Fibers of automorphic word maps and an application to composition factors

1.1 Motivation and main result

Word maps on groups have been studied intensely in recent years, resulting in substantial progress on interesting questions and a beautiful theory using tools from various areas such as representation theory and algebraic geometry; interested readers are referred to the survey article [5].

Recall that a (reduced) word w in d variables X1,…,Xd is an element of the free group F⁢(X1,…,Xd). Each such word gives, for each group G, rise to a word map wG:Gd→G induced by substitution. Studying the fibers of wG means studying the solution sets in Gd to equations of the form w=w⁢(X1,…,Xd)=g for g∈G. By Larsen and Shalev’s result [3, Theorem 1.1], for fixed w, the maximum number of solutions to such an equation in a nonabelian finite simple group S is in o⁡(|S|d) as |S|→∞. Hence for each fixed number ρ∈(0,1], for only finitely many nonabelian finite simple groups S, wS has a fiber of size at least ρ⁢|S|d.

Based on this, it is natural to ask what one can say more generally about the nonabelian composition factors of a finite group G where the word map wG has a fiber of size at least ρ⁢|G|d. In order to be able to use [3, Theorem 1.1] for this, it would be useful if one could somehow relate the maximum fiber size of wG with the maximum fiber sizes of the word maps associated with w over the composition factors of G. For example, it would be nice to have an inequality of the form Πw⁢(G)≤Πw⁢(N)⋅Πw⁢(G/N) for all finite groups G and all normal subgroups N of G, where Πw denotes the function that maps each finite group G to the maximum fiber size of wG. Unfortunately, this is not the case, even if we assume that N is characteristic in G; consider, for example, G=D2⁢o, the dihedral group of order 2⁢o, for some odd integer o≥3, N the unique cyclic subgroup of index 2 in G, and w=X12.

In this paper, we will describe a way to circumvent these difficulties and provide some strong restrictions on possible composition factors of a finite group G such that Πw⁢(G)≥ρ⁢|G|d in the form of Theorem 1.1.2 below. First, we introduce some constants:

Our main result is the following (as usual, the “untwisted Lie rank” of a Lie-type group is the Lie rank of the corresponding untwisted group):

In other words, the list of potential composition factors for such a group G consists of cyclic groups of prime order, finitely many alternating groups, the sporadic groups, and all simple Lie-type groups of bounded rank.

1.2 Main ideas and overview of the paper

The main idea for proving Theorem 1.1.2 is to make up for the above mentioned “flaw” of the function Πw by replacing it by an evaluation-wise larger function 𝔓w which satisfies the inequality 𝔓w⁢(G)≤𝔓w⁢(N)⋅𝔓w⁢(G/N) at least when N is characteristic in G and study 𝔓w instead. To this end, we generalize the notion of a word map in a certain way.

Let us first fix some notation. For a fixed reduced word w in the d variables X1,…,Xd and of length l, write

where x1,…,xl∈{X1,…,Xd} and ϵi=±1. Denote by ι the unique function {1,…,l}→{1,…,d} such that for i=1,…,l, xi=Xι⁢(i). Thus for each group G, the word map wG is just the map Gd→G sending

Hence wG(α1,…,αl) is like wG, except that in the i-th factor of the l factor product as which the evaluation wG⁢(g1,…,gd) is defined, we additionally apply αi, one of l automorphisms fixed beforehand. In particular, wG(id,…,id)=wG.

The approach of studying fibers of automorphic word maps will actually allow us to prove the following stronger form of Theorem 1.1.2:

We now give an overview of the rest of this paper:

1. In Section 2, we prove our main lemma, Lemma 2.1, which includes the inequality 𝔓w⁢(G)≤𝔓w⁢(N)⋅𝔓w⁢(G/N) for characteristic subgroups N of G. It also includes the observation that the element of G having the largest fiber size under any automorphic word map on G is the identity element of G.

