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

Maximiliano Escayola, Victor Kleptsyn

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.

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

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.

Paper Structure

This paper contains 43 sections, 45 theorems, 250 equations, 8 figures.

Table of Contents

  1. Introduction
  2. History of the question
  3. Description of results
  4. Plan of the paper
  5. Acknowledgements
  6. Preliminaries
  7. Critical regularity
  8. Generalized Kopell's Lemma and the control of distortion
  9. Stabilizer subgroups and Schreier graphs
  10. Growth of subgroups and Schreier graphs
  11. Nilpotent groups and Bass--Guivarc'h formula
  12. Main results
  13. Classes of actions associated to a given central element
  14. General critical regularity
  15. Generalisations of the Bass--Guivarc'h formula
  16. ...and 28 more sections

Key Result

Proposition 2.3

Let $f_1,\ldots,f_k$ be $C^1$-diffeomorphisms of compact one-dimensional $M$ that commute with a $C^1$-diffeomorphism $g$. Assume that $I_g$ is an interval, fixed by $g$, such that $g|_{I_g}\neq \mathrm{id}$. Assume moreover that for a certain $0 < \tau < 1$ and a sequence of indices $i_j\in \{1,\ld Then $f_1,\ldots, f_k$ cannot be all of class $C^{1+\tau}$.

Figures (8)

  • Figure 1: Estimating the derivative of $g^m$.
  • Figure 2: A support interval and its image
  • Figure 3: Extending the action of $K_{j+1}$, acting on $J_{j+1}$
  • Figure 4: A family of intervals, permuted by the action
  • Figure 5: An map $\varphi_{I',I}^{J',J}$ from the Tsuboi family: its graph over $I$ and tangent lines at the endpoints (left), and its construction (right)
  • ...and 3 more figures

Theorems & Definitions (103)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Generalized Kopell's lemma
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 93 more