We find 26 reflections in the automorphism group of the Lorentzian Leech lattice L over Z[e 2πi/3 ] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the reflection group.

We study a second example of the phenomenon studied in the article "The complex Lorentzian Leech lattice and the bimonster." We find 14 roots in the automorphism group of the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram given by the incidence graph of projective plane over F 2 . We prove that the reflections in these roots generate the automorphism group of L. The investigation is guided by an analogy with the theory of Weyl groups. There is a unique point in the quaternionic hyperbolic space fixed by the "diagram automorphisms" that we call the Weyl vector. The unit multiples of the 14 roots forming the diagram are the analogs of the simple roots. The 14 mirrors perpendicular to the simple roots are the mirrors that are closest to the Weyl vector.

Abstract. We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c's and develop enough algebraic topology in this setting to prove the Poincare duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.

Abstract. We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from C n , or complex hyperbolic space CH n , or the Hermitian symmetric space for O(2, n), and then takes the quotient by a discrete group P Γ. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if P Γ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in CH 13 . The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M , that gives geometric meaning to the generators and relations in the Conway-Simons presentation of (M × M ) : 2.