The profinite polynomial automorphism group

TL;DR

This paper introduces the profinite tame polynomial automorphism group, a topologically enriched extension of the automorphism group over finite fields, revealing its density properties and relation to bijections.

Contribution

It defines the profinite tame polynomial automorphism group and demonstrates its inclusion of known non-tame automorphisms, suggesting tame maps are densely situated within automorphisms.

Findings

Most known non-tame automorphisms are inside the profinite tame automorphism group

The profinite tame automorphism group is close to the set of bijections from endomorphisms

Tame automorphisms may be dense in the automorphism group

Abstract

We introduce an extension of the (tame) polynomial automorphism group over finite fields: the profinite (tame) polynomial automorphism group, which is obtained by putting a natural topology on the automorphism group. We show that most known candidate non-tame automorphisms are inside the profinite tame polynomial automorphism group, giving another result showing that tame maps are potentially "dense" inside the set of automorphisms. We study the profinite tame automorphism group and show that it is not far from the set of bijections obtained by endomorphisms.