Embeddings of groups \rm Aut(F_n) into automorphism groups of algebraic varieties

2021/06/03 by Popov, Vladimir L.
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.02072

Abstract

For every positive integer n, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties X,whose automorphism group \rm Aut(X) contains the automorphism group \rm Aut(F_n) of a free group F_n of rank n as a subgroup. This property implies that, for n \geqslant 2, such groups \rm Aut(X) are nonamenable, and, for n \geqslant 3, nonlinear and contain the braid group B_n on n strands. Some of these varieties X are affine, and among affine, some are rational and some are not, some are smooth and some are singular. As an application, we deduce that, for n \geqslant 3 , every Cremona group of rank \geqslant 3n contains the groups \rm Aut(F_n) and B_n as the subgroups. This bound is better than the one that follows from the paper by D. Krammer [14], where the linearity of the braid group B_n is proved.

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