On homogeneous spaces with finite anti-solvable stabilizers

TL;DR

This paper studies homogeneous spaces with finite anti-solvable stabilizers, showing they are dominated by G-torsors and have rational points in certain cases, expanding understanding of algebraic group actions.

Contribution

It introduces a new class of anti-solvable groups and proves that homogeneous spaces with these stabilizers are dominated by G-torsors, ensuring rational points in specific instances.

Findings

Homogeneous spaces with finite anti-solvable stabilizers are dominated by G-torsors.

Such spaces have rational points when G=SL_n.

The work applies to a broad family including alternating and sporadic simple groups.

Abstract

We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for $n\neq 6$ and all 26 sporadic simple groups. We prove that, if $K$ is a perfect field and $X$ is a homogeneous space of a smooth algebraic $K$-group $G$ with finite geometric stabilizers lying in this family, then $X$ is dominated by a $G$-torsor. In particular, if $G=\mathrm{SL}_n$, all such homogeneous spaces have rational points.