Embeddings of groups ${\rm Aut}(F_n)$ into automorphism groups of algebraic varieties

Abstract

For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$ of rank $n$ as a subgroup. This property implies that, for $n \geqslant 2$, such groups ${\rm Aut}(X)$ are nonamenable, and, for $n \geqslant 3$, nonlinear and contain the braid group $B_n$ on $n$ strands. Some of these varieties $X$ are affine, and among affine, some are rational and some are not, some are smooth and some are singular. As an application, we deduce that, for $n \geqslant 3 $, every Cremona group of rank $\geqslant 3n$ contains the groups ${\rm Aut}(F_n)$ and $B_n$ as the subgroups. This bound is better than the one that follows from the paper by D. Krammer [14], where the linearity of the braid group $B_n$ is proved.