The linear part of a discontinuously acting Euclidean semigroup
A subsemigroup \(\Gamma\) of the group \(O(n)\ltimes\mathbb R^n\) of isometries of \(\mathbb R^n\) will be called a Euclidean semigroup. \(\Gamma\) is called properly discontinuous if the set \(\{\gamma\in\Gamma\); \(\gamma K\cap K\neq \emptyset\}\) is finite for every compact subset \(K\) of \(\mathbb R^n\). And \(\Gamma\) is called crystallographic if it is properly discontinuous and \(\bigcup_{\gamma\in\Gamma} \gamma K =\mathbb R^n\) for some compact subset \(K\) of \(\mathbb R^n\). Margulis conjectured that every Euclidean crystallographic semigroup is actually a group. The paper under review is a first step towards proving this conjecture. Its main result says that for a Euclidean properly discontinuous semigroup \(\Gamma\) the connected component of the closure of the linear part \(\ell(\Gamma)\) of \(\Gamma\) is a reducible group. Here the linear part is the natural homomorphism \(\ell : O(n)\ltimes\mathbb R^n\to O(n)\). In the proof an important role is played by the convex cone \(C(S)\) positively spanned by the translational parts of elements \(\gamma\in\Gamma\) with regular \(\ell(\gamma)\) and a Tits type alternative, proved by using dynamical properties of linear maps.
properly discontinuous Euclidean semigroup
scientific article; zbMATH DE number 1763550