For an embeddable semigroup S, the free S-group is the unique group over which every surjective continuous S-action admits a natural extension, and left reversibility characterizes when all compact extensions factor through it.
Ceccherini-Silberstein, M
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Natural extensions of embeddable semigroup actions
For an embeddable semigroup S, the free S-group is the unique group over which every surjective continuous S-action admits a natural extension, and left reversibility characterizes when all compact extensions factor through it.