Degree growth for tame automorphisms of an affine quadric threefold

Dang, Nguyen-Bac

doi:10.2140/ant.2024.18.1

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Degree growth for tame automorphisms of an affine quadric threefold

Dang Nguyen-Bac

Vol. 18 (2024), No. 1, 1–86

DOI: 10.2140/ant.2024.18.1

Abstract

We consider the degree sequences of the tame automorphisms preserving an affine quadric threefold. Using some valuative estimates derived from the work of Shestakov and Umirbaev and the action of this group on a $CAT (0)$ , Gromov-hyperbolic square complex constructed by Bisi, Furter and Lamy, we prove that the dynamical degrees of tame elements avoid any value strictly between 1 and $\frac{4}{3}$ . As an application, these methods allow us to characterize when the growth exponent of the degree of a random product of finitely many tame automorphisms is positive.

Keywords

dynamical degrees, tame groups

Mathematical Subject Classification 2010

Primary: 14E07

Secondary: 37H15

Milestones

Received: 23 June 2019

Revised: 16 December 2021

Accepted: 1 February 2022

Published: 22 November 2023

Authors

Dang Nguyen-Bac
Universite Paris-Saclay Institut Mathématique d’Orsay Orsay France

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Vol. 18, No. 1, 2024