Distorted diffeomorphisms

Abstract: An element of a finitely-generated group is said to be distorted if the word-length of its powers grows sublinearly. An element of a general group is said to be distorted if it is distorted inside a finitely-generated subgroup. This notion was introduced by Gromov and is worth studying in many frameworks. In this talk I will be interested in diffeomorphisms groups.

Calegary and Freedman showed that many homeomorphisms are distorted, However, in general, \(C^1\) diffeomorphisms are not, for instance due to the existence of hyperbolic fixed points. Studying similar phenomena in higher regularity turns out to be interesting in the context of elliptic dynamics. In particular, we may address the following question: Given \(r>s>1\), does there exist undistorted \(C^r\) diffeomorphisms that are distorted inside the group of \(C^s\) diffeomorphisms? After a general discussion, we will focus on the 1–dimensional case of this question for \(r=2\) and \(s=1\), for which we solve it in the affirmative via the introduction of a new invariant, namely the asymptotic variation.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar