˆ χ (ˆ ι Y y − ∗ ).Let y † denote the element of { y ∗ } ± such that ˆ χ (ˆ ι Y y † ) = 0; hence, ˆ χ (ˆ τ Y y † ) = ˆ χ (ˆ ι Y y − † ) = 1.For each y ∈ Y ± −{ y † } ± , we set y ′ := y ˆ χ (ˆ ι Y y ) † · y · y − ˆ χ (ˆ τ Y y ) † , while, for each y ∈ { y † } ± , we set y ′ := y . We then set Y ′ := { y ′ | y ∈ Y } . Thus, Y ′ is a Whitehead transform of Y . Since Y is a local-minimum point for h , h ( Y ) h ( Y ′ ). It is not difficult to see from the definitionof Y ′ that, for each y ∈ Y ± −{ y † } ± , h ( Y ′| y ′ ) > h ( Y | y ). Similarly, h ( Y | y ) > h ( Y ′| y ′ ), and, hence,equality holds. Now(3.2) 0 h ( Y ′ ) − h ( Y ) = h ( Y ′| y ′† ) − h ( Y | y † ) . e next define a map ξ : Paths( X y V y Y ) → Paths( X y F y Y ′ ). It suffices to define ξ on V and on the set of length-one ( X y V y Y )-paths, and then concatenate paths.We define ξ on V by V = V ∪ V → V ∪ V · y − † ⊆ F, v ξ ( v ) := v · y − χ ( v ) † . Consider a length-one ( X y V y Y )-path of the form v · y −→− w , y ∈ Y ± . We define ξ ( v · y −→− w )to be ξ ( v ) · ξ ( v ) − · ξ ( w ): Y ′ —– · · · · · →− ξ ( w ). Notice that y ˆ χ (ˆ ι Y y ) † · y · y − ˆ χ (ˆ τ Y y ) † = y ′ δ ( y ) where δ ( y ) := ( y ∈ Y ± − { y † } ± , y ∈ { y † } ± . As ξ ( v ) = v · y − χ ( v ) † and ξ ( w ) = w · y − χ ( w ) † = v · y · y − χ ( w ) † = ξ ( v ) · y χ ( v ) † · y · y − χ ( w ) † = ξ ( v ) · y χ ( v ) − ˆ χ (ˆ ι Y y ) † · y ′ δ ( y ) · y ˆ χ (ˆ τ Y y ) − χ ( w ) † , (3.3) ξ ( v · y −→− w ) equals ξ ( v ) · y †′ χ ( v ) − ˆ χ (ˆ ιY y ) · y ′ δ ( y ) · y †′ ˆ χ (ˆ τY y ) − χ ( w ) : Y ′ ——– · · · · · · · · · · · · · · · · · · · · →− ξ ( w ) . Consider now a length-one ( X y V y Y )-path of the form v x · −→− w , x ∈ X ± . Here, ξ ( v ) = v · y − χ ( v ) † and ξ ( w ) = w · y − χ ( w ) † = x · v · y − χ ( w ) † = x · ξ ( v ) · y χ ( v ) − χ ( w ) † . We shall define ξ ( v x · −→− w ) to be ξ ( v ) x · −→− x · ξ ( v ) · y †′ χ ( v ) − χ ( w ) : Y ′ — · · · · · · →− ξ ( w )(3.4) or ξ ( v ) · y †′ χ ( v ) − χ ( w ) : Y ′ — · · · · · · →− ξ ( v ) · y †′ χ ( v ) − χ ( w ) x · −→− ξ ( w ) . Recall that χ ( v ) = χ ( w ) unless v x · ⇋ w is the disconnecting edge v ∗ x ∗ · ⇋ x ∗ · v ∗ . If χ ( v ) = χ ( w ),then ξ ( v x · −→− w ) equals ξ ( v ) x · −→− ξ ( w ). Later, we shall have enough information to choosebetween the two options and define ξ ( v ∗ x ∗ · −→− x ∗ · v ∗ ) precisely.Now ξ will convert ( X y V y Y )-paths into ( X y F y Y ′ )-paths without changing the left X ± -labels, and, hence, without changing the left F -labels. Since ξ (1) = 1 · y − χ (1) † = 1, wesee that R is now represented by closed ( X y F