On the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$

TL;DR

This paper introduces 'uber-contracting' elements to establish acylindrical hyperbolicity of groups acting on complex spaces, applying this to prove the non-simplicity of the tame automorphism group of SL_2(C).

Contribution

It develops a new criterion using uber-contracting elements for proving acylindrical hyperbolicity in non-locally compact spaces, and applies it to a specific automorphism group.

Findings

Proves the tame automorphism group of SL_2(C) is acylindrically hyperbolic.

Introduces the concept of uber-contracting elements for group actions.

Provides a local criterion for constructing uber-contracting elements.

Abstract

We introduce the notion of \"uber-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct \"uber-contracting elements for groups acting on complexes with unbounded links of vertices. As an application, we show the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$, a subgroup of the $3$-dimensional Cremona group, through its action on a CAT(0) square complex recently introduced by Bisi-Furter-Lamy.