On Shalev's conjecture for type 𝐴_𝑛 and ²𝐴_𝑛

1 Introduction

Recently, there has been considerable interest in word maps on finite, algebraic and topological groups; see, for example, [1, 7, 8, 10, 11, 14, 15, 16] and a survey [17]. Recall that a group word ω=ω⁢(x1,…,xd) is an element of a free group Fd with free generators x1,…,xd. We may write ω=xi1m1⁢…⁢xikmk, where ij∈{1,…,d} and mj are integers. We say that ω is a power word if ω=υm for m>1 and some word υ. For a group G and g1,…,gd∈G, we write

The corresponding map ω:Gd→G is called a word map, and its image is denoted by ω⁢(G). Many papers are devoted to estimate the size of ω⁢(G). In [15], Larsen and Shalev proved the following:

Notice that c depends on ω in Theorem 1.1. Later, in [16], Nikolov and Pyber found a weaker lower bound for the groups of type An and An2. In the expository article [17], Shalev conjectured that Theorem 1.1 holds for all groups of Lie type.

Furthermore, Shalev states as a problem that, in the case of non-power words, a stronger conclusion might hold.

This problem shows that power words are of considerable interest, and, in this case, one could expect the lowest bound on |ω⁢(G)|. In this article, we explore the Shalev conjecture for finite groups of type An and An2 and for power words ω=xM. It is clear that, if ω=xi1m1⁢…⁢xikmk∉Fd′ and M is the smallest positive absolute value of the sum of the exponents of a fixed variable xi in ω, then |ω⁢(G)|⩾|ω¯⁢(G)|, where ω¯=xM. In the paper, we restrict ourself to the consideration of power words of type ω=xM.

Throughout the paper, we assume that ω=xM and fix symbols ω and M for this situation.

First, we show by an example that the constant c in the conjecture depends on the word ω. Next, we prove that, for power words ω=xM, the conjecture is true (see Theorem 4.5). Moreover, we improve the bound as follows:

We believe that, for the estimate in Theorem 1.4, it is possible to prove a better bound |ω⁢(G)|⩾c⁢lnk⁡(n)n⁢|G|, where k is any fixed power. Notice also that, in view of Theorem 4.5, there exists N such that, if G=PSLnε⁡(q) and |G|⩾N, then |ω⁢(G)|⩾|G|2⁢n⁢M, and this bound could be better than the bound in Theorem 1.4.

2 Preliminaries

Our notation is standard and, in general, agrees with that of [3, 13, 5]. For integers m1,…,ms, by (m1,…,ms) or gcd⁡{m1,…,ms} we denote their greatest common divisor, and by [m1,…,ms] or lcm⁡{m1,…,ms} we denote their least common multiple. We write H⩽G and H⊴G if H is a subgroup or H is a normal subgroup of G, respectively. We recall some relations about gcd.

Recall that any finite abelian group A can be uniquely presented as a product of cyclic groups A=ℤd1×…×ℤdk, where ℤdi is a cyclic group of order di and d1 divides d2, d2 divides d3, …, dk-1 divides dk. We use the notation A=d1×…×dk and refer to it as a standard decomposition of A for brevity.

By p we always denote some prime number, and q is a power of p; 𝔽q denotes a finite field of q elements, and 𝔽¯q is an algebraic closure of 𝔽q. The symmetric group on n elements is denoted by Symn. By diag⁢(λ1,λ2,…,λn) we denote a diagonal n×n-matrix with λ1,λ2,…,λn on the diagonal. By bd⁡(T1,T2,…,Tn) we denote a block-diagonal matrix with square blocks T1,T2,…,Tn on the diagonal.

Let G¯ be a connected simple algebraic group over an algebraic closure 𝔽¯p of a finite field 𝔽p. We say that an algebraic group endomorphism σ of G¯ is a Steinberg endomorphism (or a Frobenius map), if the set G¯σ of σ-stable points of G¯ is finite. Any group G=Op′⁢(G¯σ) is called a finite group of Lie type. If G¯ is of type An, then Op′⁢(G¯σ) is a homomorphic image of either SLn+1⁡(q) or SUn+1⁡(q) for some q, and we say that G is a group Anε⁢(q) in this case, where ε∈{+,-} and An+⁢(q) is a homomorphic image of SLn+1⁡(q), while An-⁢(q) is a homomorphic image of SUn+1⁡(q). Recall that an element is regular if the dimension of its centralizer in the corresponding group over the algebraic closure of 𝔽q is equal to the Lie rank of the group G. In particular, a semisimple element is regular if and only if the connected component of its centralizer is a maximal torus. Thus every regular semisimple element belongs to exactly one maximal torus. Recall also that every semisimple element of G¯σ lies in a σ-stable maximal torus T¯ of G¯.

If T¯ is a maximal σ-stable torus of G¯, then T=T¯∩G is called a maximal torus of G, and N⁢(G,T)=NG¯⁢(T¯)∩G is called an algebraic normalizer. Clearly, N⁢(G,T)⩽NG⁢(T), and the next lemma gives a sufficient condition for the equality.

We denote the Weyl group NG¯⁢(T¯)/T¯ of An-1 by W. Since all maximal tori are conjugate, W does not depend on a particular choice of T¯ and is isomorphic to Symn. If we fix a σ-stable torus H¯ of G¯, then σ leaves NG¯⁢(H¯) invariant and so induces an automorphism on W=NG¯⁢(H¯)/H¯, and we denote this automorphism also by σ. We say that u,w∈W are σ-conjugate if there exists v∈W such that u=v-1⁢w⁢vσ. There is a bijection between the G-classes of σ-stable maximal tori of G¯ and the σ-conjugacy classes of W (see [4, Propositions 3.3.1, 3.3.3]). Moreover, by [4, Proposition 3.3.6], if a torus T=H¯g∩G corresponds to an element w of W, we have

Given w∈W, we denote a representative of the corresponding conjugacy class of maximal tori by Tw, and we denote N⁢(G,T)/T by W⁢(T). If G=Anε⁢(q), then we may choose H¯ so that σ acts trivially on NG¯⁢(H¯)/H¯ (in this case, H¯σ is of order (q-ε⁢1)n), so, in this case, there is a bijection between the G-classes of σ-stable maximal tori of G¯ and the conjugacy classes of W. In particular, for a maximal torus T=Tw, the identity W⁢(T)≃CW⁢(w) holds.

Each element of Symn can be uniquely expressed as a product of disjoint cycles. The lengths of these cycles define a set of integers, which is called the cycle-type of the element. Two permutations are conjugate if and only if they have the same cycle-type. Thus the conjugacy classes of Symn are in one-to-one correspondence with the partitions of n, and so the conjugacy classes of maximal tori in An-1ε⁢(q) are in one-to-one correspondence with the partitions of n.

The structure of maximal tori in linear groups is well known. For our purposes, we will use the description from [2].

If G is GLn⁡(q),Un⁡(q),SLn⁡(q),SUn⁡(q), a semisimple element is regular if and only if its minimal polynomial is equal to its characteristic polynomial, i.e., if and only if its characteristic polynomial has no repeated roots (see [6]).

We briefly describe the general idea of proofs of the main results. We need to compute the lower and upper bounds for the ratio |w⁢(G)||G|. We obtain the bounds only for regular semisimple elements and then apply them to obtain the bounds for the whole group. So we need to consider maximal tori of G. Since any regular semisimple element belongs to a unique maximal torus, we can take the union (and this union is disjoint) of tori, thus avoiding repetition. Finally, instead of considering all maximal tori, we could restrict ourselves to a finite number (in our paper, one or two) maximal tori and then multiply the bound by the index of normalizer.

3 Example

In this section, we give an example that the constant c in the conjecture depends on the word ω.

The following lemma is clear.

Thus we have the following:

As a corollary to Theorem 3.4, we get an example showing that the constant c in the conjecture does depend on ω.

4 Estimate for power words in SLnε⁡(q)

The following lemma shows that it is enough to prove Theorem 1.4 for SLnε⁡(q).

For any subset S of a maximal torus, denote by Sreg the set of regular semisimple elements in S.