y Y ′ )-paths. We do not claim that the right Y ′± -labels are reduced, but the image of R in Paths( X y F y Y ′ ) does give an upper boundfor h ( Y ′| y ′† ). On carefully considering (3.4) and (3.3), and noting that the y ′ δ ( y ) -terms con-tribute no y ′† -terms, we see that h ( Y ′| y ′† ) e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) + P y ∈ Y ± e h ( V − ˆ χ (ˆ ι Y y ) · y −→− V ) . Since h ( Y | y † ) = h ( Y | y ∗ ), we see from (3.2) that(3.5) 0 h ( Y ′ ) − h ( Y ) e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) − h ( Y | y ∗ ) + P y ∈ Y ± e h ( V − ˆ χ (ˆ ι Y y ) · y −→− V ) . We now consider two cases.
Case 1: d( X, Y ) rank F .Here, we assume that | V −{ }| = rank F and (cid:12)(cid:12) ( X y V y Y ) > (cid:12)(cid:12) = d( X, Y ).Since V y Y is a tree, we have P y ∈ Y | ˆ ι Y y | = | E( V y Y ) | = | V |− | Y | . For each y ∈ Y , | ˆ ι Y y | >
1; hence, | ˆ ι Y y | = 1. Here in Case 1, for each y ∈ Y ± , we write ι Y y to denote theunique element of ˆ ι Y y , and similarly for τ Y y , and analogously with X in place of Y .As an abelian group, F/ [ F, F ] is freely generated by the image of any F -basis. Hence,there exists a unique map n X,Y : X × Y → Z , ( x, y ) n x,y , such that, for each y ∈ Y , y · [ F, F ] = Q x ∈ X (cid:0) ( x · [ F, F ]) n x,y (cid:1) in F/ [ F, F ]; e set X -absupp( y ) := { x ∈ X | n x,y = 0 } . By choosing bijections from { , , . . . , rank F } to X and to Y , we may view the map n X,Y as an invertible matrix over Z , and view everybijection ϕ : X ∼ −→ Y , x ϕx , as a permutation of { , , . . . , rank F } . Then P ϕ : X ∼ −→ Y (sign( ϕ ) · Q x ∈ X n x,ϕx ) = Det( n X,Y ) ∈ { , − } . There thus exists some bijection ψ : X ∼ −→ Y such that Q x ∈ X n x,ψx = 0; we fix such a ψ throughout Case 1. Hence, x ∈ X -absupp( ψx ) for each x ∈ X .Consider any x ∗ ∈ X , and set y ∗ := ψ ( x ∗ ) ∈ Y and v ∗ := ι X x ∗ ∈ V . We say that ( v ∗ , x ∗ , y ∗ )is a second-stage Case triple . We have all the data associated to a first-stage triple.Let us first show that, for each y ∈ Y ± , ˆ χ (ˆ ι Y y ) = χ ( ι Y y ). Clearly Y ± = ∅ ; hence, if y ∈ Y ± −{ y ∗ } = Y ± , then ˆ χ (ˆ ι Y y ) = χ ( ι y y ), as desired. It remains to consider y ∗ . Nowˆ χ (ˆ ι Y y ∗ ) = 1 − ˆ χ (ˆ ι Y y − ∗ ) = 1 − χ ( ι Y y − ∗ ) = 1 − χ ( τ Y y ∗ ) , and it suffices to show that χ ( τ Y y ∗ ) = χ ( ι Y y ∗ ). Let ( x ℓ , . . . , x , x ), ℓ ∈ N , be the reduced X ± -sequence for ( τ Y y ∗ ) · ( ι Y y ∗ ) − . Then ι Y y ∗ · y ∗ · ( ι Y y ∗ ) − = ( τ Y y ∗ ) · ( ι Y y ∗ ) − = x ℓ · · · x x . Hence, y ∗ · [ F, F ] = Q ℓk =1 ( x k · [ F, F ]). Since x ∗ ∈ X -absupp( ψ ( x ∗ )) and ψ ( x ∗ ) = y ∗ , there existssome k ∈ { , , . . . , ℓ } such that { x k } ± = { x ∗ } ± . The reduced ( X y F )-path ι Y y ∗ = v x · −→− v x · −→− · · · x ℓ − · −−−→− v ℓ − x ℓ · −→− v ℓ = x ℓ · · · x · ι Y y ∗ = τ Y y ∗ is the unique reduced ( X y V )-path from ι Y y ∗ to τ Y y ∗ , and it traverses v k − x k · ⇋ v k , which is v ∗ x ∗ · ⇋ x ∗ · v ∗ , which is the disconnecting edge. Hence, χ ( ι Y y ∗ ) = χ ( τ Y y ∗ ), as desired.Since y ∗ = ψ ( x ∗ ), here in Case 1, (3.5) takes the form(3.6) 0 h ( Y ′ ) − h ( Y ) h ( X | x ∗ ) − h ( Y | ψ ( x ∗ ) ) . Since x ∗ is arbitrary, 0 h ( X | x ) − h ( Y | ψx ) for each x ∈ X . Thus,0 P x ∈ X (cid:0) h ( X | x ) − h ( Y | ψx ) (cid:1) = h ( X ) − h ( Y ) . By the interchangeability of X and Y , we then have h ( X ) − h ( Y ) = 0. It follows in turnthat h ( X | x ) − h ( Y | ψx ) = 0 for each x ∈ X . By (3.6), h ( Y ′ ) = h ( Y ), as desired.Consider the subcase where, for each y ∈ Y ± such that ι Y y = 1, the element τ Y y of V has( X y V y Y )-valence exactly two, and therefore ( X y V )-valence exactly one and ( V y Y )-valenceexactly one. The latter means that V ± −{ } = Y ± , and the former then means that V ± −{ } = X ± . We then have X ± = Y ± , which is one of the desired conclusions; here,d( X, Y ) = 0. It remains to consider the subcase where there exists some y † ∈ Y ± such that ι Y y † = 1 and τ Y y † ∈ ( X y V y Y ) > . We fix such a y † , and take y ∗ ∈ Y ∩ { y † } ± , x ∗ := ψ − ( y ∗ ),and v ∗ := ι X x ∗ . We say that ( v ∗ , x ∗ , y ∗ ) is a third-stage Case triple .By (3.3), for each y ∈ Y ± − { y † } ± , we have ξ ( ι Y y · y −→− τ Y y ) equals ξ ( ι Y y ) · y ′ −→− ξ ( τ Y y ),while ξ ( ι Y y † · y † −→− τ Y y † ) equals ξ ( ι Y y † ) · Y ′ – · · · →− ξ ( τ Y y † ).Set x † := x ( − χ ( v ∗ ) ∗ ; thus, ι X x † x † · ⇋ τ X x † equals v ∗ x ∗ · ⇋ x ∗ · v ∗ , χ ( ι X x † ) = 0, and χ ( τ X x † ) = 1 . In (3.4), for each x ∈ X ± − { x † } ± , ξ ( ι X x x · −→− τ X x ) equals ξ ( ι X x ) x · −→− ξ ( τ X x ), while we nowchoose ξ ( ι X x † x † · −→− τ X x † ) to be equal to ξ ( ι X x † ) x † · −→− τ X x † · y ′− † −−−→− ξ ( τ X x † ). et V ′ := ξ ( V ) ∪ ˆ τ X x † = V ∪ V · y − † ∪ ˆ τ X x † ⊆ F . We shall see that V ′ is an ( X, Y ′ )-trans-lator. Since ˆ τ X x † ⊆ V , we see that V ′ is a finite, 1-containing, F -generating set. Thus, | V ′ | > | V | . Since ˆ ι Y y † ⊆ V ∩ V · y − † , we see that | V ′ | = | V | and τ X x † V ∪ V · y − † .It is clear that ξ (cid:0) Paths( X y V y Y ) (cid:1) ⊆ Paths( X y V ′ y Y ′ ). Let us examine the graphs X y V ′ y Y ′ , X y V ′ , and V ′ y Y ′ . From the form that ξ takes here, we see that X y V ′ y Y ′ is obtainedfrom X y V y Y by first subdividing the edge ι X x † x † · ⇋ τ X x † , and secondly collapsing the edge ι Y y † · y † ⇋ τ Y y † . The graph X y V ′ is thus obtained from the tree X y V by first removing an edge,leaving two components with vertex-sets V and V , secondly identifying one vertex of V with one vertex of V , and thirdly attaching one new vertex and one new edge incident tothe new vertex and an old vertex. Hence, X y V ′ is a tree. The graph V ′ y Y ′ is obtained fromthe tree V y Y by first collapsing one edge identifying its vertices, and secondly attaching onenew vertex and one new edge incident to the new vertex and an old vertex. Thus, V ′ y Y ′ isa tree. Hence, V ′ is an ( X, Y ′ )-translator.Finally, (cid:12)(cid:12) ( X y V y Y ) > (cid:12)(cid:12) > (cid:12)(cid:12) ( X y V ′ y Y ′ ) > (cid:12)(cid:12) , since the newly created vertex has ( X y V ′ y Y ′ )-va-lence two, while the two old vertices which become identified are τ Y y † ∈ ( X y V y Y ) > and ι Y y † = 1. Hence, d( X, Y ) > d( X, Y ′ ). Case 2: d( X, Y ) > rank F .Here, we assume that | V −{ }| = d( X, Y ). Hence, | V −{ }| > rank F = | Y | .Since V y Y is a tree, P y ∈ Y | ˆ ι Y y | = | E( V y Y ) | = | V |− > | Y | . There then exists some y ∗ ∈ Y ± such that | ˆ ι Y y ∗ | >
2. The tree X y (ˆ ι Y y ∗ ) must then contain some edge v ∗ x ∗ · ⇋ x ∗ · v ∗ ,and then the tree X y (ˆ τ Y y ∗ ) = (cid:0) X y (ˆ ι Y y ∗ ) (cid:1) · y ∗ contains the edge v ∗ · y ∗ x ∗ · ⇋ x ∗ · v ∗ · y ∗ , giving adiagram v ∗ · y ∗ x ∗ · −−−→− x ∗ · v ∗ · y ∗· y ∗ x x · y ∗ v ∗ x ∗ · −−−→− x ∗ · v ∗ of length-one ( X y V y Y )-paths. We say that ( v ∗ , x ∗ , y ∗ ) is a second-stage Case triple . Wenow have all the data associated with a first-stage triple.If y − ∗ ∈ Y , then ˆ χ (ˆ ιy − ∗ ) = χ ( v ∗ · y ∗ ), because v ∗ · y ∗ ∈ ˆ ιy − ∗ . If y − ∗ ∈ Y ± , then,by definition, ˆ χ (ˆ ιy − ∗ ) = χ F ( v ∗ · y ∗ ) = χ ( v ∗ · y ∗ ). This proves that ˆ χ (ˆ ιy − ∗ ) = χ ( v ∗ · y ∗ ); hence,ˆ χ (ˆ ιy ∗ ) = 1 − χ ( v ∗ · y ∗ ).Let west ( v ∗ ,x ∗ ) (ˆ ι Y y ∗ ) and east ( v ∗ ,x ∗ ) (ˆ ι Y y ∗ ) denote the vertex-sets of the components of (cid:0) X y (ˆ ι Y y ∗ ) (cid:1) − { v ∗ x ∗ · ⇋ x ∗ · v ∗ } which contain v ∗ and x ∗ · v ∗ respectively. Let proper( v ∗ , x ∗ , y ∗ )denote the intersection of ˆ ι Y y ∗ with the component of (cid:0) X y F (cid:1) − { v ∗ x ∗ · ⇋ x ∗ · v ∗ } which does not contain v ∗ · y ∗ and, hence, intersects V in V − χ ( v ∗ · y ∗ ) . Since ˆ χ (ˆ ιy ∗ ) = 1 − χ ( v ∗ · y ∗ ),proper( v ∗ , x ∗ , y ∗ ) = ˆ ι Y y ∗ ∩ V ˆ χ (ˆ ι Y y ∗ ) ∈ { west ( v ∗ ,x ∗ ) (ˆ ι Y y ∗ ) , east ( v ∗ ,x ∗ ) (ˆ ι Y y ∗ ) } .Let south ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) and north ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) denote the vertex-sets of the components of (cid:0) (ˆ ι X x ∗ ) y Y (cid:1) − { v ∗ · y ∗ ⇋ v ∗ · y ∗ } which contain v ∗ and v ∗ · y ∗ respectively. It is not difficult toshow that south ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) = { v ∗ } if and only if v ∗ has ( X y V )-valence one, if and only if Y ± = ∅ . e now consider an arbitrary y ∈ Y ± . Thus, ˆ χ (ˆ ι Y y ) = χ F ( v ∗ · y − ). We have a diagram v ∗ · y ∗ x ∗ · −−−→− x ∗ · v ∗ · y ∗· y ∗ x x · y ∗ v ∗ x ∗ · −−−→− x ∗ · v ∗· y y y · y v ∗ · y x ∗ · −−−→− x ∗ · v ∗ · y of length-one ( X y V y Y )-paths. Notice that ( v ∗ · y, x ∗ , y − ) is a second-stage Case 2 triple,and that proper( v ∗ · y, x ∗ , y − ) is then the intersection of ˆ ι Y y − with that component of (cid:0) X y F (cid:1) − { v ∗ · y x ∗ · ⇋ x ∗ · v ∗ · y } which does not contain v ∗ . On right multiplying by y − ,we see that (cid:0) proper( v ∗ · y, x ∗ , y − ) (cid:1) · y − is the intersection of ˆ ι Y y with that componentof (cid:0) X y F (cid:1) − { v ∗ x ∗ · ⇋ x ∗ · v ∗ } which does not contain v ∗ · y − and, hence, intersects V in V − χ F ( v ∗ · y − ) . Since ˆ χ (ˆ ι Y y ) = χ F ( v ∗ · y − ), (cid:0) proper( v ∗ · y, x ∗ , y − ) (cid:1) · y − = ˆ ι Y y ∩ V − ˆ χ (ˆ ι Y y ) .Now P y ∈ Y ± e h ( V − ˆ χ (ˆ ι Y y ) · y −→− V )= e h ( V − ˆ χ (ˆ ι Y y ∗ ) · y ∗ −→− V ) + P y ∈ Y ± e h ( V − ˆ χ (ˆ ι Y y ) · y −→− V )= h ( Y | y ∗ ) − e h ( V ˆ χ (ˆ ι Y y ∗ ) · y ∗ −→− V ) + P y ∈ Y ± e h (cid:16)(cid:0) proper( v ∗ · y, x ∗ , y − ) (cid:1) · y − · y −→− V (cid:17) = h ( Y | y ∗ ) − e h (cid:0) proper( v ∗ , x ∗ , y ∗ ) · y ∗ −→− V (cid:1) + P y ∈ Y ± e h (cid:0) proper( v ∗ · y, x ∗ , y − ) · y − −−→− V (cid:1) . Thus, here in Case 2, (3.5) takes the form0 h ( Y ′ ) − h ( Y ) e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) − e h (cid:0) proper( v ∗ , x ∗ , y ∗ ) · y ∗ −→− V (cid:1) (3.7) + P y ∈ Y ± e h (cid:0) proper( v ∗ · y, x ∗ , y − ) · y − −−→− V (cid:1) . In particular, e h (cid:0) proper( v ∗ , x ∗ , y ∗ ) · y ∗ −→− V (cid:1) e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) + P y ∈ Y ± e h (cid:0) proper( v ∗ · y, x ∗ , y − ) · y − −−→− V (cid:1) . Since e h (cid:0) south ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) x ∗ · −→− V (cid:1) = e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) + P y ∈ Y ± e h (cid:0) south ( v ∗ · y,y − ) (ˆ ι X x ∗ ) x ∗ · −→− V (cid:1) , it may be