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Abstract

It is known that, under a necessary non-compactness assumption, the
Brauer-Manin obstruction is the only one to strong approximation on homogeneous
spaces $X$ under a linear group $G$ (or under a connected algebraic group,
under assumption of finiteness of a suitable Tate-Shafarevich group), provided
that the geometric stabilizers of $X$ are connected. In this work we prove,
under similar assumptions, that the \'etale-Brauer-Manin obstruction to strong
approximation is the only one for homogeneous spaces with arbitrary
stabilisers. We also deal with some related questions, concerning strong
approximation outside a finite set of valuations. Finally, we prove a
compatibility result, suggested to be true by work of Cyril Demarche, between
the Brauer-Manin obstruction pairing on quotients $G/H$, where $G$ and $H$ are
connected algebraic groups and $H$ is linear, and certain abelianization
morphisms associated with these spaces.