Whitehead's algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group Fn. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead. It is still unknown if Whitehead's algorithm has polynomial time complexity.
Statement of the problem
Let be a free group of rank with a free basis. The automorphism problem, or the automorphic equivalence problem for asks, given two freely reduced words whether there exists an automorphism such that.Thus the automorphism problem asks, for whether.
For one has if and only if, where are conjugacy classes in of accordingly. Therefore, the automorphism problem for is often formulated in terms of -equivalence of conjugacy classes of elements of.
For an element, denotes the freely reduced length of with respect to, and denotes the cyclically reduced length of with respect to. For the automorphism problem, the length of an input is measured as or as, depending on whether one views as an element of or as defining the corresponding conjugacy class in.
History
The automorphism problem for was algorithmically solved by J. H. C. Whitehead in a classic 1936 paper, and his solution came to be known as Whitehead's algorithm. Whitehead used a topological approach in his paper. Namely, consider the 3-manifold, the connected sum of copies of. Then, and, moreover, up to a quotient by a finite normal subgroup isomorphic to, the mapping class group of is equal to ; see. Different free bases of can be represented by isotopy classes of "sphere systems" in, and the cyclically reduced form of an element, as well as the Whitehead graph of, can be "read-off" from how a loop in general position representing intersects the spheres in the system. Whitehead moves can be represented by certain kinds of topological "swapping" moves modifying the sphere system.Subsequently, Rapaport, and later, based on her work, Higgins and Lyndon, gave a purely combinatorial and algebraic re-interpretation of Whitehead's work and of Whitehead's algorithm. The exposition of Whitehead's algorithm in the book of Lyndon and Schupp is based on this combinatorial approach. Culler and Vogtmann, in their 1986 paper that introduced the Outer space, gave a hybrid approach to Whitehead's algorithm, presented in combinatorial terms but closely following Whitehead's original ideas.
Whitehead's algorithm
Our exposition regarding Whitehead's algorithm mostly follows Ch.I.4 in the book of Lyndon and Schupp, as well as.Overview
The automorphism group has a particularly useful finite generating set of Whitehead automorphisms or Whitehead moves. Given the first part of Whitehead's algorithm consists of iteratively applying Whitehead moves to 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 of, we check if. If then are not automorphically equivalent in.If, we check if there exists a finite chain of Whitehead moves taking to so that the cyclically reduced length remains constant throughout this chain. The elements are not automorphically equivalent in if and only if such a chain exists.
Whitehead's algorithm also solves the search automorphism problem for. Namely, given, if Whitehead's algorithm concludes that, the algorithm also outputs an automorphism such that. Such an element is produced as the composition of a chain of Whitehead moves arising from the above procedure and taking to.
Whitehead automorphisms
A Whitehead automorphism, or Whitehead move, of is an automorphism of of one of the following two types:There is a permutation of such that for
There is an element, called the multiplier, such that for every
Often, for a Whitehead automorphism, the corresponding outer automorphism in Out| is also called a Whitehead automorphism or a Whitehead move.
Examples
Let.Let be a homomorphism such that
Then is actually an automorphism of, and, moreover, is a Whitehead automorphism of the second kind, with the multiplier.
Let be a homomorphism such that
Then is actually an inner automorphism of given by conjugation by, and, moreover, is a Whitehead automorphism of the second kind, with the multiplier.
Automorphically minimal and Whitehead minimal elements
For, the conjugacy class is called automorphically minimal if for every we have.Also, a conjugacy class is called Whitehead minimal if for every Whitehead move we have.
Thus, by definition, if is automorphically minimal then it is also Whitehead minimal. It turns out that the converse is also true.
Whitehead's "Peak Reduction Lemma"
The following statement is referred to as Whitehead's "Peak Reduction Lemma", see Proposition 4.20 in and Proposition 1.2 in:Let. Then the following hold:
If is not automorphically minimal, then there exists a Whitehead automorphism such that .
Suppose that is automorphically minimal, and that another conjugacy class is also automorphically minimal. Then if and only if and there exists a finite sequence of Whitehead moves such that
and
Part of the Peak Reduction Lemma implies that a conjugacy class is Whitehead minimal if and only if it is automorphically minimal.
The automorphism graph
The automorphism graph of is a graph with the vertex set being the set of conjugacy classes of elements. Two distinct vertices are adjacent in if and there exists a Whitehead automorphism such that. For a vertex of, the connected component of in is denoted.Whitehead graph
For with cyclically reduced form, the Whitehead graph is a labelled graph with the vertex set, where for there is an edge joining and with the label or "weight" which is equal to the number of distinct occurrences of subwords read cyclically in.If is a Whitehead automorphism, then the length change can be expressed as a linear combination, with integer coefficients determined by, of the weights in the Whitehead graph. See Proposition 4.6 in Ch. I of. This fact plays a key role in the proof of Whitehead's peak reduction result.
