On Equivalent Words in the Free Group on Two Generators Bobbe J. Cooper Taylor University Eric S. Rowland University of California Santa Cruz Advisor: Professor Dennis J. Garity∗ Oregon State University August 15, 2002 Abstract We characterize minimal words in the free group on two generators and prove various results for words that are effectively "new" with respect to word length. Several properties of single-word equivalence classes are given. The size and behavior of equivalence classes is also discussed in relation to the automorphisms between words of an equivalence class. 1 Introduction In 1936 J.H.C. Whitehead proved that if two words in a free group are equivalent under an automorphism, then they are equivalent under a finite sequence of a certain class of au- tomorphisms [5],[6]. Furthermore he showed that the lengths of the words obtained after applying each such "Whitehead automorphism" in this sequence are strictly decreasing un- til the minimal length is attained, after which the automorphisms leave the length fixed. Whitehead’s theorem has led to several studies of equivalent words. The following notation and definitions will be used in discussing these pursuits. Fn = F (x1,x2,...,xn) denotes the free group on the generators x1,x2,...,xn (in • which no relation exists except the trivial one between an element and its inverse). We typically write the explicit generators of Fn as a, b, . and their inverses a, b,... ∗Much thanks goes to Professor Garity for his comments and suggestions throughout the development of this paper. 52 Awordisanelementw Fn. The identity word is represented by 1. • ∈ w v designates equivalence between two words w, v Fn under some automorphism • ∼ ∈ S Aut Fn. ∈ w denotes the length of the word w (after any adjacent inverses are cancelled). The •|length| of the identity word is 0. The set a,b,...,xn, a, b,...,xn of generators and their inverses is referred to as Ln. • © ª Definition 1.1 A cycle is an inner automorphism S Aut Fn.Acyclic word w rep- resents the equivalence class of w under all cycles, i.e.∈ the final n w letters of w can be fronted to obtain the same cyclic word. (The initial letter is thus≤ adjacent| | to the final letter.) We begin studying equivalence classes by examining cyclic words, as this reduces the number of words we must consider. We also restrict ourselves to minimal words, for each word in Fn has a representation of minimal length. Definition 1.2 A cyclic word w Fn is minimal if w S (w) S Aut Fn. ∈ | | ≤ | | ∀ ∈ Automorphisms that fix the length of a word are useful in finding equivalence classes of minimal words. Definition 1.3 An automorphism S is level on a word w if S (w) = w . | | | | Definition 1.4 A Whitehead Type I automorphism is a permutation S Aut Fn acting ∈ on Ln such that S (x)=S (x) x Ln. ∀ ∈ We typically refer to Type I automorphisms simply as permutations. Definition 1.5 A cyclic permutation is a cycle composed with a permutation. A cyclic permutation of a word w is the image of w under a cyclic permutation. [w] denotes a cyclic permutation of w. Using Type I automorphisms we now create a more expansive equivalence relation for words. Definition 1.6 Aminimalwordw is reduced under cyclic permutations, i.e. RCP, if, of all cyclic permutations [w] of w, w itself appears first in the lexicographic ordering specified by Ln = a,b,...,xn, a, b,...,xn . Note that we do© not consider nonminimalª words to be RCP. Example 1.7 In F3, aabcbc = cababc ,andmoreoveraabcbc is RCP. Lau [1] notes the following. £ ¤ 53 Remark 1.8 An RCP word begins with the string an, and furthermore this is the longest single-letter substring in the word. Definition 1.9 Let x Ln and A Ln. S =(A, x) represents a Whitehead Type II ∈ ⊂ automorphism,defined on y Ln as ∈ yx if y A, y/A, y/ x, x , xy if y/∈ A, y ∈ A, y/∈ {x, x} , S (y)= ∈ ∈ ∈ { } xyx if y, y A, y otherwise.∈ Our definition of Type II automorphisms is slightly simpler than that used by past researchers because membership of x in A is immaterial in the actual images under S;we therefore omit the conditions x A and x/A. Generally, we take x, x/A. ∈ ∈ ∈ Definition 1.10 A one-letter automorphism is a Type II automorphism S =(A, x) with the set A containing only a single letter. Example 1.11 The one-letter automorphism ( a ,b) maps a ab and a ba and leaves b, b fixed. { } → → Example 1.12 Let S =( a, b, a, c ,b) Aut F3. S (bac)=bbabbc = ac.