Automorphic forms, fake monster algebras and hyperbolic reflection groups

We construct 2 families of automorphic forms related to twisted fake monster algebras and calculate their Fourier expansions. This gives a new proof of their denominator identities and shows that they define automorphic forms of singular weight. We also obtain new infinite product identities which are the denominator identities of generalized Kac-Moody superalgebras. Finally we describe the reflection groups of the root lattices of these algebras.