The Linear Part of a Discontinuously Acting Euclidean Semigroup

AbstractLet Rn be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of Rn. We say that a semigroup S acts discontinuously on Rn if the subset {s∈S:sK∩K≠∅} is finite for any compact set K of Rn. The main results of this work areTheorem.If S is a Euclidean semigroup which acts discontinuously on Rn, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group.Corollary.Let S be a Euclidean semigroup acting discontinuously on Rn; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n).These results are the first step in the proof of the followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group