Centralizers of automorphisms permuting free generators

Abstract By σ ∈ Skm we denote a permutation of the cycle-type km and also the induced automorphism permuting subscripts of free generators in the free group Fkm. It is known that the centralizer of the permutation σ in Skm is isomorphic to a wreath product Zk ≀ Sm and is generated by its two subgroups: the first one is isomorphic to Zkm $\begin{array}{} \displaystyle Z_k^m \end{array}$, the direct product of m cyclic groups of order k, and the second one is Sm. We show that the centralizer of the automorphism σ ∈ Aut(Fkm) is generated by its subgroups isomorphic to Zkm $\begin{array}{} \displaystyle Z_k^m \end{array}$ and Aut(Fm).


Introduction
This paper was inspired by a question of Vitaly Sushchanskyy who asked about the structure of centralizers of automorphisms permuting free generators in a free group.
Another motivation of the present paper are numerous papers in which the authors investigate centralizers of nite subgroups of both Aut(Fn) and Out(Fn). K. Vogtmann's paper [1] is the vast survey on this and related topics with many references. We present here a few results. If G is a nite subgroup of Aut(Fn), then E = Fn G is a nite extension of Fn. In general, if E is given by the short exact sequence → Fn → E → K → , then conjugation action of E on Fn induces a homomorphism θ : K → Out(Fn). There exists a connection between the group of outer automorphisms of E and the centralizer of θ(K). For example, it is known (see in [2,3]) that Out(E) is nite if and only if the centralizer of θ(K) is nite. D. Boutin in [2] and M. Pettet in [3] describe nite extensions E of a free group Fn for which the group Out(E) is nite. In the present paper, if an automorphism of Fn is induced by a permutation σ of the cycle-type k m (m cycles of length k), then the centralizer of σ in Aut(Fn) is nite if and only if σ is a long cycle (e.g. it is the cycle without xed points). Since the automorphism induced by the permutation σ of the cycle-type k m has no xed points, there are no non-trivial inner automorphisms in the centralizer C(σ). Hence the subgroup C(σ) < Aut(Fn) is isomorphic to its image in Out(Fn). Y. Algom-K r and C. Pfa study in [4] a centralizer of a subgroup of Out(Fn) cyclically generated by lone axis fully irreducible outer automorphisms. The centralizer of such group is in nite cyclic group. It follows from the main theorem of the present paper that the centralizer of an automorphism induced by a permutation of a cycle-type k m is either nite cyclic or in nite non-abelian. S. Krstič proves in [5] (Theorem 2) that the centralizer of a nite subgroup of Aut(Fn) is nitely presentable.
Let σ be a permutation of the cycle-type k m (m cycles of length k). We use the same letter σ for the induced automorphism of the free group F km with standard basis X = {x , . . . , x km }. The automorphism σ acts on the basis X by permuting subscripts of generators, which we write as: We describe here the structure of the centralizer C(σ) ⊆ Aut(F km ) of the automorphism σ induced by the permutation σ. It is known that the centralizer of σ in S km is isomorphic to the wreath product Z k Sm of a cyclic group of order k and a symmetric group Sm (see for example [6,7] for details) and it is generated by subgroups isomorphic to Z m k and Sm (in fact C(σ) ∼ = Z m k Sm). We show that the centralizer of the automorphism induced by σ in the free group F km has a similar structure and is generated by subgroups isomorphic to Z m k and Aut(Fm) but unfortunately in this case Z m k is not a normal subgroup of Aut(Fm).

Notation
Since the structure of the centralizer C(σ) does not depend on the contents of cycles of the permutation σ, we shall assume: The elements of the basis X of the free group F km can be written as the entries of the k×m-matrix with columns denoted by X i so that the columns correspond to the σ-orbits.
Note that the subscripts of the generators in columns are the elements of orbits of the permutation σ. Note also, that the rst row of this matrix de nes the other rows by applying the automorphism σ.

