## Abstract

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

Original language | English |
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Pages (from-to) | 647-663 |

Number of pages | 17 |

Journal | Journal of Algebra |

Volume | 250 |

Issue number | 2 |

DOIs | |

State | Published - 15 Apr 2002 |

### Bibliographical note

Funding Information:1Partially supported by Germany-Israel Foundation grant G-454-213.06/95 and Israeli Academy of Sciences grant 8007/99-01.

### Funding

1Partially supported by Germany-Israel Foundation grant G-454-213.06/95 and Israeli Academy of Sciences grant 8007/99-01.

Funders | Funder number |
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Germany-Israel Foundation | G-454-213.06/95 |

Israeli Academy of Sciences | 8007/99-01 |