Centralizers in the Jonquières group
type: Item
instance of: scholarly article
title: Centralizers in the Jonquières group
zbMATH Open document ID: 1273.14028
author: Julie Déserti
author: Dominique Cerveau
published in: Michigan Mathematical Journal
publication date: 2013-01-16T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.mmj/1353098512
review text: There is a strong dichotomy among polynomial automorphisms of the complex plane, due to \textit{S. Friedlander} and \textit{J. Milnor} [Ergodic Theory Dyn. Syst. 9, No. 1, 67--99 (1989; Zbl 0651.58027)]: if \(f\) is such an automorphism, then either \(\lambda(f)>1\) or \(\lambda(f)=1\). In the first case, \(f\) is conjugate to a product of Henon maps, and its centralizer is an extension of \(\mathbb{Z}\) by a finite group \(\mathbb{Z}/N\mathbb{Z}\), hence countable. If \(\lambda(f)=1\), \(f\) is conjugate to an elementary automorphism and its centralizer is uncountable. For birational transformations of the projective plane, the dynamical classification is due to \textit{J. Diller} and \textit{C. Favre} [Am. J. Math. 123, No. 6, 1135--1169 (2001; Zbl 1112.37308)] and goes as follows: the sequence \((\lambda(f^ n))_{n>0}\) is either asymptotically bounded, linear, quadratic or exponential. In the bounded case but for transformations of infinite order, the classification of the centralizers has been achieved by \textit{J. Blanc} and \textit{J. Déserti} [``Degree growth of birational maps of the plane'', \url{arxiv:1109.6810}]. In the quadratic case, standard considerations about elliptic surfaces allow to prove that the centralizers of such birational maps are virtually abelian. Recently, \textit{J. Blanc} and \textit{S. Cantat} [``Dynamical degrees of birational transformations of projective surfaces'', \url{arxiv:1307.0361}] proved that the centralizers of elements of exponential type were infinite and virtually cyclic (i.e., they contain \(\mathbb{Z}\) as a finite-index subgroup). The aim of the present paper is to complete the description of the centralizers of elements in the Cremona group by dealing with the linear case. Recall that birational maps with linear degree growth preserve a rational fibration. If \(\varphi\) is such a map and \(\mathrm{Ab}(\varphi)\) is the maximal abelian subgroup containing \(\varphi\) and preserving the \(\varphi\)-invariant fibrat…
reviewed by: Julien Grivaux
Mathematics Subject Classification ID: 14E07
zbMATH DE Number: 6126101
zbMATH Keywords: Cremona group
zbMATH Keywords: Jonquiŕes group
MaRDI profile type: Publication
OpenAlex ID: W1971340229
cites work: Q5589098
cites work: Elements and cyclic subgroups of finite order of the Cremona group
cites work: On birational transformations of pairs in the complex plane
cites work: The \(C^1\) generic diffeomorphism has trivial centralizer
cites work: On the groups of birational transformations of surfaces
cites work: Analytic actions of mapping class groups on surfaces
cites work: Feuilletages et Transformations Périodiques
cites work: Groupes d'automorphismes de $({\bbfC},0)$ et équations différentielles $ydy+\cdots =0$
cites work: Experiments on some birational quadratic transformations
cites work: Dynamics of bimeromorphic maps of surfaces
cites work: Q3737734
cites work: Trivial centralizers for axiom A diffeomorphisms
cites work: Trivial centralizers for codimension-one attractors
cites work: Dynamical properties of plane polynomial automorphisms
cites work: Q3864641
cites work: Über ganze birationale Transformationen der Ebene.
cites work: The Tits alternative for \(\text{Aut}[\mathbb C^2]\)
cites work: Q4160102
cites work: Centralizers of Anosov diffeomorphisms on tori
cites work: Rigidity of centralizers of diffeomorphisms
cites work: Non linearizable holomorphic dynamics having an uncountable number of symmetries
DOI: 10.1307/MMJ/1353098512
Recommended article: Centralizers of the Riordan group
Recommended article: Q3770706
Recommended article: Centralizers in pseudo-finite groups
Recommended article: Centralizers in finite solvable groups
Recommended article: Centralizers of gap groups
Recommended article: Centralizers in groups with finiteness conditions
Recommended article: Centralizers in \(\tilde A_2\) groups
Recommended article: Q5441351
Recommended article: Centralisers of Galois group actions
Recommended article: On centralizers in locally finite groups
description: scientific article; zbMATH DE number 6126101