On the shortest identity in finite simple groups of Lie type

Abstract

Abstract. We prove that the length of the shortest identity in a finite simple group of Lie type of rank r defined over F q is bounded (from above and below) by explicit polynomials in q and r.

Asymptotic growth and least common multiples in groups

“…In Section 4, we extend this approach by generalizing least common multiples to finitely generated groups (a similar approach was also taken in the article of Hadad [7]). Indeed with this analogy, Theorem 1.2 and the upper bound of n 3 established in [1], [11] can be viewed as a weak Prime Number Theorem for free groups since the Prime Number Theorem yields F Z (n) ≃ log(n).…”

“…We call G Γ,X (n) the residual girth function and relate G Γ,X (n) to F Γ,X and w Γ,X (n) for a class of groups containing non-elementary hyperbolic groups; Hadad [7] studied group laws on finite groups of Lie type, a problem that is related to residual girth and the girth of a Cayley graph for a finite group. Specifically, we obtain the following inequality (see Section 4 for a precise description of the class of groups for which this inequality holds):…”

“…The asymptotic growth of F Γ,X (n), G Γ,X (n), and related functions arise in quantifying residual finiteness, a topic introduced in [1] (see also the recent articles of the authors [2], Hadad [7], KassabovMattucci [8], and Rivin [11]). Quantifying residual finiteness amounts to the study of so-called divisibility functions.…”

Suzuki groups as expanders

“…This follows from the proof of [17], Lemma 5, but for the convenience of the reader we sketch the argument in Appendix A. We remark that the correct bound in this lemma is probably cq: see the remarks following Lemma 4.1 of [13].…”

No iterated identities satisfied by all finite groups

“…, x n in G. This notion, introduced by Bou-Rabee in [5] is called the normal residual finiteness growth function, see also [6,9,14,26], and it is called residual finiteness growth function in Thom [35] and Bradford and Thom [8] (not to be confused with residual finiteness growth function in terminology of [7], that measures the size of finite, not necessary normal subgroups, not containing a given element), who have proven the lower bound ě Cn 3{2 { log 9{2`ǫ n, which holds for any ǫ ą 0. Kassabov and Matucci suggested in [26] that the argument of Hadad [21] can give a close upper bound for normal residual finiteness growth, function, namely n 3{2 . A known upper bound so far is n 3 [5], which is a corollary of the estimate for SLp2, Zq, using imbedding of a free group to this group.…”