Faithful actions of automorphism groups of free groups on algebraic varieties

TL;DR

This paper constructs algebraic varieties with faithful automorphism group actions of free group automorphisms, revealing nonlinearity, nonrationality, and bounds on embedding dimensions of braid and automorphism groups.

Contribution

It introduces a criterion for faithfulness of automorphism actions on varieties, producing new examples with specific properties and establishing bounds on embedding dimensions of braid and free group automorphisms.

Findings

Automorphism groups of free groups embed into automorphism groups of certain algebraic varieties.

The minimal dimension for varieties containing braid groups or automorphism groups of free groups is at most 3n.

Existence of varieties with faithful automorphism actions that are nonrational or rational, with specified dimensions.

Abstract

Considering a certain construction of algebraic varieties $X$ endowed with an algebraic action of the group ${\rm Aut}(F_n)$, $n<\infty$, we obtain a criterion for the faithfulness of this action. It gives an infinite family $\mathscr F$ of $X$'s such that ${\rm Aut}(F_n)$ embeds into ${\rm Aut}(X)$. For $n\geqslant 3$, this implies nonlinearity, and for $n\geqslant 2$, the existence of $F_2$ in ${\rm Aut}(X)$ (hence nonamenability of the latter) for $X\in \mathscr F$. We find in ${\mathscr F}$ two infinite subfamilies ${\mathscr N}$ and $\mathscr R$ consisting of irreducible affine varieties such that every $X\in {\mathscr N}$ is nonrational (and even not stably rational), while every $X\in \mathscr R$ is rational and $3n$-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties $Z$, for which ${\rm Aut}(Z)$ contains the braid group $B_n$ on $n\geqslant 3$ strands, does not exceed $3n$. This upper bound strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of $B_n$ was proved (this latter bound is quadratic in $n$). The same upper bound also holds for ${\rm Aut}(F_n)$. In particular, it shows that the minimal rank of the Cremona groups containing ${\rm Aut}(F_n)$, does not exceed $3n$, and the same is true for $B_n$ if $n\geqslant 3$. This paper is a major revision of [Po21].