Whitehead's algorithm

The automorphism group ${\displaystyle \operatorname {Aut} (F_{n})}$ has a particularly useful finite generating set ${\displaystyle {\mathcal {W}}}$ of Whitehead automorphisms or Whitehead moves. Given ${\displaystyle w,w'\in F_{n}}$ the first part of Whitehead's algorithm consists of iteratively applying Whitehead moves to ${\displaystyle w,w'}$ to take each of them to an ``automorphically minimal" form, where the cyclically reduced length strictly decreases at each step. Once we find automorphically these minimal forms ${\displaystyle u,u'}$ of ${\displaystyle w,w'}$, we check if ${\displaystyle \|u\|_{X}=\|u'\|_{X}}$. If ${\displaystyle \|u\|_{X}\neq \|u'\|_{X}}$ then ${\displaystyle w,w'}$ are not automorphically equivalent in ${\displaystyle F_{n}}$.

If ${\displaystyle \|u\|_{X}=\|u'\|_{X}}$, we check if there exists a finite chain of Whitehead moves taking ${\displaystyle u}$ to ${\displaystyle u'}$ so that the cyclically reduced length remains constant throughout this chain. The elements ${\displaystyle w,w'}$ are not automorphically equivalent in ${\displaystyle F_{n}}$ if and only if such a chain exists.

Whitehead's algorithm also solves the search automorphism problem for ${\displaystyle F_{n}}$. Namely, given ${\displaystyle w,w'\in F_{n}}$, if Whitehead's algorithm concludes that ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'}$, the algorithm also outputs an automorphism ${\displaystyle \varphi \in \operatorname {Aut} (F_{n})}$ such that ${\displaystyle \varphi (w)=w'}$. Such an element ${\displaystyle \varphi \in \operatorname {Aut} (F_{n})}$ is produced as the composition of a chain of Whitehead moves arising from the above procedure and taking ${\displaystyle w}$ to ${\displaystyle w'}$.

A Whitehead automorphism, or Whitehead move, of ${\displaystyle F_{n}}$ is an automorphism ${\displaystyle \tau \in \operatorname {Aut} (F_{n})}$ of ${\displaystyle F_{n}}$ of one of the following two types:

(i) There is a permutation ${\displaystyle \sigma \in S_{n}}$ of ${\displaystyle \{1,2,\dots ,n\}}$ such that for ${\displaystyle i=1,\dots ,n}$

${\displaystyle \tau (x_{i})=x_{\sigma (i)}^{\pm 1}}$

Such ${\displaystyle \tau }$ is called a Whitehead automorphism of the first kind.

(ii) There is an element ${\displaystyle a\in X^{\pm 1}}$, called the multiplier, such that for every ${\displaystyle x\in X^{\pm 1}}$

${\displaystyle \tau (x)\in \{x,xa,a^{-1}x,a^{-1}xa\}.}$

Such ${\displaystyle \tau }$ is called a Whitehead automorphism of the second kind. Since ${\displaystyle \tau }$ is an automorphism of ${\displaystyle F_{n}}$, it follows that ${\displaystyle \tau (a)=a}$ in this case.

Often, for a Whitehead automorphism ${\displaystyle \tau \in \operatorname {Aut} (F_{n})}$, the corresponding outer automorphism in ${\displaystyle \operatorname {Out} (F_{n})}$ is also called a Whitehead automorphism or a Whitehead move.

Let ${\displaystyle F_{4}=F(x_{1},x_{2},x_{3},x_{4})}$.

Let ${\displaystyle \tau :F_{4}\to F_{4}}$ be a homomorphism such that

${\displaystyle \tau (x_{1})=x_{2}x_{1},\quad \tau (x_{2})=x_{2},\quad \tau (x_{3})=x_{2}x_{3}x_{2}^{-1},\quad \tau (x_{4})=x_{4}}$

Then ${\displaystyle \tau }$ is actually an automorphism of ${\displaystyle F_{4}}$, and, moreover, ${\displaystyle \tau }$ is a Whitehead automorphism of the second kind, with the multiplier ${\displaystyle a=x_{2}^{-1}}$.

