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 normeddivision algebras K=R,C,H,O, respectively. A crucial role is played by integrallattices of the division algebras and associated discrete matrix groups. Ourfindings can be summarized by saying that the even subgroups, W^+, of theKac-Moody Weyl groups, W, are isomorphic to generalized modular groups over Kfor the simply laced algebras, and to certain finite extensions thereof for thenon-simply laced algebras. This hints at an extended theory of modular formsand functions.
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