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For all classical groups (and for their analogs in infinite dimension or overgeneral base fields or rings) we construct certain contractions, called"homotopes". The construction is geometric, using as ingredient involutions ofassociative geometries. We prove that, under suitable assumptions, the groupsand their homotopes have a canonical semigroup completion.

Associative Geometries. II: Involutions, the classical torsors, and their homotopes