The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by Lorimer [11], Richardson and Rubinstein [17] and Best [2].

Abstract.It is well known that regular maps exist on the projective plane but not on the Klein bottle, nor the non-orientable surface of genus 3. In this paper several infinite families of regular maps are constructed to show that such maps exist on non-odentable surfaces of over 77 per cent of all possible genera.Mathematics Subject Classification (1991): 05C25.

By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic −1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.The n-dimensional manifold Siegel problem. Determine the minimum possible volume obtained by an orientable hyperbolic n-manifold.All hyperbolic manifolds in this paper will be complete Riemannian manifolds of constant sectional curvature −1. The "full" Siegel problem refers to the problem above for orbifolds rather than manifolds. Nevertheless, the manifold Siegel problem is one with a long and venerable history. This paper describes our solution when n = 6. But first, an overview and some background. The Euler characteristic χ creates a big difference between even and odd dimensions. When n is even, the Gauss-Bonnet theorem gives vol(M ) = κ n χ(M ), with κ n = (−2π) n 2 /(n − 1)!! for the volume of an n-dimensional hyperbolic manifold M . As χ(M ) ∈ Z, the most obvious place to look for solutions to the problem is when |χ| = 1. A compact orientable M satisfies |χ(M )| ∈ 2Z, so the minimum volume is most likely achieved by a non-compact manifold. When n is odd, χ(M ) = 0, and a different approach must be found. For these reasons progress in even dimensions has been more rapid.

It is shown that any finitely generated, non-elementary Fuchsian group has among its homomorphic images all but finitely many of the alternating groups A n . This settles in the affirmative a long-standing conjecture of Graham Higman.

We define a homology theory for a certain class of posets equipped with a pre-sheaf of modules. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the "cube" complex defined by Khovanov.