Half-regular and regular points in compact polygons
type: Item
instance of: scholarly article
title: Half-regular and regular points in compact polygons
zbMATH Open document ID: 0813.51011
DOI: 10.1007/BF01263994
author: Andreas E. Schroth
author: Hendrik Van Maldeghem
published in: Geometriae Dedicata
publication date: 1994-11-07T00:00:00Z
review text: 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}_ …
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 663760
zbMATH Keywords: compact polygon
zbMATH Keywords: half-regular point
zbMATH Keywords: regular point
zbMATH Keywords: symplectic quadrangles
zbMATH Keywords: split Cayley hexagons
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01263994
OpenAlex ID: W1996802442
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description: scientific article; zbMATH DE number 663760