FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES

• Published in: Transformation groups 2023-09, Vol.28 (3), p.1277-1297
• Main Author: POPOV, VLADIMIR L.
• Format: Article
• Language: English
• Subjects: Algebra; Automorphisms; Braid theory; Group theory; Lie Groups; Mathematics; Mathematics and Statistics; Topological Groups; Upper bounds
• ISSN: 1083-4362; 1531-586X
• DOI: 10.1007/s00031-023-09819-y

Considering a certain construction of algebraic varieties X endowed with an algebraic action of the group Aut( F n ), n < ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family F of X s such that Aut( F n ) embeds into Aut( X ). For n ≥ 3, this implies nonlinearity, and for n ≥ 2, the existence of F 2 in Aut( X ) (hence nonamenability of the latter) for X ∈ F . We find in F two infinite subfamilies N and R consisting of irreducible affine varieties such that every X ∈ N is nonrational (and even not stably rational), while every X ∈ F is rational and 3 n -dimensional. As an application, we show that the minimal dimension of affine algebraic varieties Z , for which Aut( Z ) contains the braid group B n on n strands, does not exceed 3 n . This upper bound significantly strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of B n was proved (this latter bound is quadratic in n ). The same upper bound also holds for Aut( F n ). In particular, it shows that the minimal rank of the Cremona groups containing Aut( F n ), does not exceed 3 n , and the same is true for B n .