On the Classification of Hyperbolic Root Systems of the Rank Three. Part Iii

See Parts I and II in alg-geom/9711032 and 9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector ρ with ρ < 0 (i. e. of the hyperbolic type). We give classification of all reflective of hyperbolic type elementary hyperbolic lattices of the rank three. For elliptic (when ρ > 0) and parabolic (when ρ = 0) types it was done in Parts I and II. We apply the same arguments as for elliptic and parabolic types: the method of narrow parts of polyhedra in hyperbolic spaces, and class numbers of central symmetries. But we should say that for the hyperbolic type all considerations are much more complicated and required much more calculations and time. These results are important, for example, for Theory of Lorentzian Kac–Moody algebras and some aspects of Mirror Symmetry. We also apply these results to prove boundedness of families of algebraic surfaces with almost finite polyhedral Mori cone (see math.AG/9806047 about this subject).