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Degree growth for tame automorphisms of an affine quadric threefold

Dang Nguyen-Bac

Vol. 18 (2024), No. 1, 1–86
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 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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