On the Smallest Non-trivial Action of SAut(Fn) for Small n

TL;DR

This paper studies the minimal non-trivial actions of the subgroup SAut(Fn) on small sets, showing that for n ≥ 5, any action on fewer than 18 elements must be trivial, thus identifying the smallest non-trivial actions.

Contribution

It improves previous results by precisely determining the minimal size of non-trivial actions of SAut(Fn) for small n using computational methods.

Findings

Actions on fewer than 18 elements are trivial for n ≥ 5

Established the smallest non-trivial actions for small n

Extended previous bounds on group actions

Abstract

In this paper we investigate actions of $\operatorname{SAut}(F_n)$, the unique index $2$ subgroup of $\operatorname{Aut}(F_n)$, on small sets, improving upon results by Baumeister--Kielak--Pierro for several small values of $n$. Using a computational approach for $n \geqslant 5$, we show that every action of $\operatorname{SAut}(F_n)$ on a set containing fewer than $18$ elements is trivial.