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

Vladimir L. Popov

doi:10.1134/S0081543823010121

paper

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

arXiv:2207.13072 · doi:10.1134/S0081543823010121

Abstract

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group $Aut (F_{n})$ 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 $n ⩾ 3$ , and are nonamenable for $n ⩾ 2$ . As an application, it is proved that for $n ⩾ 3$ , every Cremona group of rank $⩾ 3 n - 1$ contains the groups $Aut (F_{n})$ 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.

arXiv comment14 pages