On some finite hyperbolic spaces. (Q2864460)

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 tetrahedron from a finite projective space of dimension three, hyperbolic planes with the same parameters are isomorphic.