The Linear Part of a Discontinuously Acting Euclidean Semigroup,

Let n be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of n. We say that a semigroup S acts discontinuously on n if the subset {s S:sK ∩ K ≠ ︀} is finite for any compact set K of n. The main results of this work are Theorem. If S is a Euclidean semigroup which acts discontinuously on n, 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 n; 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 following Margulisʹ Conjecture. If S is a crystallographic Euclidean semigroup, then S is a group.