On some hyperbolic planes from finite projective planes

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≥m2+m). Consider the substructure Π0=(P0,L0,I0) with P0=P\{X∈P|XIl,  l∈L′}, L0=L\L′ where I0 stands for the restriction of I to P0×L0. It is shown that every Π0 is a hyperbolic plane, in the sense of Graves, if n≥m2+m+1+m2+m+2. Also we give some combinatorial properties of the line classes in Π0 hyperbolic planes, and some relations between its points and lines.