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

Abstract

For each integer n > 0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F n ) of a free group F n of rank n. For n 2, such groups Aut(X) are nonamenable, and for n 3, they are nonlinear and contain the braid group B n . Some of these varieties X are affine, and among affine, some are rational and some are not, some are smooth and some are singular. The byproduct is that for n 3, each Cremona group of rank 3n contains Aut(F n ) and the braid group B n .

Embeddings of automorphism groups of free groups into automorphism groups of affine algebraic varieties

“…The trend of the last decade has been the study of abstract-algebraic, topological, algebro-geometric, and dynamical properties of automorphism groups of algebraic varieties. This paper is related to this topic and continues the research started in author's paper [11].…”

“…], [16, 2.2, 2.3]). In view of (11), it is Tinvariant. The set Θ 0 of all elements of the form (12) with x α = 0 for each α ∈ Φ has the same properties.…”

“…In [11], an infinite series of irreducible algebraic varieties is constructed in whose automorphism group embeds the automorphism group of a free group F n of rank n. This has applications to the problems of linearity and amenability of automorphism groups of algebraic varieties and that of the embeddability of various groups into Cremona groups. To formulate the results obtained in this paper, we recall the construction introduced in [11].…”

“…For some (but not all) G and R, homomorphism ( 5) is an embedding. Namely, in [11] it is proved that (a) in the following cases, homomorphism ( 5) is an embedding:…”

“…The main result of the present paper is Theorem 1.1, in which the next step after [11] is taken towards answering the question posed: we add one more class to the triples of the specified type found in [11]: Theorem 1.1. Let G be a connected semisimple algebraic group and let R be a closed subgroup of its maximal torus.…”