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On the critical regularity of nilpotent groups acting on the interval: the metabelian case

Part of: Special aspects of infinite or finite groups Low-dimensional dynamical systems Smooth dynamical systems: general theory

Published online by Cambridge University Press:  24 September 2024

MAXIMILIANO ESCAYOLA  and
CRISTÓBAL RIVAS
MAXIMILIANO ESCAYOLA*
Affiliation:
IRMAR - UMR CNRS 6625, Université de Rennes, Rennes, France
CRISTÓBAL RIVAS
Affiliation:
Dpto de Matemáticas, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile (e-mail: cristobalrivas@u.uchile.cl)
*
e-mail: maximiliano.escayola@univ-rennes.fr
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Abstract

Let G be a torsion-free, finitely generated, nilpotent and metabelian group. In this work, we show that G embeds into the group of orientation-preserving $C^{1+\alpha }$-diffeomorphisms of the compact interval for all $\alpha < 1/k$, where k is the torsion-free rank of $G/A$ and A is a maximal abelian subgroup. We show that, in many situations, the corresponding $1/k$ is critical in the sense that there is no embedding of G with higher regularity. A particularly nice family where this happens is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity is equal to $1+1/n$.

Keywords

nilpotent groupsHeisenberg groupsdiffeomorphism groupsHölder continuitycritical regularity

MSC classification

Primary: 20F18: Nilpotent groups 37C05: Smooth mappings and diffeomorphisms
Secondary: 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 4 , April 2025 , pp. 1177 - 1198
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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