We introduce the notion of quantifying the extent to which a finitely generated group is residually finite. We investigate this behavior for examples that include free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as SL n (Z). In the context of finite nilpotent quotients, we find a new characterization of finitely generated nilpotent groups.

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.

In this short article, we study the extremal behavior F Γ (n) of divisibility functions D Γ introduced by the first author for finitely generated groups Γ. These functions aim at quantifying residual finiteness and bounds give a measurement of the complexity in verifying a word is non-trivial. We show that finitely generated subgroups of GL(m, K) for an infinite field K have at most polynomial growth for the function F Γ (n). Consequently, we obtain a dichotomy for the growth rate of log F Γ (n) for finitely generated subgroups of GL(n, C). We also show that if F Γ (n) log log n, then Γ is finite. In contrast, when Γ contains an element of infinite order, log n F Γ (n). We end with a brief discussion of some geometric motivation for this work.

The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth n dim(G) .

Let G be a virtually special group. Then the residual finiteness growth of G is at most linear. This result cannot be found by embedding G into a special linear group. Indeed, the special linear group SL k (Z), for k > 2, has residual finiteness growth n k−1 .