Description:
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<jats:title>Abstract</jats:title>
<jats:p> For every integer <jats:inline-formula><jats:tex-math>$$n>0$$</jats:tex-math></jats:inline-formula>, we construct a new infinite series of rational affine algebraic varieties such that their automorphism groups contain the automorphism group <jats:inline-formula><jats:tex-math>$$\mathrm{Aut}(F_n)$$</jats:tex-math></jats:inline-formula> of the free group <jats:inline-formula><jats:tex-math>$$F_n$$</jats:tex-math></jats:inline-formula> of rank <jats:inline-formula><jats:tex-math>$$n$$</jats:tex-math></jats:inline-formula> and the braid group <jats:inline-formula><jats:tex-math>$$B_n$$</jats:tex-math></jats:inline-formula> on <jats:inline-formula><jats:tex-math>$$n$$</jats:tex-math></jats:inline-formula> strands. The automorphism groups of such varieties are nonlinear for <jats:inline-formula><jats:tex-math>$$n\geq 3$$</jats:tex-math></jats:inline-formula> and are nonamenable for <jats:inline-formula><jats:tex-math>$$n\geq 2$$</jats:tex-math></jats:inline-formula>. As an application, we prove that every Cremona group of rank <jats:inline-formula><jats:tex-math>$${\geq}\,3n-1$$</jats:tex-math></jats:inline-formula> contains the groups <jats:inline-formula><jats:tex-math>$$\mathrm{Aut}(F_n)$$</jats:tex-math></jats:inline-formula> and <jats:inline-formula><jats:tex-math>$$B_n$$</jats:tex-math></jats:inline-formula>. This bound is <jats:inline-formula><jats:tex-math>$$1$$</jats:tex-math></jats:inline-formula> better than the bound published earlier by the author; with respect to <jats:inline-formula><jats:tex-math>$$B_n$$</jats:tex-math></jats:inline-formula>, 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 <jats:inline-formula><jats:tex-math>$$(G,R,n)$$</jats:tex-math></jats:inline-formula>, where <jats:inline-formula><jats:tex-math>$$G$$</jats:tex-math></jats:inline-formula> is a connected semisimple algebraic group and <jats:inline-formula><jats:tex-math>$$R$$</jats:tex-math></jats:inline-formula> is a closed subgroup of its maximal torus. </jats:p>
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