Abstract
Let ℝn be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometrics 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 are Theorem. 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 following Margulis' Conjecture. If S is a crystallographic Euclidean semigroup, then S is a group.
| Original language | English |
|---|---|
| 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 |
|---|---|
| Germany-Israel Foundation | G-454-213.06/95 |
| Israeli Academy of Sciences | 8007/99-01 |