On Some Finite Hyperbolic Spaces

Abstract

Let $\pi$ be a finite projective plane of order $n$. Consider the substructure ${\pi }_{n+2}$ obtained from $\pi$ by removing $n+2$ lines (including all points on them) no three of which are concurrent. In this paper, firstly, it is shown that ${\pi }_{n+2}$ is a B-L plane and it is also homogeneous. Let $PG(3,2)$ be a finite projective $3$-space of order $n$. The substructure obtained from $PG(3,2)$ by removing a tetrahedron that is four planes of $PG(3,n)$ no three of which are collinear is a finite hyperbolic $3$-space (Olgun-Ozgir [10]). Finally, we prove that any two hyperbolic planes with the same parameters are isomorphic in this hyperbolic $3$-space. These results appeared in the second author’s MSc thesis.