Regular polyhedra related to projective linear groups

AbstractFor each odd prime p, there is a regular polyhedron Πp of type {3,p} with 12(p2−1) vertices whose rotation group is PSL(2,p); its complete group is PSL(2,p) × Z2 or PGL(2,p) as p ≡ 1 or 3 (mod 4). If p ≡ 1 (mod 4), then the group of Πp contains a central involution, and identification of antipodal vertices under this involution yields another regular polyhedron Πp2 of type {3,p} with 14(p2)−1) vertices and group PSL(2,p). Realizations of the polyhedra in euclidean spaces are briefly described

Finite groups acting on homology manifolds

In this paper we study homology manifolds T admitting the action of a finite group preserving the structure of a regular CW-complex on T. The CW-complex is parameterized by a poset and the topological properties of the manifold are translated into a combinatorial setting via the poset. We concentrate on n-manifolds which admit a fairly rigid group of automorphisms transitive on the n-cells of the complex. This allows us to make yet another translation from a combinatorial into a group theoretic setting. We close by using our machinery to construct representations on manifolds of the Monster, the largest sporadic group. Some of these manifolds are of dimension 24, and hence candidates for examples to Hirzebruch's Prize Question in [HBJ], but unfortunately closer inspection shows the A^-genus of these manifolds is 0 rather than 1, so none is a Hirzebruch manifold