  Starting from Borcherds' fake monster Lie algebra we construct a sequence of
six generalized Kac-Moody algebras whose denominator formulas, root systems and
all root multiplicities can be described explicitly. The root systems decompose
space into convex holes, of finite and affine type, similar to the situation in
the case of the Leech lattice. As a corollary, we obtain strong upper bounds
for the root multiplicities of a number of hyperbolic Lie algebras, including
AE3.
