Properties of Reflection Geometries and the Corresponding Group Spaces

Abstract

To any reflection group, i.e. a pair $(G, \mathfrak {D})$ consisting of a group G and a set $\mathfrak {D}$ of involutory generators satisfying the “Three Reflection Axiom” there can be associated a planar (called reflection geometry) and a spatial (called group space) incidence structure. In the spatial case the elements $\alpha$ of $\mathfrak {D}^2$ are called “points” and via the notion pencil and proper pencil we define certain subsets of $\mathfrak {D}^2$ as lines and projective lines. Let $\mathfrak {G}$ denote the set of all lines and $\mathfrak {G}_0$ the subset of all projective lines. Then $(G,\mathfrak {G})$ is an incidence space which can be provided with two parallelism $\parallel _l$ and $\parallel _r$ such that $(G,\mathfrak {G},\parallel _l,\parallel _r)$ becomes a double space. Between a point $\alpha$ and the elements $\varepsilon$ of $\mathfrak {D}^3$ we define an incidence relation by $\varepsilon \cdot \alpha \in \mathfrak {D}$. Then $\langle \varepsilon \rangle := \{\xi \in \mathfrak {D}^2 \ | \ \varepsilon \cdot \xi \in \mathfrak {D} \}$ is a 2-dimensional subspace of $(G,\mathfrak {G})$, i.e. a plane. In this paper we show: If $L \in \mathfrak {G}_0$ is a projective line and $\varepsilon \in \mathfrak {D}^3$ then $L \cap \langle \varepsilon \rangle \ne \emptyset$. If $|L \cap \langle \varepsilon \rangle | \ge 2$ then $L \subseteq \langle \varepsilon \rangle$. If $\alpha \in \mathfrak {D}^2 {\setminus } L$ then there is exactly one $\delta \in \mathfrak {D}^3$ with $\alpha \cup L \subseteq \langle \delta \rangle$. There is exactly one line $M \subseteq \langle \varepsilon \rangle$ with $M \parallel _l L$ denoted by $(\varepsilon \parallel _l, L) := M$ (resp. $M \parallel _r L$ denoted by $(\varepsilon \parallel _r, L)$) (cf. Theorem 3.10).