Bounding the residual finiteness of free groups

Abstract

Abstract. 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 w n of length n in non-abelian free groups with the property that w n is the identity on all finite quotients of size ∼ n 2/3 or less. This improves on a previous result of BouRabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.

Full residual finiteness growths of nilpotent groups

“…The full residual finiteness growth of the discrete Heisenberg group is presented in [8]. Compare full residual finiteness growth to the concept of residual finiteness growth, which measures how well individual elements are detected by finite quotients, appearing in [5], [7], [3], [11], [18], [6], [12]. Also compare this with Sarah Black's growth function defined and studied in [2].…”

Asymptotic growth and least common multiples in groups

“…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). Recently, Kassabov-Matucci [8] improved the lower bound of n 1/3 to n 2/3 . A reasonable guess is that F F m ,X (n) ≃ n, though presently neither the upper or lower bound is known.…”

“…A reasonable guess is that F F m ,X (n) ≃ n, though presently neither the upper or lower bound is known. We refer the reader to [8] for additional questions and conjectures.…”

“…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.…”

Zariski closures and subgroup separability

“…Recently, several effective separability results have been established; see [2][3][4][5][6]8,9,[12][13][14][15][20][21][22][23]26]. Most relevant here are the papers [9,20] where bounds on the index of the separating subgroups for free and surface groups given.…”