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

Abstract

Regular maps on linear fractional groups PSL(2, q) and PGL(2, q) have been studied for many years and the theory is well-developed, including generating sets for the associated groups. This paper studies the properties of self-duality, self-Petrie-duality and Möbius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5, 5). The final section includes an enumeration of the PSL(2, q) maps for q ≤ 81 and a list of all the PSL(2, q) maps which have any of these special properties for q ≤ 49.

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

“…We note that if p e ≡ ±1 (mod 10), the group PSL(2, p e ) contains (up to conjugacy) two exceptional pairs R, S as above for (k, ℓ) = (5, 5) with the property that R, S ∼ = A 5 ; this case (omitted from [8, Theorem 2.2]) is addressed in [6]. However, this situation does not apply in what follows.…”

“…Necessary and sufficient conditions for self-duality and self-Petrie-duality of the maps M = (G k,ℓ ; R, S) from Proposition 1 were established in [6]. As they are also quite complex we present here only a simple sufficient condition appearing as Corollary 4.3 in [6] which (in terms and notation of Proposition 1) can be re-stated as follows.…”

“…Here we completely settle the problem by showing that for every odd k ≥ 5 there exists a regular, self-dual and self-Petrie-dual map of valency k. Our strategy is to establish this result first for every prime k ≥ 5 and for k = 9 by algebraic methods motivated by those used in [8] and applied to more detailed results of [6] on regular maps defined on linear fractional groups. We then extend this to non-prime odd values of k ≥ 5 by an analogue of a covering tool from [1].…”