On Equivalent Words in the Free Group on Two Generators

[Pages:27]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 automorphisms [5],[6]. Furthermore he showed that the lengths of the words obtained after applying each such "Whitehead automorphism" in this sequence are strictly decreasing until 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.

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? A word is an element w 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.

Definition 1.1 A cycle is an inner automorphism S Aut Fn. A cyclic word w represents 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 A minimal word w is reduced under cyclic permutations, i.e. RCP, if, of all cyclic pe?rmutations [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 nonminimal words to be RCP.

?

?

Example 1.7 In F3, aabcbc = cababc , and moreover aabcbc is RCP.

Lau [1] notes the following.

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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

S (y)

=

yx xy xyx y

if y A, y / A, y / {x, x} , if y / A, y A, y / {x, x} , if y, y A, 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 inequality 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 characterizing 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 v and v-1 in w.

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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. (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 d?eterm?ine, for example, that (aa)w = 1, (bb)w = 1, (ab)w = 1, (ab)w = 2, and (bbaa)w aabb w = 1. Additionally (w) = 2.

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, we prove the 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, we have

{xy}w + {xy}w = {yx}w + {yx}w {yx}w + {yx}w = {xy}w + {xy}w .

By definition, {xy}w + {yx}w = (xy)w and {yx}w + {xy}w = (yx)w. We use these 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), where w is a cyclic word. Then

(yy)v = (yxy)w (yx)v = (yx)w + (yy)w (yx)v = (yx)w - (yxy)w (xx)v = (yx)w - (yxy)w + (xx)w - (yxx)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)w (yx)w + (yy)w.

The following result characterizes level automorphisms.

Lemma 2.9 ({y} , x) is a level automorphism on w if and only if (yx)w = (yx)w + (yy)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)w. Similarly ({y} , x) causes additions to w in the syllables yx, and xy, yy, and yy, totaling (yx)w + (yy)w. A level 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.

Remark 2.10 The automorphism ({y, y} , x) cycles by one place words of the forms yevxe and xevye for e {1, -1} and leaves all other words fixed.

Because two-letter automorphisms are cycles, it suffices to consider one-letter automorphisms when determining the minimality of a word in F2. There are eight one-letter automorphisms, but each of these can be written as the product of a cycle and another one-letter automorphism:

Lemma 2.11 ({y} , x) = ({y, y} , x) ({y} , x).

?? Therefore we need only consider the four automorphisms ({a} , b), {a} , b , ({b} , a), and ({b} , a) in characterizing minimal words. Sanchez [3] gives a characterization in the

same vein of thought as the following, but the current presentation is a much easier test to

implement in practice.

Theorem

2.12

w

is

minimal

if

and

only

if

??(ab)w

-

?? ab w

??

min

((aa)w

,

(bb)w).

??

? ?Proof. By Rapaport's theorem, w is minimal if and only if (ab)w ab w + (aa)w,

ab w gives

??(ab()awb)-w

+?a(ba?aw)??w, (b(aa)aw)wa(nbda)|w(b+a)(wbb-)w(,baa)nwd|

(ba)w (bb)w

(ba)w + (bb)w. Combining . By Corollary 2.6, these

these yield

the result.

Corollary 2.13 (ab)w min ((aa)w , (bb)w) for minimal words w with no inverses.

Corollary 2.14 If w and v are minimal words with the same initial letter, then wv is minimal.

Proof. The inequality of Theorem 2.12 holds for w and for v, so it holds for wv by addition because the syllables included are preserved.

56

3 Root Words

A result of Corollary 2.14 was observed by Virnig [4], namely that if w is RCP, then aw is also RCP. This motivates the following definition.

Definition 3.1 An RCP word v is a descendant of an RCP word w if v = [an [w]], n 1, where [w] begins with an a-string.

A descendant is simply a word obtained by increasing the length of an x-string in another word (and possibly applying a cyclic permutation to achieve RCP form). (As we consider descendants we also have ancestors, but words do not necessarily have unique ancestors. For example, aabbaabb has ancestors aabbabb and aabaabb.) Descendants are necessarily minimal by Corollary 2.14.

