Regular polyhedra related to projective linear groups (Q1182578)

For each odd prime \(p\) the author constructs a regular map of type \(\{3,p\}\) on an orientable surface with \((p^ 2-1)/2\) vertices. For \(p=3,5,7\) this leads to the tetrahedron, the icosahedron, and Klein's map \(\{3,7\}_ 8\), respectively. If \(p\equiv 1(\bmod 4)\) there is a 2-fold quotient of the same type \(\{3,p\}\) on a nonorientable surface with \((p^ 2-1)/4\) vertices. The automorphism group in this case is \(PSL(2,p)\). The construction involves specific algebraic properties of this group. Finally realizations in high dimensional euclidean space are discussed. These are based on appropriate orthogonal representations of \(PSL(2,p)\) or \(PGL(2,p)\).