Whitehead's minimization algorithm
Whitehead's minimization algorithm, given a freely reduced word, finds an automorphically minimal such thatThis algorithm proceeds as follows. Given, put. If is already constructed, check if there exists a Whitehead automorphism such that. If such exists, put and go to the next step. If no such exists, declare that is automorphically minimal, with, and terminate the algorithm.
Part of the Peak Reduction Lemma implies that the Whitehead's minimization algorithm terminates with some, where, and that then is indeed automorphically minimal and satisfies.
Whitehead's algorithm for the automorphic equivalence problem
Whitehead's algorithm for the automorphic equivalence problem, given decides whether or not.The algorithm proceeds as follows. Given, first apply the Whitehead minimization algorithm to each of to find automorphically minimal such that and. If, declare that and terminate the algorithm. Suppose now that. Then check if there exists a
finite sequence of Whitehead moves such that
and
This condition can be checked since the number of cyclically reduced words of length in is finite. More specifically, using the breadth-first approach, one constructs the connected components of the automorphism graph and checks if.
If such a sequence exists, declare that, and terminate the algorithm. If no such sequence exists, declare that and terminate the algorithm.
The Peak Reduction Lemma implies that Whitehead's algorithm correctly solves the automorphic equivalence problem in. Moreover, if, the algorithm actually produces an automorphism such that.
Computational complexity of Whitehead's algorithm
- If the rank of is fixed, then, given, the Whitehead minimization algorithm always terminates in quadratic time and produces an automorphically minimal cyclically reduced word such that. Moreover, even if is not considered fixed, the Whitehead minimization algorithm on an input terminates in time.
- If the rank of is fixed, then for an automorphically minimal constructing the graph takes time and thus requires a priori exponential time in. For that reason Whitehead's algorithm for deciding, given, whether or not, runs in at most exponential time in.
- For, Khan proved that for an automorphically minimal, the graph has at most vertices and hence constructing the graph can be done in quadratic time in. Consequently, Whitehead's algorithm for the automorphic equivalence problem in, given runs in quadratic time in.
Applications, generalizations and related results
- Whitehead's algorithm can be adapted to solve, for any fixed, the automorphic equivalence problem for m-tuples of elects of and for m-tuples of conjugacy classes in ; see Ch.I.4 of and
- McCool used Whitehead's algorithm and the peak reduction to prove that for any the stabilizer is finitely presentable, and obtained a similar results for -stabilizers of m-tuples of conjugacy classes in. McCool also used the peak reduction method to construct of a finite presentation of the group with the set of Whitehead automorphisms as the generating set. He then used this presentation to recover a finite presentation for, originally due to Nielsen, with Nielsen automorphisms as generators.
- Gersten obtained a variation of Whitehead's algorithm, for deciding, given two finite subsets, whether the subgroups are automorphically equivalent, that is, whether there exists such that.
- Whitehead's algorithm and peak reduction play a key role in the proof by Culler and Vogtmann that the Culler–Vogtmann Outer space is contractible.
- Collins and Zieschang obtained analogs of Whitehead's peak reduction and of Whitehead's algorithm for automorphic equivalence in free products of groups.
- Gilbert used a version of a peak reduction lemma to construct a presentation for the automorphism group of a free product.
- Levitt and Vogtmann produced a Whitehead-type algorithm for saving the automorphic equivalence problem in a group where is a closed hyperbolic surface.
- If an element chosen uniformly at random from the sphere of radius in , then, with probability tending to 1 exponentially fast as, the conjugacy class is already automorphically minimal and, moreover,. Consequently, if are two such ``generic" elements, Whitehead's algorithm decides whether are automorphically equivalent in linear time in.
- Similar to the above results hold for the genericity of automorphic minimality for ``randomly chosen" finitely generated subgroups of.
- Lee proved that if is a cyclically reduced word such that is automorphically minimal, and if whenever both occur in or then the total number of occurrences of in is smaller than the number of occurrences of, then is bounded above by a polynomial of degree in. Consequently, if are such that is automorphically equivalent to some with the above property, then Whitehead's algorithm decides whether are automorphically equivalent in time.
- The Garside algorithm for solving the conjugacy problem in braid groups has a similar general structure to Whitehead's algorithm, with "cycling moves" playing the role of Whitehead moves.
- Clifford and Goldstein used Whitehead-algorithm based techniques to produce an algorithm that, given a finite subset decides whether or not the subgroup contains a of that is an element of a free generating set of
- Day obtained analogs of Whitehead's algorithm and of Whitehead's peak reduction for automorphic equivalence of elements of right-angled Artin groups.