Realizations of Symmetric Maps by Symmetric Polyhedra

Representations by polyhedra of several regular maps of positive genus are developed. The groups of symmetries of some of these polyhedra are proper subgroups of their groups of combinatorial symmetries, but we show that faithful representations can be obtained if appropriately general polyhedra are admitted. The general definition of polyhedra, which may be useful in other contexts as well, is also discussed, as are some of the reasons for its absence from the literature. Received February 3, 1997, and in revised form August 5, 1997.