We are interested, then, in characterizing words that cannot be obtained through descendancy. These will be the essentially new minimal words of a given length. Since they provide the basis of all other words we refer to them as "root" words.

Definition 3.2 A root word is a minimal word that is not a descendant of any minimal word.

The definition of root words is natural enough, but it is not immediately evident that

they will be simple to work with. After all, in order to verify that a word is a root word

from the definition we must check that shortening any x-string in the word does not result

in a minimal word. However, there is a useful characterization of root words: They are

simply the words for which Theorem 2.12 holds for equality.

Theorem

3.3

w

is

a

root

word

if

and

only

if

??(ab)w

-

?? ab w

??

=

(aa)w

= (bb)w.

Proof.

By

Theorem

2.12,

we

have

??(ab)w

- ?ab?w??

(aa)w

and

??(ab)w

-

?? ab w

??

(bb)w.

w is a root word if and only if decrementing (aa)w or (bb)w by shortening some a-string or

b-string of length 2 in w causes w to lose minimality, so one of the inequalities fails under

these conditions. Therefore both hold for equality if and only if w is a root word.

Corollary 3.4 w is a root word if and only if |(yx)w - (yx)w| = (xx)w = (yy)w for any x, y L2, x / {y, y}.

From this characterization we can derive many properties of root words.

Corollary 3.5 If w is a root word, then the weights of the generators in w are equal.

Proof. The only two-letter syllables in w with unequal generator weights are aa, aa, bb, and bb, but by the previous theorem (aa)w = (bb)w.

Corollary 3.6 w is a root word if and only if wn is a root word.

Proof. Multiplying w by itself preserves equality in the Theorem 3.3. Likewise, taking nth roots of wn preserves equality.

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Theorem 3.7 If w is a root word, then |w| is divisible by 4.

?? ??

Proof. (aa)w + (bb)w + (ab)w + ab w + ab w + (ab)w = |w| because these syllables

and their inverses constitute the set of two-l?ette?r syllables. By Corollary 2.5 and Theorem

3.3 this (aa)w =

s??i(mabp)lwifi-es?atob?2w

??(,aaso)w2+??(a2b()aw?b-)w??+ab?2w

??a+b

w

?2

= |w|. (a?b)w +

?By? Theorem 2 ab w = |w|.

3.3 we also?ha?ve If (ab)w ab w

then 4 (ab)w = |w|, and if (ab)w < ab w then 4 ab w = |w|. In either case |w| is divisible

by 4.

Remark 3.8 If w is a minimal alternating word, then w contains a or b.

Proof. Ass?ume w? contains no instances of either a or b. Then w = (ab)n. The automorphism {a} , b maps w an, precluding the minimality of w.

We now show that all minimal alternating words are root words.

Theorem 3.9 The following are equivalent:

(1) w is a minimal alternating word. (2) w is an alternating root word. (3) All four one-letter automorphisms are level on w.

Proo?f. ?Assume (1). If w is a minimal alternating word, then (aa)w = (bb)w = 0, so

(ab)w = ab w by Theorem 2.12. Therefore, by Theorem 3.3, w is a root word, so we have

(2).

??

Assume (2) and let S = ({x} , y). Because (aa)w = (bb)w = 0 and (ab)w = ab w,

we have (xy)w = (xy)w and (xx)w = 0 x, y L2, x / {y, y}. Therefore the number of

cancellations caused by S is equal to the number of additions, and the length of w does not

change. Thus we have (3).

Let a?ll o?ne-letter au?tom?orphisms be level on w in a?cco?rdance with (3)?. ?This implies (ab)w - ab w = (aa)w, ab w - (ab)w = (aa)w, (ab)w - ?ab?w = (bb)w, and ab w - (ab)w = (bb)w by Lemma 2.9, so (aa)w = (bb)w = 0 and (ab)w = ab w. Therefore w is minimal and alternating, so (1) holds.

We can also say something about non-alternating root words.

Lemma 3.10 If w is a non-alternating root word, then exactly two of the four one-letter

automorphisms are level on w.

