Faithful actions of automorphism groups of free groups on algebraic varieties

Transformation Groups. 2023. Vol. 28. No. 3. P. 1277–1297.

Popov V.

Considering a certain construction of algebraic varieties X endowed with an
algebraic action of the group Aut(F_n), n < \infty, we obtain a criterion for the faithfulness
of this action. It gives an in nite family F of Xs such that Aut(F_n) embeds into Aut(X).
For n > 2, this implies nonlinearity, and for n > 1, the existence of F_2 in Aut(X) (hence
nonamenability of the latter) for X in F. We find in F two in finite subfamilies N
and R consisting of irreducible ane varieties such that every X in N is nonrational
(and even not stably rational), while every X in 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.

Keywords: алгебраическое многообразие, группа автоморфизмов, algebraic variety, Automorphism Group Action, automorphism group, действие группы автомррфизмов