  We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended'
type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed
division algebras K=R,C,H,O, respectively. A crucial role is played by integral
lattices of the division algebras and associated discrete matrix groups. Our
findings can be summarized by saying that the even subgroups, W^+, of the
Kac-Moody Weyl groups, W, are isomorphic to generalized modular groups over K
for the simply laced algebras, and to certain finite extensions thereof for the
non-simply laced algebras. This hints at an extended theory of modular forms
and functions.
