The paper is devoted to the study of recently discovered generic isomorphisms between the Weyl groups of hyperbolic Kac-Moody algebras and modular groups over integer domains in normed division algebras. As the authors declare, this over fifty pages long work is rather a first step towards the comprehensive and complete theory. It contains numerous new results. The topic is of great interest, also due to the potential applications in quantum gravity and M-theory. The authors find realizations of the group action on generalized upper half-planes over the division algebras and construct explicitly automorphic forms for quaternionic and octonionic cases. Such generalized upper half-planes are associated with the modular groups appearing in the isomorphisms. Among the main new results is the explicit octavian realization of the even Weyl group \(W^+(E_{10})\) and then of \(W^+(E_{8})\) as its special case.
The paper consists of six sections and four appendices. From the contents: Introduction; Weyl groups as modular groups; \(D_4\) and its hyperbolic extension; \(E_8\) and its hyperbolic extension; Automorphic functions on \(\mathcal{H}(\mathbb{A})\).