  We show using Borcherds products that for any fixed-point free automorphism
of the Leech lattice satisfying a "no massless states" condition, the
corresponding cyclic orbifold of the Leech lattice vertex operator algebra is
isomorphic to the Monster vertex operator algebra. This induces an "orbifold
duality" bijection between algebraic conjugacy classes of fixed-point free
automorphisms of the Leech lattice satisfying this condition and algebraic
conjugacy classes of non-Fricke elements in the Monster. We use the duality to
show that non-Fricke Monstrous Lie algebras are Borcherds-Kac-Moody Lie
algebras, and prove a refinement of Norton's Generalized Moonshine conjecture:
the ambiguous constants relating generalized moonshine Hauptmoduln under
conjugation and modular transformations are necessarily roots of unity. We also
describe a class of rank 2 Borcherds-Kac-Moody Lie algebras attached to the
Conway group.
