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

TL;DR

This paper determines the maximum regularity for faithful actions of finitely generated torsion-free nilpotent groups on one-dimensional manifolds, linking algebraic subgroup growth to geometric action smoothness.

Contribution

It provides a new algebraic formula for critical regularity based on subgroup growth, generalizing the Bass-Guivarc'h formula for nilpotent groups.

Findings

Derived an explicit algebraic expression for critical regularity.

Generalized the Bass-Guivarc'h formula for subgroup growth.

Connected subgroup growth to the smoothness of group actions.

Abstract

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.