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

Abstract

The main result of D. Archdeacon, M. Conder and J.Širáň [Trans. Amer. Math. Soc. 366 (2014) 8, 4491-4512] implies existence of a regular, self-dual and self-Petrie dual map of any given even valency. In this paper we extend this result to any odd valency ≥ 5. This is done by algebraic number theory and maps defined on the groups PSL(2, p) in the case of odd prime valency ≥ 5 and valency 9, and by coverings for the remaining odd valencies.

Self-dual, self-Petrie-dual and Möbius regular maps on linear fractional groups

“…In [2], Archdeacon, Conder andŠiráň proved the existence of such a map for any even valency. The results in this paper allow Fraser, Jeans andŠiráň [6] to prove the existence of a self-dual, self-Petriedual regular map for any given odd valency k ≥ 5.…”

Self-dual, self-Petrie-dual and Möbius regular maps on linear fractional groups

“…At this point we note that there is an exception for maps of type (5,5), which is addressed in Section 5.…”

“…Adrianov's [1] enumeration of regular hypermaps on P SL(2, q) includes a constant which deals with the special case which occurs for maps of type (5,5).…”

“…For us to be considering a map of type (k, l) we must have 2k|q ± 1 and 2l|q ± 1, and it is known, see [9], that P SL(2, q) has subgroup A 5 when q ≡ ±1 mod 10. The constant in Adrianov's enumeration, which is 2 for maps of type (5,5) and zero otherwise, is subtracted to account for the cases when the group R, S collapses into the subgroup A 5 ≤ P SL(2, q).…”

“…In Section 4 we address the necessary and sufficient conditions for a map of type (k, l) to be self-dual, self-Petrie-dual and Möbius regular respectively. Section 5 highlights a special case, namely maps of type (5,5) whose orientation-preserving automorphism groups turn out to be isomorphic to A 5 and Section 6 comments on and lists examples of maps with some or all of these properties.…”