Let ${\displaystyle \tau ':F_{4}\to F_{4}}$ be a homomorphism such that

${\displaystyle \tau '(x_{1})=x_{1},\quad \tau '(x_{2})=x_{1}^{-1}x_{2}x_{1},\quad \tau '(x_{3})=x_{1}^{-1}x_{3}x_{1},\quad \tau '(x_{4})=x_{1}^{-1}x_{4}x_{1}}$

Then ${\displaystyle \tau '}$ is actually an inner automorphism of ${\displaystyle F_{4}}$ given by conjugation by ${\displaystyle x_{1}}$, and, moreover, ${\displaystyle \tau '}$is a Whitehead automorphism of the second kind, with the multiplier ${\displaystyle a=x_{1}}$.

For ${\displaystyle w\in F_{n}}$, the conjugacy class ${\displaystyle [w]}$ is called automorphically minimal if for every ${\displaystyle \varphi \in \operatorname {Aut} (F_{n})}$ we have ${\displaystyle \|w\|_{X}\leq \|\varphi (w)\|_{X}}$. Also, a conjugacy class ${\displaystyle [w]}$ is called Whitehead minimal if for every Whitehead move ${\displaystyle \tau \in \operatorname {Aut} (F_{n})}$ we have ${\displaystyle \|w\|_{X}\leq \|\tau (w)\|_{X}}$.

Thus, by definition, if ${\displaystyle [w]}$ is automorphically minimal then it is also Whitehead minimal. It turns out that the converse is also true.

The following statement is referred to as Whitehead's "Peak Reduction Lemma", see Proposition 4.20 in [8] and Proposition 1.2 in:[10]

Let ${\displaystyle w\in F_{n}}$. Then the following hold:

(1) If ${\displaystyle [w]}$ is not automorphically minimal, then there exists a Whitehead automorphism ${\displaystyle \tau \in \operatorname {Aut} (F_{n})}$ such that ${\displaystyle \|\tau (w)\|_{X}<\|w\|_{X}}$.

(2) Suppose that ${\displaystyle [w]}$ is automorphically minimal, and that another conjugacy class ${\displaystyle [w']}$ is also automorphically minimal. Then ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'}$ if and only if ${\displaystyle \|w\|_{X}=\|w'\|_{X}}$ and there exists a finite sequence of Whitehead moves ${\displaystyle \tau _{1},\dots ,\tau _{k}\in \operatorname {Aut} (F_{n})}$ such that

${\displaystyle \tau _{k}\cdots \tau _{1}(w)=w'}$

and

${\displaystyle \|\tau _{i}\cdots \tau _{1}(w)\|_{X}=\|w\|_{X}{\text{ for }}i=1,\dots ,k.}$

Part (1) of the Peak Reduction Lemma implies that a conjugacy class ${\displaystyle [w]}$ is Whitehead minimal if and only if it is automorphically minimal.

The automorphism graph ${\displaystyle {\mathcal {A}}}$ of ${\displaystyle F_{n}}$ is a graph with the vertex set being the set of conjugacy classes ${\displaystyle [u]}$ of elements ${\displaystyle u\in F_{n}}$. Two distinct vertices ${\displaystyle [u],[v]}$ are adjacent in ${\displaystyle {\mathcal {A}}}$ if ${\displaystyle \|u\|_{X}=\|v\|_{X}}$ and there exists a Whitehead automorphism ${\displaystyle \tau }$ such that ${\displaystyle [\tau (u)]=[v]}$. For a vertex ${\displaystyle [u]}$ of ${\displaystyle {\mathcal {A}}}$, the connected component of ${\displaystyle [u]}$ in ${\displaystyle {\mathcal {A}}}$ is denoted ${\displaystyle {\mathcal {A}}[u]}$.

For ${\displaystyle 1\neq w\in F_{n}}$ with cyclically reduced form ${\displaystyle u}$, the Whitehead graph ${\displaystyle \Gamma _{[w]}}$ is a labelled graph with the vertex set ${\displaystyle X^{\pm 1}}$, where for ${\displaystyle x,y\in X^{\pm 1},x\neq y}$ there is an edge joining ${\displaystyle x}$ and ${\displaystyle y}$ with the label or "weight" ${\displaystyle n(\{x,y\};[w])}$ which is equal to the number of distinct occurrences of subwords ${\displaystyle x^{-1}y,y^{-1}x}$ read cyclically in ${\displaystyle u}$. (In some versions of the Whitehead graph one only includes the edges with ${\displaystyle n(\{x,y\};[w])>0}$.)

