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

TL;DR

The paper constructs new algebraic varieties whose automorphism groups contain free group automorphisms and braid groups, revealing complex nonlinear structures and implications for Cremona groups.

Contribution

It introduces a novel series of algebraic varieties with automorphism groups embedding free group automorphisms and braid groups, improving bounds on Cremona group embeddings.

Findings

Automorphism groups are nonlinear and contain braid groups for n≥3.

Cremona groups of rank ≥3n-1 contain free group automorphisms and braid groups.

Bound on embedding size is improved by one over previous results.

Abstract

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are nonlinear and contain the braid group $B_n$ on $n$ strands for $n\geqslant 3$, and are nonamenable for $n\geqslant 2$. As an application, it is proved that for $n\geqslant 3$, every Cremona group of rank $\geqslant 3n-1$ contains the groups ${\rm Aut}(F_n)$ and $B_n$. This bound is 1 better than the one published earlier by the author; with respect to $B_n$ the order of its growth rate is one less than that of the bound following from the paper by D. Krammer. The basis of the construction are triplets $(G, R, n)$, where $G$ is a connected semisimple algebraic group and $R$ is a closed subgroup of its maximal torus.