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

Abstract

Abstract 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.