seen by induction on | south ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) | that e h (cid:0) proper( v ∗ , x ∗ , y ∗ ) · y ∗ −→− V (cid:1) e h (cid:0) south ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) x ∗ · −→− V (cid:1) . Let us write e h -west := e h (cid:0) west ( v ∗ ,x ∗ ) (ˆ ι Y y ∗ ) · y ∗ −→− V (cid:1) and e h -south := e h (cid:0) south ( v ∗ ,y ∗ ) (ˆ ι X x ∗ ) x ∗ · −→− V (cid:1) , nd similarly for e h -east and e h -north. We have shown thatmin { e h -west , e h -east } e h (cid:0) proper( v ∗ , x ∗ , y ∗ ) · y ∗ −→− V (cid:1) e h -south . Replacing ( v ∗ , x ∗ , y ∗ ) with the second-stage Case 2 triple ( v ∗ · y ∗ , x ∗ , y − ∗ ) interchanges southand north, and we find that min { e h -west , e h -east } e h -north. Hence,min { e h -west , e h -east } min { e h -south , e h -north } . Interchanging X and Y interchanges south and west, as well as north and east, and we find(3.8) min { e h -south , e h -north } min { e h -west , e h -east } e h (cid:0) proper( v ∗ , x ∗ , y ∗ ) · y ∗ −→− V (cid:1) . We now choose a third-stage Case 2 triple as follows. Consider the preceding x ∗ .Thus, (ˆ ι X x ∗ ) y Y is a finite tree that has at least one edge and, hence, at least twovalence-one vertices. There then exists a valence-one (cid:0) (ˆ ι X x ∗ ) y Y (cid:1) -vertex v ∗ such that e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) h ( X | x ∗ ) /
2. Taking v ∗ x ∗ · ⇋ x ∗ · v ∗ as the disconnecting edge determines a map χ : V → { , } . If χ ( v ∗ ) = 0, we fix this x ∗ and this v ∗ . If χ ( v ∗ ) = 1, we replace ( x ∗ , v ∗ ) with( x − ∗ , x ∗ · v ∗ ), and then fix this new x ∗ and v ∗ ; then χ ( v ∗ ) = 0. Now χ ( x ∗ · v ∗ ) = 1. Let y ∗ denotethe element of Y ± such that v ∗ · y ∗ ⇋ v ∗ · y ∗ is the unique edge of (ˆ ι X x ∗ ) y Y that is incident to v ∗ .Now ( v ∗ , x ∗ , y ∗ ) is a second-stage Case 2 triple, Y ± = ∅ , e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) h ( X | x ∗ ) / χ ( v ∗ ) = 0, and χ ( x ∗ · v ∗ ) = 1; we say that ( v ∗ , x ∗ , y ∗ ) is a third-stage Case triple . Since Y ± = ∅ , e h -south = e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) h ( X | x ∗ ) − e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) = e h -north;thus, e h ( v ∗ x ∗ · −→− x ∗ · v ∗ ) = min { e h -south , e h -north } . Also, since Y ± = ∅ , it follows from (3.7)and (3.8) that h ( Y ′ ) = h ( Y ), as desired.Set V ′ := ξ ( V ) = V ∪ V · y − † . It suffices to show that V ′ is an ( X, Y ′ )-translator with | V ′ | < | V | .Since Y ± = ∅ , (3.3) says that ξ ( v · y −→− v · y ) equals ξ ( v ) · y ′ −→− ξ ( v · y ) if y ∈ Y ± − { y ∗ } ± and v ∈ ˆ ι Y y, and that ξ ( v · y ∗ −→− v · y ∗ ) = ξ ( v ) · Y ′ – · · · →− ξ ( v · y ∗ ) if v ∈ V ˆ χ (ˆ ι Y y ∗ ) ∩ ˆ ι Y y ∗ ,ξ ( v ) · y ′∗ −→− ξ ( v · y ∗ ) if v ∈ V − ˆ χ (ˆ ι Y y ∗ ) ∩ ˆ ι Y y ∗ . By (3.4), ξ ( v x · −→− x · v ) equals ξ ( v ) x · −→− ξ ( x · v ) if x ∈ X ± , v ∈ ˆ ι X x, and v x · ⇋ x · v is not equalto v ∗ x ∗ · ⇋ x ∗ · v ∗ . It remains to specify ξ ( v ∗ x ∗ · −→− x ∗ · v ∗ ). Clearly, v ∗ · y ∗ x ∗ · ⇋ x ∗ · v ∗ · y ∗ is not equalto v ∗ x ∗ · ⇋ x ∗ · v ∗ ; hence, χ ( x ∗ · v ∗ · y ∗ ) = χ ( v ∗ · y ∗ ) = 1 − ˆ χ (ˆ ι Y y ∗ ) ∈ { , } . We then have two sub-cases.If χ ( v ∗ · y ∗ ) = χ ( x ∗ · v ∗ · y ∗ ) = 1 − ˆ χ (ˆ ι Y y ∗ ) = 1, then y † = y ∗ and ξ ( v ∗ · y ∗ ) = v ∗ · y ∗ · y − † = v ∗ , ξ ( x ∗ · v ∗ · y ∗ ) = x ∗ · v ∗ · y ∗ · y − † = x ∗ · v ∗ ,ξ ( v ∗ ) = v ∗ , ξ ( x ∗ · v ∗ ) = x ∗ · v ∗ · y − † = x ∗ · v ∗ · y − ∗ . Here, ξ : V → V ′ is not injective, and we define ξ ( v ∗ x ∗ · −→− x ∗ · v ∗ ) to be ξ ( v ∗ ) x ∗ · −→− ξ ( x ∗ · v ∗ · y ∗ ) · y ′− ∗ −−−→− ξ ( x ∗ · v ∗ )in Paths( X y V ′ y Y ′ ); notice that ξ ( x ∗ · v ∗ · y ∗ ) · y ′− ∗ −−−→− ξ ( x ∗ · v ∗ ) equals ξ (cid:0) x ∗ · v ∗ · y ∗ · y − ∗ −−→− x ∗ · v ∗ (cid:1) . f χ ( v ∗ · y ∗ ) = χ ( x ∗ · v ∗ · y ∗ ) = 1 − ˆ χ (ˆ ι Y y ∗ ) = 0, then y † = y − ∗ and ξ ( v ∗ · y ∗ ) = v ∗ · y ∗ , ξ ( x ∗ · v ∗ · y ∗ ) = x ∗ · v ∗ · y ∗ ,ξ ( v ∗ ) = v ∗ , ξ ( x ∗ · v ∗ ) = x ∗ · v ∗ · y − † = x ∗ · v ∗ · y ∗ . Here, ξ : V → V ′ is not injective, and we define ξ ( v ∗ x ∗ · −→− x ∗ · v ∗ ) to be ξ ( v ∗ ) · y ′∗ −→− ξ ( v ∗ · y ∗ ) x ∗ · −→− ξ ( x ∗ · v ∗ )in Paths( X y V ′ y Y ′ ); notice that ξ ( v ∗ ) · y ′∗ −→− ξ ( v ∗ · y ∗ ) equals ξ (cid:0) v ∗ · y ∗ −→− v ∗ · y ∗ (cid:1) .Since ξ : V → V ′ is surjective, but not injective, | V ′ | < | V | . Notice also that y ∗ ∈ h V ′ i ;hence, V ′ generates F .Let us examine the graphs X y V ′ y Y ′ , X y V ′ , and V ′ y Y ′ . From the form that ξ takes here, wesee that X y V ′ y Y ′ is obtained from X y V y Y by first removing one edge, secondly reattachingit elsewhere, and thirdly collapsing various edges. The graph X y V ′ is thus obtained fromthe tree X y V by first removing an edge, leaving components with vertex-sets V and V ,secondly reattaching the edge elsewhere, and thirdly identifying one or more vertices of V with vertices of V . Hence, X y V ′ is connected, and therefore a tree. The graph V ′ y Y ′ isobtained from the tree V y Y by collapsing edges; hence, V ′ y Y ′ is a tree. Thus, V ′ is an( X, Y ′ )-translator, and d( X, Y ) > d( X, Y ′ ). (cid:3) References
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