Properties of automorphisms commuting with σ.
If A := {a , a , ..., a km } is another basis in F km , then the action of σ on a i is de ned by its action on X. We are interested in the case, when the basis A is also a union of σ-orbits.
De nition 1. The basis A := {a , a , ..., a km } in F km is called a σ-basis if it can be ordered so that that is for σ of the form ( ) the σ-basis A can be written as We formulate now a criterion for an automorphism α ∈ Aut(Fn) to centralize the automorphism σ ∈ Aut(Fn).

commutes with the automorphism σ if and only if A is a σ-basis.
Proof. Let α : . So the equality ασ = σα holds if and only if (a i ) σ = a σ(i) as required.

Corollary 1. An automorphisms α is in C(σ) if and only if it maps each orbit
Example 2. An automorphism δ, acting as σ in only one σ-orbit commutes with σ, hence it is the σautomorphism and is in C(σ).
To de ne the notion of a Nielsen transformation, we recall that in matrix theory, to switch i-th and j-th rows in a matrix M, we multiply M from the left by the identity matrix with switched i-th and j-th rows. The action of the Nielsen transformations is similar, namely: de nes a Nielsen transformation Nα acting on another m-tuple (b , b , ..., bm) by multiplication from the left, as in (7), the group of Nielsen transformations and the group of automorphisms Aut(Fn) are anti-isomorphic [8] (sec 3.2, (11)) and the following equality holds The elementary m-tuples (6) de ne the so called elementary automorphisms and elementary Nielsen transformations.

SE-automorphisms
Let Fm , F σ m , · · · , F σ k− m be the free subgroups generated by the elements of the rows in the matrix ( ), then If the same elementary automorphism τ acts on every row (in every F σ i m ), then it de nes a so-called simultaneous elementary automorphism which we address as a SE-automorphism in F km [9].
Note that if the same τ acts on every row, then it acts on the row of columns, which generate the free group Fm of rank m Fm := X , X , . . . , Xm .
So the elementary automorphism τ in Fm induces the above SE-automorphism in F km . We can write this as Corollary 2. An elementary automorphism τ in the free m-generator group Fm = X , X , . . . , Xm de nes the SE-automorphism in F km = Fm * · · · * F σ k− m acting as τ in each of k factors F σ i m .

Proposition 2. The SE-automorphisms (a), (b), (c) in Aut(F km ) generate a subgroup in the centralizer C(σ) isomorphic to Aut(Fm).
Proof. Since by [8] (Theorem 3.2 on page 131) the elementary automorphisms generate the full automorphism group, it follows by Corollary 2 that the group generated by all SE-automorphisms in F km = Fm * F σ m *· · ·* F σ k− m is isomorphic to the group Aut( Fm) and hence to Aut(Fm). Each elementary automorphism in Fm has one of the forms (a), (b), (c), changes the basis X into a σ-basis X τ , and by Proposition 1 is in the centralizer of σ, which nishes the proof.
The subgroup generated by the SE-automorphisms is the proper subgroup in the centralizer C(σ), since it does not contain automorphisms δ i , which maps each generator a in the i − th column to a σ (vertical permutations in i − th columns). So we have to consider also this type of σ-automorphisms mentioned in Example 4.
(d) for a xed i and all y ∈ X i , y δ i := y σ , that is δ i acts on X i as a cyclic permutation of order k, while the other orbits are xed, X δ i := X , X , . . . , X σ i , . . . , X j , . . . , Xm .

Proposition 3. The automorphisms of the type (d) form a subgroup isomorphic to the direct product Z m k of m cyclic groups of order k.
Note The σ-automorphism (a), (b), (c) act in rows of the basis-matrix, while (d) permutes vertically elements from di erent rows.
Our goal now is to show that the SE-automorphisms (a), (b), (c), and(d) generate the centralizer of σ in Aut(F km ). To proceed we shall use the Nielsen transformations on the m-tuples of orbits.

Transformations for m-tuples of σ-orbits
Let F km be a free group with the σ-basis A which is the union of m σ-orbits. (A , A , . . . , A i , . . . , Am) .

