or less. This improves on a previous result of Bou- Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.
Testo completo
In this short note, we will focus on the free group F k on k generators. Bou- Rabee [2] and Rivin [9] have shown that F Fk
Recently, Bou-Rabee and McReynolds [1] have shown (see Corollary 11) that F Fk
Theorem 1. F Fk
Applying Lemma 2 twice with G = H, we see that we can drop the decoration S in F G S (n). Moreover, this implies that the growth functions for all non-abelian finitely generated free groups are all equivalent. Hence, we will restrict to study the function F F2
Theorem 4. F Zk
Therefore, for any w ∈ F k \[F k , F k ] one has k Fk
Theorem 4 can be generalized to nilpotent and soluble groups, but one needs to replace the logarithmic bound of Theorem 4 with a polylogarithmic one. This suggests that the words w in F k , where k Fk
[F k : F k,n ] ≤ n aCn
most log[F k : F k,n ] ≤ e 4k log2
any finite group of size at most n, therefore k Fk
• if α k (Γ) > l for some finite group Γ of order n, then F k,n does not contain any words of length l, which implies that F Fk
• if a word w ∈ F k is an identity in any finite group of order at most n then k Fk
These two observations allow us to obtain upper and lower bounds for F Fk
Remark 8. Let w be a word in the free group of length n. By the above result w is not an identity in the group PSL 2 (F p ) for any prime p > 3n + 1, therefore w is not in the kernel of some map F k → PSL 2 (F p ), i.e., k F2
Thus, we have F Fk
Remark 9. It seems the methods [4] can be used to show that the shortest identity satisfied in all groups of the form SL 2 (R), where R is a finite commutative ring of size at most N , has length CN 2 . If this is indeed the case then one can improve the upper bound F Fk
On the other side, it is easy see that w = x n! is an identity in any group of order at most n. This can be used to obtain a lower bound for F Fk
Lemma 10. Let r 1 > . . . > r m be a finite sequence of integers. There exist a nontrivial word w = w r1
X r1
Sketch of the proof: The nontrivial word w = w r1
[x r1
w := [[x r1
This lemma gives the following lower bound for the function F Fk
F Fk
Therefore F Fk
Question 18. Is it true that F Fk
Let F 2,n max denote the intersection of all maximal normal subgroup in F 2 of index at most n. It can be shown that [F 2 : F 2,n max ] e n4/3
We define the divisibility function of the free group D Fk
D Fk
Question 20. Does there exist a C = C(k) > 0 such that D Fk
Theorem 21. max |w|≤n D Fk
Theorem 22. D Fk
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