  In this paper, we consider the degree sequences of the tame automorphisms
preserving an affine quadric threefold. Using some valuatives estimates derived
from the work of Shestakov-Umirbaev and the action of this group on a CAT(0),
Gromov-hyperbolic square complex constructed by Bisi-Furter-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.
