Algebraical surfaces with planes of skew symmetry

V. F. Ignatenko

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 5 (1998) no. 1, pp. 35-48

Résumé

Let $G$ be an infinite group generated by skew reflections with respect to hyperplanes in a real space $E^m$; $\mu_j$-planes $\Pi^{\mu_j}=\Pi^{d_j}\oplus\Pi^{\gamma_j}$ ($j=\overline{0, 3}$; $\gamma_0\ge\gamma_1\ge\gamma_2\ge\gamma_3$) be linear envelopes of the $G({\mathbf u})$-orbits of directions of symmetry ${\mathbf u}({\mathbf u}{\not\,\parallel}\Pi^{\gamma_j})$. We consider a case where $\dim\sum_k\Pi^{\gamma_k}=\sum_k{\gamma_k}$ and $\dim\left( \Pi^{\gamma_3}\cap\sum_k\Pi^{\gamma_k}\right)>0$ ($k=0,1,2$). It is proved that for any disposition of $\Pi^{\gamma_j}$ there exists the such an invariant of a certain $G$, the symmetry group of which is non-extended.