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

V. L. Popov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 287-297

Résumé

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.