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(F-n), n < infinity, 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 is an element of F. We find in F two infinite subfamilies N and R consisting of irreducible affine varieties such that every X 2 N is nonrational (and even not stably rational), while every X is an element of R 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 B-n 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 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 3n, and the same is true for B-n.