On the spectrum of residual finiteness growth functions

In [K. Bou-Rabee, B. Seward, J. Reine Angwe. Math. 2016] Bou-Rabee and Seward constructed examples of finitely generated residually finite groups $G$ whose residual finiteness growth function $\mathcal{F}_G$ can be at least as fast as any prescribed function. In this note we describe a modified version of their construction, which allows us to give a complementary upper bound on $\mathcal{F}_G$. As such, every nondecreasing function at least $\exp ( n \log (n)^2 \log \log (n)^{1+\epsilon} )$ is close to the residual finiteness growth function of some finitely generated group. We also have similar result for the full residual finiteness growth function and for the divisibility function.