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Does there exist a model of hyperbolic geometry such that only finite number of distinct parallel lines through a point which does not intersect given line?

Edit (Misha): I usually do not edit other people's questions, but I do in this case, since I think there is an interesting question behind the post, which is likely to be closed.

My (somewhat vague) question is: Are there "interesting" hyperbolic geometries over finite fields? Are any of them of any use in algebra, say, in finite group theory and number theory? What are (if any) references on this topic?

As Doug noted, it all depends on what one finds "interesting" and how many axioms/properties of hyperbolic geometry one is willing to drop. Note that, for instance, for finite fields there is no invariant notion of order (and, hence, betweenness), but there is a natural cyclic order (for the primary field). Can one exploit this to define a finite analogue of the metric hyperbolic geometry (which has triangle inequalities)?

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    $\begingroup$ This depends on what you want to give up. If you retain the axiom of betweenness, then between two lines which don't intersect the line you can find infinitely many. If you give up the axiom of betweenness, then you can view some block designs like most Steiner triple systems as finite hyperbolic planes. However, these are too flexible and it is hard to impose any structure on them, or say that they are really geometries. $\endgroup$
    – Douglas Zare
    Commented Jan 8, 2014 at 4:34
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    $\begingroup$ I do not understand what "a model of hyperbolic geometry" means precisely. If the geometry is fixed and the model only changes (Klein model vs. Poincaré model vs. hyperboloïd model etc.) then the number of distinct parallel lines cannot be anything than infinite. $\endgroup$
    – Benoît Kloeckner
    Commented Jan 8, 2014 at 14:28

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Here is how you can define a hyperbolic plane over a finite field $F$, I do not know if it is sufficiently useful or interesting though.

Let $V$ be a 3-dimensional vector space over $F$; let $Q$ be a nondegenerate
isotropic bilinear form on $V$, i.e., such that the null-cone $N$ of $Q$ is nonzero: $$ N=\{n\in V: Q(n,n)=0\}. $$

Define the "positive cone" in $V$ to be the set $C$ consisting of vectors $v\notin N$ so that $Q$ restricted to $v^\perp$ is anisotropic. Then define a finite hyperbolic plane $H$ to be the projection of $C$ to the projective plane $PV$. The group $PO(V,Q)$ serves as the natural transformation group for $H$. Lines in $H$ are projections of planes $W\subset V$ such that $Q$ restricted to $W$ is nondegenerate and isotropic. Lastly, define "distance" between points in $H$ as $$ d([u], [v])= \frac{Q(u,v)^2}{Q(u,u) Q(v,v)}. $$

See for instance here for a detailed discussion of quadratic forms over finite fields.

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