Embeddings of Automorphism Groups of Free Groups into Automorphism Groups of Affine Algebraic Varieties

For every integer $n>0$, we construct a new infinite series of rational affine algebraic varieties such that their automorphism groups contain the automorphism group $\mathrm {Aut}(F_n)$ of the free group $F_n$ of rank $n$ and the braid group $B_n$ on $n$ strands. The automorphism groups of such varieties are nonlinear for $n\geq 3$ and are nonamenable for $n\geq 2$. As an application, we prove that every Cremona group of rank ${\geq }\,3n-1$ contains the groups $\mathrm {Aut}(F_n)$ and $B_n$. This bound is $1$ better than the bound 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 a paper by D. Krammer. The construction is based on triples $(G,R,n)$, where $G$ is a connected semisimple algebraic group and $R$ is a closed subgroup of its maximal torus.