[Paper Review] Degree growth for tame automorphisms of an affine quadric threefold

Nguyen-Bac Dang|arXiv (Cornell University)|Oct 22, 2018

Mathematical Dynamics and Fractals36 references4 citations

TL;DR

This paper investigates the degree growth of tame automorphisms on an affine quadric threefold defined by $xt - yz - 1 = 0$, using geometric group actions on a CAT(0) square complex. It proves that the first dynamical degree $\lambda_1(f)$ of any such automorphism is either 1 or at least $4/3$, and characterizes when random products of these automorphisms exhibit positive degree growth exponent, resolving a question by Urech in this setting.

ABSTRACT

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.

Motivation & Objective

• To understand the asymptotic degree growth of iterated tame automorphisms on an affine quadric threefold over an algebraically closed field of characteristic zero.
• To determine the possible values of the first dynamical degree $\lambda_1(f)$ for tame automorphisms of the threefold.
• To characterize when the degree growth exponent of a random product of finitely many tame automorphisms is positive.
• To resolve a question by Urech (2016) on the existence of fibrations or linear conjugacy for automorphisms with $\lambda_1(f) = 1$.

Proposed method

• Utilizes valuative estimates derived from Shestakov-Umirbaev theory to analyze degree sequences of automorphisms.
• Applies the action of the tame automorphism group on a $\operatorname{CAT}(0)$, Gromov-hyperbolic square complex constructed by Bisi-Furter-Lamy.
• Employs the structure of the group action to classify automorphisms into three types based on degree growth: bounded, linear, or exponential.
• Uses the drift of random walks on the square complex and associated Bass-Serre trees to analyze the growth exponent of degree sequences.
• Applies the law of large numbers for random walks on groups and trees to estimate the asymptotic degree growth of random products.
• Relies on isometric group actions and distance estimates in the complex to bound degree growth from below using the constant $4/3$.

Experimental results

Research questions

• RQ1 What values can the first dynamical degree $\lambda_1(f)$ take for tame automorphisms of the affine quadric threefold?
• RQ2 Under what conditions does a random product of tame automorphisms exhibit positive degree growth exponent?
• RQ3 When is a tame automorphism with $\lambda_1(f) = 1$ conjugated to a linear map or preserving a fibration?
• RQ4 Can the degree growth of iterated tame automorphisms be fully classified via geometric group actions?
• RQ5 How does the structure of the group action on the $\operatorname{CAT}(0)$ square complex relate to the asymptotic degree behavior?

Key findings

• The first dynamical degree $\lambda_1(f)$ of any tame automorphism $f$ on the affine quadric threefold satisfies $\lambda_1(f) \in \{1\} \cup [4/3, \infty)$, with no values strictly between 1 and $4\!\!\:/\!\!\!3$.
• For any tame automorphism $f$, the degree sequences $\deg(f^n)$ and $\deg(f^{-n})$ are either bounded, grow linearly, or grow at least exponentially with rate at least $\left(\frac{4}{3}\right)^n$.
• If $\lambda_1(f) = 1$, then $f$ either preserves a fibration or is conjugated to a linear map, and both $\deg(f^n)$ and $\deg(f^{-n})$ are either bounded or grow linearly.
• The degree growth exponent $\lambda$ of a random product of finitely many tame automorphisms is positive if and only if the group generated is non-elementary in the square complex or contains non-commuting hyperbolic isometries in the associated Bass-Serre tree.
• The growth exponent $\lambda$ is zero if the group is elementary or if all elements have uniformly bounded degree, and positive otherwise, as shown via random walk drift on the geometric complexes.
• The paper confirms a conjecture of Urech (2016) in this special case: automorphisms with $\lambda_1(f) = 1$ either preserve a fibration or are conjugated to linear maps.