We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form Γ × Γ where Γ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if P is a finitely generated, residually finite group with P ∼ = Γ × Γ then there is an embedding P ֒→ Γ × Γ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for P .

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.

For X = R, C, or H, it is well known that cusp cross-sections of finite volume X -hyperbolic (n + 1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n + 1)-dimensional Heisenberg group N 2n+1 or the (4n + 3)-dimensional quaternionic Heisenberg group N 4n+3 (H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X -hyperbolic (n + 1)-orbifold.A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.

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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.