Transformation of a σ-basis A to the standard basis X
We have to show that each σ-basis A written as (5) can be transformed to the σ-basis X written as (3) by the sequence Nτ , ..., Nτ k of elementary Nielsen transformations, applied to the m-tuples of orbits, which are the columns in the matrices. The proof is similar to that for Theorem 3.1 in [8] and uses the following terminology for the words in the free group Fn = x , x , ..., xn , n = km: -A word is freely reduced if no cancellations in it is possible.
-An x-length |a| of a word a is a number of x ± i in its freely reduced form. -The x-length of a basis A is the sum of x-lengths of all its elements. -A major initial (major terminal) segment of a word has a minimally bigger length then half of the word's length. -A major initial (major terminal) segment of a word a ∈ A is isolated if no word b ∈ A has such an initial (or terminal) segment. Note that otherwise |a − b| < |b| (or |ba − | < |b| ). -A subset S in the free group Fn is Nielsen-reduced if the major initial and major terminal segments of each a ∈ S are isolated and for an element a of even length either its left or its right half is isolated. -The basis X in F km is Nielsen-reduced and has the minimal x-length, which is equal to n = km. Since by De nition 1, a σ-basis A in general case has a form and the permutation σ has no xed points, it follows that every σ-orbit } is a Nielsen reduced set. Proof. Let A = (A , A , . . . , Am) be the freely reduced σ-basis in F km written as the matrix (9). Since elements in columns have di erent rst (last) letters x i , we have that the subsets in each column are Nielsen-reduced, while the whole basis A need not be Nielsen-reduced.
1. Assume rst, that the x-length of the σ-basis A is equal km. Then its entries are of the form x ε i , ε = ± . Since (x ε i ) σ = (x σ i ) ε , all elements in a σ-orbit have the same value of ε. Then by the Nielsen-transformations of the type (a) we can eliminate all ε = − . Now by permuting (cyclically) elements in the columns by transformations (d), we get the proper order inside each column. Then the permutations of the columns (transformations (c) ) lead to the basis X, as required. 2. Let now the x-length of the σ-basis A be greater than km. Then by Lemma 3.1 in [8] the basis A is not Nielsenreduced. It follows that there is a word a = uv in some column A i in (9) which has a major (say initial) segment u (|u| > |v|), equal to the initial segment of some word b = uw in some other column A j in (9). We apply the Nielsen transformations of the type (d) to these columns to get the top elements in i-th and j-th columns respectively: uv and uw Then the i-th and j-th columns consist of elements uv, (uv) σ , . . . and uw, (uw) σ , . . . respectively. Now we change A i to A − i by transformation (a) and apply transformation (b) to change A j to A − i A j . We get the top elements in i-th and j-th columns (uv) − and v − w.
Since |u| > |v|, we get |uw| > |vw| ≥ |v − w|. It diminishes the length of the j-th column at least for m and hence diminishes the x-length of the σ-basis A. Similarly we can isolate the halves of the entries. By repeating these steps we get the Nielsen reduced σ-basis of minimal length equal to km, which was considered in the part 1 of the proof. So the proof is complete.

The Main Theorem
Theorem 2. The centralizer C(σ) in Aut(F km ) for the automorphism σ with m orbits of order k is generated by the SE-automorphisms (a), (b), (c) and (d).
Proof. Let an automorphism α ∈ C(σ) map X → A,  which shows the similarity with the results from [6,7] for permutation σ ∈ S km and C(σ) ⊆ S km but in the case of automorphism permuting generators Z m k is not a normal subgroup of C(σ).
Theorem 3. The centralizer C(σ) in Aut(F km ) for the automorphism σ with m cycles of order k can be generated by the automorphism cyclically permuting elements in the rst σ-orbit (the type (d)), and in addition, by two automorphisms if m ≥ and by three automorphisms for m = , . Moreover, C(σ) is nitely presented.