Critical regularity of nilpotent groups acting on one-dimensional compact manifolds

Given a finitely generated, torsion-free nilpotent group, we find the maximum possible (critical) regularity for its faithful actions by diffeomorphisms of the closed or half-open interval and of the circle. Our result gives an expression for its value in purely algebraic terms (using the relative growth of appropriate subgroups), generalizing many preceding works. As an intermediate step, we generalize the Bass-Guivarc'h formula, obtaining a formula for the relative growth of subgroups of nilpotent groups, as well as for the growth of the corresponding Schreier graphs.