Asymptotic growth and least common multiples in groups

Abstract

In this article we relate word and subgroup growth to certain functions that arise in the quantification of residual finiteness. One consequence of this endeavor is a pair of results that equate the nilpotency of a finitely generated group with the asymptotic behavior of these functions. The second half of this article investigates the asymptotic behavior of two of these functions. Our main result in this arena resolves a question of Bogopolski from the Kourovka notebook concerning lower bounds of one of these functions for nonabelian free groups.

Full residual finiteness growths of nilpotent groups

“…Recall that the word growth, w G , of a finitely generated group G is the growth of the function w X G (n) = B X G (n) , which is independent of X. Gromov [9] has characterized nilpotent groups in the class of finitely generated groups as those for which w G is polynomial. By applying this theorem, it is shown in [4] (see Theorem 1.3 there) that full residual finiteness growth enjoys the same conclusion. In spite of this similarity, these two growths rarely coincide.…”

“…The concept of full residual finiteness growth was first studied by Ben McReynolds and K.B. in [4]. The full residual finiteness growth of the discrete Heisenberg group is presented in [8].…”

“…A subset S ⊆ G is fully detected by a group Q if there exists a homomorphism ϕ : G → Q such that ϕ| S is injective. For a natural number n, set Φ X G (n) to be the minimal order of a group Q that fully detects the ball of radius n in G (first studied in [4]). The full residual finiteness growth of G with respect to X is the growth of the function Φ X G , that is, its equivalence class under the equivalence relation defined by f ≈ g if and only if there is a constant C so that f (n) ≤ Cg(Cn) and g(n) ≤ C f (Cn) for all natural numbers n. The growth of Φ X G is independent of choice of generating set X (see Lemma 1.1).…”

Local commensurability graphs of solvable groups

“…While the article, [BRSc], began the study of local commensurability graphs with the goal of drawing fundamental group properties from residual invariants (a direction of much activity: [KT], [BRK12], [BRM11], [GK], [BRHP], [BRSa], [KM11], [Riv12], [Pat13], [LS03]), the study of graphs given by relations between subgroups goes back to work of B. Csákány and G. Pollák in the 1960's [CP69]. The p-local commensurability graphs are weighted pieces of what are called the intersection graphs of a group (see, for instance, [AHM15]).…”

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.…”