Real approximation for homogeneous spaces with finite stabilizers

TL;DR

This paper advances the understanding of real approximation on homogeneous spaces with finite stabilizers, leveraging recent progress in Brauer--Manin obstruction theory, and establishes new cases where approximation holds.

Contribution

It introduces new results on real approximation for homogeneous spaces with finite stabilizers using recent developments in Brauer--Manin obstruction theory.

Findings

Finite k-groups split by 2-primary extensions satisfy real approximation.

Provides proofs of known results not previously documented in literature.

Describes the current state of research on real approximation for these spaces.

Abstract

We prove some new cases of real appoximation for homogeneous spaces with finite stabilizers and describe the state of the art around this question, giving proofs that are well-known to experts but that, to our knowledge, cannot be found in the literature. Our main new result needs the latest advances in the topic of the Brauer--Manin obstruction for homogeneous spaces with supersolvable stabilizers. It states that any finite $k$-group that is split by a $2$-primary extension satisfies real approximation.