Half-regular and regular points in compact polygons
The authors consider compact generalized \(n\)-gons whose point rows and line pencils are manifolds. \textit{N. Knarr} [Forum Math. 2, No. 6, 603-512 (1990; Zbl 0711.51002)] showed that in this case there are positive integers \(p\) and \(q\) such that every point row is homeomorphic to the Euclidean \(p\)-sphere \(S_ p\) and every line pencil is homeomorphic to \(S^ q\). Such generalized \(n\)-gons are said to be of order \((p,q)\). Furthermore, such \(n\)-gons exist only for \(n=2,3,4,6\). Since \(n=3\) yields compact projective planes, the authors are mainly concerned with \(n=4,6\). The main results are geometric characterizations of the symplectic quadrangles over \(\mathbb{R}\) or \(\mathbb{C}\) and the split Cayley hexagons over \(\mathbb{R}\) or \(\mathbb{C}\). Furthermore, the concepts of half-regular and regular points and derivations at such points are defined. The notion of a (half- ) regular point was introduced for finite generalized hexagons by the second author and \textit{I. Bloemen} [Eur. J. Comb. 14, No. 6, 593-604 (1993; Zbl 0804.51006)]. A generalized \(n\)-gon is called (half-) regular if every point is (half-) regular. Let \({\mathcal S}_ m (x)\) denote the set of all elements in the generalized \(n\)-gon of distance \(m\) from \(x\). A half-regular point is a point \(p\) such that for all pairs \(x\), \(y\) in \({\mathcal S}_ 2(p)\) with \(p\), \(x\) and \(y\) being not collinear the set \({\mathcal S}_ 2 (p)\cap {\mathcal S}_{n-2} (z)\) is independent of the choice of \(z\in {\mathcal S}_{n-2} (x)\cap {\mathcal S}_{n-2} (y)\cap {\mathcal S}_ n (p)\); then \({\mathcal S}_ 2 (p)\cap {\mathcal S}_{n-2} (z)\) is called an ideal line. (This definition agrees with the usual definition of a regular generalized quadrangle. However, half- regular generalized quadrangles are also regular in the new sense below.) The derivation \({\mathcal A}_ p\) at a point \(p\) has point set all points collinear to \(p\) and line set all ideal lines \({\mathcal S}_ 2 (p)\cap {\mathcal S}_{n-2} (q)\) for \(q\in {\mathcal S}_ n (p)\) together with all ordinary lines through \(p\). In fact, half-regular points \(p\) can be characterized by the property that the derivation \({\mathcal A}_ p\) is a linear space. It is then shown that the derivation of a compact generalized \(n\)-gon of order \((s,t)\) at a half-regular point is a topological projective plane of and only if \(s=t\). The symplectic quadrangle over \(\mathbb{R}\) or \(\mathbb{C}\) is characterized, among other equivalent conditions, as a locally compact connected generalized quadrangle of finite dimension such that for every point the derived structure \({\mathcal A}_ p\) is a projective plane. The characterizations listed in the paper under review slightly extend characterizations of the symplected quadrangle over \(\mathbb{R}\) or \(\mathbb{C}\) by the first author [Arch. Math. 58, No. 1, 98-104 (1992; Zbl 0761.51004)]. A point \(p\) is called regular if it is half-regular and if for every \(u\in {\mathcal S}_ 4 (p)\) the set \({\mathcal S}_ 2 (y)\cap {\mathcal S}_{n-2} (z)\) is independent of the choice of \(y\in {\mathcal S}_ 2 (p)\cap {\mathcal S}_ 2 (u)\) and \(z\in {\mathcal S}_{n-2} (p)\cap {\mathcal S}_{n-2}(u)\cap {\mathcal S}_ n (y)\). The derivation \({\mathcal S}_ p\) at a regular point \(p\) is defined rather differently. The points are the ideal planes through \(p\) (these are only defined for hexagons; roughly speaking, an ideal plane through \(p\) is the collection of all ideal lines through \(p\)) together with the sets \(x^ \perp= {\mathcal S}_ 2 (x\cup \{x\})\) for \(x\) collinear to \(p\); the lines are all ordinary and ideal lines through \(p\). In the case of a generalized quadrangle the derivations \({\mathcal S}_ p\) and \({\mathcal A}_ p\) at a regular point \(p\) are isomorphic. It is shown that the derivation of a locally compact generalized hexagon of order \((s,t)\) at a regular point is a compact quadrangle if and only if \(s=t\). The split Cayley hexagon over \(\mathbb{R}\) or \(\mathbb{C}\) is characterized, among other equivalent conditions, as a compact connected generalized hexagon where point row and line pencils are manifolds such that for every point the derived structure \({\mathcal A}_ p\) is a projective plan or such that for every point the derived structure \({\mathcal S}_ p\) is a quadrangle. These characterizations heavily rely on \textit{M. A. Ronan's} [Invent. Math. 57, 227-262 (1980; Zbl 0429.51002)] reconstruction of subhexagons for the group \(G_ 2\).
scientific article; zbMATH DE number 663760