On some finite hyperbolic spaces.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1289.05035
author: Şükrü Olgun
author: Mustafa Saltan
publication date: 2013-12-06T00:00:00Z
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51E26
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 6236457
zbMATH Keywords: finite hyperbolic space
zbMATH Keywords: hyperoval
MaRDI profile type: Publication
title: On some finite hyperbolic spaces.
published in: Ars Combinatoria
review text: A hyperbolic plane \(\pi \) is a linear space satisfying three particular axioms:NEWLINENEWLINE(H1) For every non incident point-line pair \((X,l)\) there exist at least two lines incident with \(X\) not meeting \(l\).NEWLINENEWLINE(H2) There exists a set of four points no three of which are collinear.NEWLINENEWLINE(H3) If a subset \(S\) of the points of \(\pi \) contains three points not on a line and contains all points on the lines incident with any pair of points of \(S\), then \(S\) contains all the points of \(\pi \).NEWLINENEWLINEA hyperbolic plane is called regular of order \((k,r)\) if all lines are incident with exactly \(k\) points and all points are incident with exactly \(r\) lines. The short paper under review deals mainly with finite hyperbolic planes. The authors present a construction of a finite hyperbolic plane. Their construction is comparable to a construction described by \textit{T. G. Ostrom} in [Am. Math. Mon. 69, 899--901 (1962; Zbl 0111.17502)]. However, the authors restrict themselves to projective planes of even order \(n \geq 8\) and used hyperovals instead of ovals to construct \((n/2,n+1)\)-regular hyperbolic planes as follows. Consider the \(n+2\) lines of a dual hyperoval. The lines of the hyperbolic plane \(\pi _{n+2}\) are the lines of the projective plane different from the \(n+2\) lines. The points of the hyperbolic plane are the points of the projective plane different from the points contained in the \(n+2\) lines. This construction answers a question of \textit{R. J. Bumcrot} in [Atti Convegno Geom. combinat. Appl. Perugia 1970, 113--130 (1971; Zbl 0226.50019)]. The authors also show that their constructed plane is homogeneous, i.e., the collineation group acts transitively on any pair of points, provided the hyperoval is a conic plus its nucleus. NEWLINENEWLINENEWLINEThe final section of the paper is devoted to hyperbolic spaces of dimension three. The authors show that in a hyperbolic space, obtained from removing a tetrahed…
description: scientific article; zbMATH DE number 6236457