2. Having gained a basic understanding of automorphic word maps in Section 2, the next goal is to extend, as far as necessary, Larsen and Shalev’s result [3, Theorem 1.1] on fibers of word maps on nonabelian finite simple groups mentioned above to fibers of automorphic word maps. This will be done in Section 3, see Theorem 3.1.2.

3. Section 4 consists of the proof of Theorem 1.2.2 based on the results developed so far.

4. Finally, in Section 5, we give some concluding remarks concerning further extensions of Larsen and Shalev’s techniques to automorphic word maps and an interesting consequence thereof.

1.3 Notation

We denote by ℕ the set of natural numbers (including 0) and by ℕ+ the set of positive integers. Euler’s constant is denoted by e, which is to be distinguished from the variable e. The image and preimage of a set M under a function f are denoted by f⁢[M] and f-1⁢[M], respectively. When fi:Xi→Yi for i=1,…,n, then we denote by f1×⋯×fn the product of the maps fi, i.e., the map

The n-fold product of a map f with itself is denoted by f(n). These last two notations will be used in the proofs of Lemma 4.4 and of the implication “Conjecture 5.2⇒ Conjecture 5.3” in Section 5.

For a group G and an element g∈G, we denote by conj⁡(g) the conjugation by g on G, i.e., the inner automorphism of G of the form x↦g⁢x⁢g-1. The automorphism group of G is denoted by Aut⁡(G), and the inner automorphism group of G by Inn⁡(G). For a finite set X, we denote by 𝒮X the symmetric group on X; for a positive integer n, 𝒮n and 𝒜n denote the symmetric and alternating group on {1,…,n}, respectively.

For a prime power q, the finite field with q elements is denoted by 𝔽q. For n∈ℕ+ and a prime power q, Matn⁡(q) denotes the ring of (n×n)-matrices over 𝔽q. For a vector space Δ over some field F, we denote by EndF⁡(Δ) the endomorphism ring of Δ, i.e., the ring of F-linear maps Δ→Δ.

At some points in our arguments, we will not consider all possible automorphic word maps wG(α1,…,αl) over some finite group G, but only those where the αi are from a certain subset of Aut⁡(G). Also, we sometimes want to talk about the maximum fiber size of a particular element of G under an automorphic word map or about the proportion of a fiber of an (automorphic) word map associated with w within the entire argument set Gd, rather than the actual size of the fiber. We therefore introduce the following notation that supplements the notation already introduced:

3.1 Larsen and Shalev’s result and the main result of this section

The following theorem is an equivalent reformulation of [3, Theorem 1.1]:

The proof of this theorem in [3] is split into three parts (note that the sporadic groups can be ignored here, as the fiber size bound only needs to be shown for large enough S):

1. First, the bound is established for large enough alternating groups by means of a certain combinatorial construction.

2. Next, the bound is established for all simple Lie-type groups of sufficiently high rank, where the lower bound on the rank is so large that only classical groups need to be considered in this case. As Larsen and Shalev say themselves, the argument is conceptually similar to the one for alternating groups.

3. Finally, the simple Lie-type groups of bounded rank are treated by means of an argument using results of algebraic geometry.

It turns out that Larsen and Shalev’s arguments in the first two cases can be modified to prove the following, which is the main result of this section:

Whether such bounds can also be established for the simple Lie-type groups of “small” rank is open; see Section 5 for some more remarks on this.

3.2 Reduction of Theorem 3.1.2 to Theorem 3.2.6

Similarly to [3], the main part of the argument for Theorem 3.1.2 will not provide upper bounds on 𝔓w⁢(S) for the simple groups S in question directly, but on 𝔓w(A)⁢(G), where G is a finite group “closely related with S” and A a certain subgroup of Aut⁡(G). That we can do this without loss of generality is justified by the following lemma, a modification of [3, Lemma 2.1], which served the same purpose:

Lemmata 3.2.3 and 3.2.5 below will allow us to reduce the proof of Theorem 3.1.2 to the proof of a theorem concerning fibers of automorphic word maps in slightly different classes of groups, Theorem 3.2.6. For example, for the alternating groups, these “closely related” groups will be just the symmetric groups. To make the formulations of the lemmata shorter, let us first introduce the following terminology:

The following lemma allows us to reduce Theorem 3.1.2 (1) to the study of automorphic word map fibers in symmetric groups:

As for the classical groups of Lie type X⁢(q) with which we are concerned in Theorem 3.1.2 (2), the groups “closely related” with them which we will study are, just as in [3, beginning of Section 3], the isometry groups of trivial, perfect symmetric, perfect anti-symmetric or perfect Hermitian pairings (depending on the case) of a vector space over either the field 𝔽q or (only in the Hermitian case) its degree 2 extension 𝔽q2. Set E:=𝔽q, and moreover, set F:=E except in the Hermitian case, where F:=𝔽q2.

In the notation of Kleidman and Liebeck’s book [2, Section 2.1], the classical simple Lie-type group X⁢(q) is Ω¯ and is the projective version of a subgroup Ω of the associated isometry group, which is denoted by I. Note that I, in turn, is contained as a normal subgroup in some group A which (by its conjugation action on I) may be viewed as a subgroup of Aut⁡(I) and is just the group Γ of collineations over I except when I=GLn⁡(q) is the isometry group of a trivial form, in which case A is the subgroup of Aut⁡(I) generated by Γ and the inverse-transpose automorphism of I. For later purposes, we set A⁡(I):=A.

It follows from [2, Theorem 2.1.4] that if the untwisted Lie rank of Ω¯ is at least 5 (this is just to exclude the groups C2⁡(2f)=Sp4⁡(2f) and D4⁡(q)=Ω8+⁢(q)), then every automorphism of Ω¯ is induced by the restriction to Ω of an automorphism of I from A=A⁡(I), as required in Lemma 3.2.1. Furthermore, as Larsen and Shalev observe in [3, beginning of Section 3], we always have |I|≤(|F|-1)⁢(r+1)⁢|Ω¯|, where r is the untwisted Lie rank of Ω¯. They also observe that for every ϵ>0, |Ω¯|ϵ≥(r+1)⁢(|F|-1) if r is sufficiently large. This “sufficiently large” can be made explicit.

We can now show the following:

We now give the aforementioned theorem to which Theorem 3.1.2 reduces:

Let us actually derive Theorem 3.1.2 from this.

3.3 First part of the proof of Theorem 3.2.6: Symmetric groups and isometry groups other than general linear groups

We now turn to the proof of Theorem 3.2.6, which as mentioned before, is a modification of an argument by Larsen and Shalev from [3]. Let us first make some general observations which will be used in the proof.

Note that each of the abstract groups G with which Theorem 3.2.6 deals can actually be viewed as a permutation group, acting on a set Δ, in a natural way: each symmetric group 𝒮n through its natural action on the set {1,…,n}, and each isometry group through its action on the corresponding vector space. Larsen and Shalev also exploited this fact, and their argument consisted essentially in investigating to what extent a relation of the form w⁢(g1,…,gd)=g for g∈G fixed imposes restrictions on g1,…,gd∈G when viewed as maps Δ→Δ.

In our setting, this gets more complicated because we are actually considering not a word equation in g1,…,gd, but a word equation in various images of the gi under fixed automorphisms of G from some subgroup A⁡(G)≤Aut⁡(G). Hence it would be useful if, from some single piece of mapping information of the form α⁢(gi)⁢(x)=y, α∈A⁡(G) and x,y∈Δ, we could derive such a condition on gi itself. It turns out that this is actually possible for the G with which we are concerned except for the case G=GLn⁡(q), which will require some separate treatment.

In this subsection, we deal with the G not isomorphic with any GLn⁡(q).

The point behind Notation 3.3.1 is that in each of the cases considered, the automorphism α of G can be seen as the restriction of the conjugation by t(α)-1∈𝒮Δ to the subgroup G of 𝒮Δ. Hence the following is clear:

At last, we are now ready to discuss the proofs of Theorem 3.2.6 (1) and of Theorem 3.2.6 (2) except for general linear groups; we will present these proofs one after the other.