Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties

Abstract

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are nonlinear and contain the braid group $B_n$ on $n$ strands for $n\geqslant 3$, and are nonamenable for $n\geqslant 2$. As an application, it is proved that for $n\geqslant 3$, every Cremona group of rank $\geqslant 3n-1$ contains the groups ${\rm Aut}(F_n)$ and $B_n$. This bound is 1 better than the one published earlier by the author; with respect to $B_n$ the order of its growth rate is one less than that of the bound following from the paper by D. Krammer. The basis of the construction are triplets $(G, R, n)$, where $G$ is a connected semisimple algebraic group and $R$ is a closed subgroup of its maximal torus.