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 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≥ 3 and are nonamenable for n≥ 2 . As an application, we prove that every Cremona group of rank ≥ 3n-1 contains the groups 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.