Nonorientable regular maps over linear fractional groups

Abstract

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.

Enumeration of regular maps of given type on twisted linear fractional groups

“…In our earlier notation, it follows that $$k,\ell$$ satisfying (13) are contained $${\cal D}_f\cup {\cal O}_f$$ and so $$M(q^2)$$ supports an orientably regular map of type $$(k,\ell )$$. But the conditions (13) also imply that both $$k$$ and $$\ell$$ divide $$q^2-1$$ and at least one of them does not divide $$(q^2-1)/2$$, which, by [13, Theorem 2.2], means that the group $${\rm PGL}(2,q^2)$$ also supports an orientably regular map of type $$(k,\ell )$$.…”

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

“…Classification of all orientably-regular maps with orientation-preserving automorphism group isomorphic to PSL(2, q) or PGL(2, q) follows from [9] and can be found in a somewhat more explicit form in [10]; the latter was re-interpreted and extended to regular maps (orientable or not) in [5]. Since we will be interested only in the special case of odd valency and face length we just reproduce the corresponding part of the classification result here (the cases when one of the entries in the type of the map is even are more involved and we refer to [8] for details).…”

“…For the norm of our element g ∈ O we thus have N(g) = t (3σ t (h) + 2), the product being taken over all t between 1 and (k − 1)/2, coprime to k. The ϕ(k)/2 images σ t (h) appearing in this product are precisely the roots of the minimal polynomial Ψ(x) of degree ϕ(k)/2 for h = α + α −1 , see e.g. [8]. So, if Ψ(x) = t (x − σ t (h)) = j a j x j where j ranges from 0 to ϕ(k)/2, then the integral coefficients a j will also appear in the expansion of the above product.…”

“…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].…”