The Linear Part of a Discontinuously Acting Euclidean Semigroup |
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Authors: | G. A. Soifer |
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Affiliation: | Department of Mathematics and Computer Science, Bar-Ilan University, 52900, Ramat-Gan, Israelf1 |
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Abstract: | 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 areTheorem.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 followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group. |
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