Faithful actions of automorphism groups of free groups on algebraic varieties

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].