(bac is therefore nonminimal.) { } ∈ We now give Whitehead’s theorem. Theorem 1.13 (Whitehead) If w, v Fn such that w v and v is minimal, then there ∈ ∼ exists a sequence S1,S2, ,Sm of Type I and Type II automorphisms such that ··· Sm S2S1 (w)=v and for k m, Sk S2S1 (w) Sk 1 S2S1 (w) , with strict in- ··· ≤ | ··· | ≤ | − ··· | equality unless Sk 1 S2S1 (w) is minimal. − ··· 2 Minimality Definition 2.1 A syllable is a substring of a cyclic word. We write a word in syllable form as a list of its two-letter syllables, where the last syllable is composed of the last letter and the initial letter. Every letter in a word is thus a member of exactly two such syllables. Syllable form allows letter adjacencies to be counted easily, and it is useful in char- acterizing minimality. When considering the effect of an automorphism on the syllable representation of a word, only additions and cancellations between the two letters of each syllable are counted (to avoid redundancy). Notation 2.2 (xy)w denotes the number of occurrences of the two-letter syllables xy and yx in a word w. More generally, (v)w denotes the number of occurrences of the substrings 1 v and v− in w. 54 Definition 2.3 An x-string is a syllable of a word w of the form xn that is not a substring of xn+1 in w (i.e. its length is maximal). A minimal word is alternating if the length of its longest x-string is 1. λ (w) denotes the length of the longest x-string in a cyclic word w. Example 2.4 The syllable form of w = aabbabab is (aa) ab bb ba (ab)(ba) ab ba . One can determine, for example, that (aa)w =1, (bb)w =1, (ab)w =1, (ab)w =2,and (bbaa) aabb =1. Additionally λ (w)=2. ¡ ¢¡ ¢¡ ¢ ¡ ¢¡ ¢ w ≡ w ¡ ¢ We now restrict ourselves to F2 for the remainder of the paper and adopt the convention that x, y L2, x/ y, y . By observing that each a-string and a-string has as its neighbors b or b and∈ that each∈ { b-string} and b-string likewise has as its neighbors a or a,weprovethe following theorem. Theorem 2.5 In a cyclic word w, (xy)w =(yx)w. Proof. Each occurrence of the syllables xy and xy in the syllable representation of w must be followed by either yx or yx (with possibly an intermediate y-string). Similarly each occurrence of the syllables yx and yx in w must be followed by either xy or xy (with possibly an intermediate x-string). Thus, letting xy w be the number of occurrences of the syllable xy in w,wehave { } xy w + xy w = yx w + yx w {yx} + {yx} = {xy} + {xy} . { }w { }w { }w { }w By definition, xy + yx =(xy) and yx + xy =(yx) .Weusethese { }w { }w w { }w { }w w relations and add the above equations to obtain (xy)w =(yx)w. Corollary 2.6 If w is a cyclic word, then (ab)w = ab w and ab w =(ab)w. The following theorem describes the effect of a one-letter¡ ¢ automorphism¡ ¢ on a word. Theorem 2.7 Let S =( y ,x) and v = S (w),wherew is a cyclic word. Then { } (yy)v =(yxy)w (yx)v =(yx)w +(yy)w (yx)v =(yx)w (yxy)w (xx) =(yx) − (yxy) +(xx) (yxx) . v w − w w − w Proof. yy and yy only appear in v as a result of cancellations in yxy and yxy respectively in w. yx and xy remain fixed under S but also arise from yy and yy respectively. Similarly yx and xy remain fixed unless they appear in yxy and yxy respectively. Finally, xx and xx arise from yx and xy respectively unless they appear in yxy and yxy respectively but also stay fixed unless they appear yxx and xxy respectively. The following is a simplified version of a more general theorem (Theorem 7) given by Rapaport [2]. The left side of the inequality counts the number of cancellations in w under the automorphism ( y ,x), while the right side counts the number of additions under that automorphism. { } 55 Theorem 2.8 (Rapaport) w is minimal if and only if for all x, y L2 with x/ y, y , (yx) (yx) +(yy) . ∈ ∈ { } w ≤ w w The following result characterizes level automorphisms. Lemma 2.9 ( y ,x) is a level automorphism on w if and only if (yx) =(yx) +(yy) . { } w w w Proof. The automorphism ( y ,x) causes cancellations in w only in the syllables yx and xy. The total number of cancellations{ } is therefore (yx) . Similarly ( y ,x) causes w { } additions to w in the syllables yx,andxy, yy,andyy, totaling (yx)w +(yy)w.Alevel automorphism fixes the length of the word, so (yx)w =(yx)w +(yy)w, and conversely this equality implies that ( y ,x) is a level automorphism.