If ${\displaystyle \tau \in \operatorname {Aut} (F_{n})}$ is a Whitehead automorphism, then the length change ${\displaystyle \|\tau (w)\|_{X}-\|w\|_{X}}$ can be expressed as a linear combination, with integer coefficients determined by ${\displaystyle \tau }$, of the weights ${\displaystyle n(\{x,y\};[w])}$ in the Whitehead graph ${\displaystyle \Gamma _{[w]}}$. See Proposition 4.16 in Ch. I of.[8] This fact plays a key role in the proof of Whitehead's peak reduction result.

Whitehead's minimization algorithm, given a freely reduced word ${\displaystyle w\in F_{n}}$, finds an automorphically minimal ${\displaystyle [v]}$ such that ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})v.}$

This algorithm proceeds as follows. Given ${\displaystyle w\in F_{n}}$, put ${\displaystyle w_{1}=w}$. If ${\displaystyle w_{i}}$ is already constructed, check if there exists a Whitehead automorphism ${\displaystyle \tau \in \operatorname {Aut} (F_{n})}$ such that ${\displaystyle \|\tau (w_{i})\|_{X}<\|w_{i}\|_{X}}$. (This condition can be checked since the set of Whitehead automorphisms of ${\displaystyle F_{n}}$ is finite.) If such ${\displaystyle \tau }$ exists, put ${\displaystyle w_{i+1}=\tau (w_{i})}$ and go to the next step. If no such ${\displaystyle \tau }$ exists, declare that ${\displaystyle [w_{i}]}$ is automorphically minimal, with ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w_{i}}$, and terminate the algorithm.

Part (1) of the Peak Reduction Lemma implies that the Whitehead's minimization algorithm terminates with some ${\displaystyle w_{m}}$, where ${\displaystyle m\leq \|w\|_{X}}$, and that then ${\displaystyle [w_{m}]}$ is indeed automorphically minimal and satisfies ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w_{m}}$.

Whitehead's algorithm for the automorphic equivalence problem, given ${\displaystyle w,w'\in F_{n}}$ decides whether or not ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'}$.

The algorithm proceeds as follows. Given ${\displaystyle w,w'\in F_{n}}$, first apply the Whitehead minimization algorithm to each of ${\displaystyle w,w'}$ to find automorphically minimal ${\displaystyle [v],[v']}$ such that ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})v}$ and ${\displaystyle \operatorname {Aut} (F_{n})w'=\operatorname {Aut} (F_{n})v'}$. If ${\displaystyle \|v\|_{X}\neq \|v'\|_{X}}$, declare that ${\displaystyle \operatorname {Aut} (F_{n})w\neq \operatorname {Aut} (F_{n})w'}$ and terminate the algorithm. Suppose now that ${\displaystyle \|v\|_{X}=\|v'\|_{X}=t\geq 0}$. Then check if there exists a finite sequence of Whitehead moves ${\displaystyle \tau _{1},\dots ,\tau _{k}\in \operatorname {Aut} (F_{n})}$ such that

${\displaystyle \tau _{k}\dots \tau _{1}(v)=v'}$

and

${\displaystyle \|\tau _{i}\dots \tau _{1}(v)\|_{X}=\|v\|_{X}=t{\text{ for }}i=1,\dots ,k.}$

This condition can be checked since the number of cyclically reduced words of length ${\displaystyle t}$ in ${\displaystyle F_{n}}$ is finite. More specifically, using the breadth-first approach, one constructs the connected components ${\displaystyle {\mathcal {A}}[v],{\mathcal {A}}[v']}$ of the automorphism graph and checks if ${\displaystyle {\mathcal {A}}[v]\cap {\mathcal {A}}[v']=\varnothing }$.

If such a sequence exists, declare that ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'}$, and terminate the algorithm. If no such sequence exists, declare that ${\displaystyle \operatorname {Aut} (F_{n})w\neq \operatorname {Aut} (F_{n})w'}$ and terminate the algorithm.

The Peak Reduction Lemma implies that Whitehead's algorithm correctly solves the automorphic equivalence problem in ${\displaystyle F_{n}}$. Moreover, if ${\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'}$, the algorithm actually produces (as a composition of Whitehead moves) an automorphism ${\displaystyle \varphi \in \operatorname {Aut} (F_{n})}$ such that ${\displaystyle \varphi (w)=w'}$.