Degree growth for tame automorphisms of an affine quadric threefold

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 $\mathrm{CAT}<!--FUNCTION\; APPLICATION-->(0)$, 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.