Bounding the Residual Finiteness of Free Groups

We find a lower bound to the size of finite groups detecting a given word in the free group. More precisely we construct a word wn of length n in non-abelian free groups with the property that wn is the identity on all finite quotients of size ∼ n2/3 or less. This improves on a previous result of BouRabee and McReynolds quantifying the lower bound of the residual finiteness of free groups. A group G is called residually finite if for any w ∈ G, w = 1, there exists a finite group H and a homomorphism φ : G → H such that φ(w) = 1. We will say that such a group H detects the element w. One way to quantify this property is to look at the minimal size of a finite group H which detects a given word, and study the behavior of this function. We define the following natural growth function to measure the residual finiteness of a group (introduced by Bou-Rabee in [2]): kG(w) := min{|H| | there exists π : G → H, π(w) = 1} and F G(n) := max{kG(w) | |w|S ≤ n}, where S is a generating set of the group G and |w|S denotes the word length of w with respect to the generating set S. We also write f1 f2 to mean that there exists a C such that f1(n) ≤ Cf2(Cn) for all n, and we write f1 f2 to mean f1 f2 and f2 f1. It is easy to see that if G is finitely generated, the growth type of the function F G does not depend on the set S, assuming that it is finite. In this short paper, we will focus on the free group Fk on k generators. BouRabee [2] and Rivin [9] have shown that FFk(n) n. Both proofs are obtained by embedding the free group Fk into SL2(Z) and then finding a suitable prime p such that a given word does not vanish in the quotient SL2(Z/pZ). For a slightly different proof see Remark 8. Recently, Bou-Rabee and McReynolds [1] have shown (see Corollary 11) that FFk(n) n. We improve this lower bound, establishing the following result: Theorem 1. FFk(n) n. Received by the editors March 3, 2010. 2010 Mathematics Subject Classification. Primary 20F69; Secondary 20E05, 20E07, 20E26.