?

Pro?of.

From

Theorem

3.3,

??(ab)w

-

?? ab w

??

=

(aa)w

= (bb)w.

?? If (ab)w > ?ab?w, then

{a} , b and ({b} , a) are level automorphisms of w by Lemma 2.9. If (ab)w < ab w, then

({a} , b) and ({b} , a) are level automorphisms of w.

The following result gives an upper bound on the length of x-strings in root words.

Lemma

3.11

If

w

is

a

root

word,

then

(w)

|w| 4

+ 1.

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Proof. Let w be x-string is an a-string,

a root word with (w) >

so (aa)w

|w| 4

+?1.

?(ab)w +

|w|

?4

+?

1.

a?b w >

??(Wabe)wca-n

?aa?sbs?uwm???e?=th(aaat)twhe

longest

|w| 4

+

1,

so substituting for (aa)w and (ab)w + ab w into 2 (aa)w + (ab)w + ab w = |w| (from the

proof of Theorem 3.7) gives |w| + 4 |w|. This is a contradiction.

Furthermore,

there

exists

a

root

word

of

length

4n

that

achieves

(w)

=

|w| 4

+ 1,

namely

an+1 (ba)n-1 bn+1, which can be shown to be a root word by Theorem 3.3.

Of all properties of root words discussed this far, the following result perhaps most

concretely justifies their study, for regardless of how they are represented (i.e. in RCP form

or any other), a root word is still a root word.

Theorem 3.12 If w is a root word, then all v w are root words for |v| = |w|.

Proof. Let S = ({y} , x) be a level automorphism on w. (We are only concerned

with level automorphisms because if an automorphism S increases the length of w, then

S (w) is not minimal.) Let v = S (w). By Theorem 2.7, (yy)v = (yxy)w and (xx)v = (yx)w -(yxy)w +(xx)w -(yxx)w, so (yy)v -(xx)v = (yxy)w -(yx)w +(yxy)w -(xx)w +(yxx)w, which simplifies by (yx)w = (yxy)w + (yxy)w + (yxx)w to (yy)v - (xx)v = (yx)w - (yx)w - (xx)w. By (yy)w = (xx)w from Corollary 3.4 and (yx)w = (yx)w + (yy)w from Lemma 2.9, (yx)w - (yx)w - (xx)w = 0, so (yy)v = (xx)v. By Theorem 2.7, | (yx)v - (yx)v| = | (yx)w + (yy)w - (yyx)w + (yxy)w| = | (yx)w - (yx)w + (yyx)w - (yx)w + (yxy)w| = | (yxy)w| = (yy)v. Therefore, by Corollary 3.4, v is a root word. Root words thus remain root words under level one-letter automorphisms. By Whitehead's Theorem, each such

v w is connected to w by a chain of one-letter automorphisms, cycles, and permutations

that leaves the length of w unchanged. Therefore each v is a root word.

4 Equivalence Classes

Throughout this section, the term "equivalence class" is taken to mean "equivalence class of RCP elements." The size of the equivalence class of a word w is therefore the number of RCP words equivalent to w.

Lemma 4.1 If ({y} , x) is a level automorphism on a minimal word w, then the following automorphisms are also level on w:

(1) ({y} , x) if and only if (xx)w = 0, (2) ({x} , y) if and only if w is a root word, (3) ({x} , y) if and only if w is an alternating root word.

Proof. Since ({y} , x) is level on w, we have (z) (yx)w = (yx)w + (yy)w by Lemma 2.9. Suppose ({y} , x) is level on w. Then (yx)w = (yx)w + (yy)w by Lemma 2.9. Adding this to (z) yields (yy)w = 0, and reversing this argument gives (1). Suppose ({x} , y) is level on w. Then (xy)w = (xy)w + (xx)w by Lemma 2.9. We use Theorem 2.7 to obtain (yx)w = (yx)w + (xx)w. Subtracting this from (z) yields (yy)w = (xx)w, and we have |(yx)w - (yx)w| = (yy)w from (z), so w is a root word. Reversing this argument gives (2).

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