On some hyperbolic planes from finite projective planes

Abstract

Abstract. Let Π = (P ,L,I) be a finite projective plane of order n, and let Π = (P ,L ,I ) be a subplane of Π with order m which is not a Baer subplane (i.e., n ≥ m 2 +m). Consider the substructure, where I 0 stands for the restriction of I to P 0 ×L 0 . It is shown that every Π 0 is a hyperbolic plane, in the sense of Graves, if n ≥ m 2 +m+1+ m 2 + m + 2. Also we give some combinatorial properties of the line classes in Π 0 hyperbolic planes, and some relations between its points and lines.