Regular self-dual and self-Petrie-dual maps of arbitrary valency

The existence of a regular, self-dual and self-Petrie-dual map of any given even valency has been proved by D. Archdeacon, M. Conder and J. Siran (2014). In this paper we extend this result to any odd valency ≥ 5. This is done using algebraic number theory and maps defined on the groups PSL(2, p) in the case of odd prime valency ≥ 5 and valency 9, and using coverings for the remaining odd valencies

Nonorientable regular maps over linear fractional groups

It is well known that for any given hyperbolic pair (k, m) there exist infinitely many regular maps of valence k and face length m on an orientable surface, with automorphism group isomorphic to a linear fractional group. A nonorientable analogue of this result was known to be true for all pairs (k, m) as above with at least one even entry. In this paper we establish the existence of such regular maps on nonorientable surfaces for all hyperbolic pairs