FAITHFUL ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON ALGEBRAIC VARIETIES

Considering a certain construction of algebraic varieties X endowed with an algebraic action of the group Aut(Fn), n < ∞, we obtain a criterion for the faithfulness of this action. It gives an infinite family 𝔉 of Xs such that Aut(Fn) embeds into Aut(X). For n ≥ 3, this implies nonlinearity, and for n ≥ 2, the existence of F2 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 3n-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties Z, for which Aut(Z) contains the braid group Bn on n strands, does not exceed 3n. This upper bound significantly strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of Bn was proved (this latter bound is quadratic in n). The same upper bound also holds for Aut(Fn). In particular, it shows that the minimal rank of the Cremona groups containing Aut(Fn), does not exceed 3n, and the same is true for Bn.