Regular Hypermaps Over Projective Linear Groups

Abstract

An enumeration result for orientably regular hypermaps of a given type with automorphism groups isomorphic to PSL(2, q) or PGL(2, q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to nonorientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.2000 Mathematics subject classification: primary 57M15; secondary 05C25, 20F05. Keywords and phrases: hypermap, regular hypermap, triangle group, projective linear group. IntroductionA regular hypermap H is a pair (r, s) of permutations generating a regular permutation group on a finite set, and provides a generalization of the geometric notion of a regular map on a surface, by allowing edges to be replaced by 'hyperedges'. The cycles of r, s and r s correspond to the hypervertices, hyperedges and hyperfaces of H, which determine the embedding of the underlying (and connected) hypergraph into the surface, and their orders give the type of H, say {k, l, m}. The group G generated by r and s induces a group of automorphisms of this hypergraph, preserving the embedding, and acting transitively on the flags (incident hypervertex-hyperedge pairs) of H. When one of the parameters k, l, m is 2, the hypergraph is a graph, and the hypermap is a regular map. The theory of such objects is well developed, and has been thoroughly explained in [10,11]. Without going into too much detail, we need to make a few basic observations. First, the group G has a presentation of the formand (so) is a finite quotient of the ordinary (k, l, m) triangle group. For simplicity, we will say that such a group G has type (k, l, m), provided that k, l, m are the true orders of the corresponding elements r, s, t. There is a bijective correspondence between isomorphism classes of regular hypermaps of a given type {k, l, m} and torsionfree normal subgroups of the ordinary (k, l, m) triangle group (k, l, m), and the number of those with a given group G as 'rotational symmetry group' (or quotient of (k, l, m)) is equal to the number of ways of generating G by a (k, l, m)-triple (r, s, t) up to equivalence under Aut(G). For further details about representing hypermaps in the form of cellular decomposition of closed two-dimensional surfaces and visualizing the rotational symmetries, and also their association with Riemann surfaces and algebraic number fields (through Grothendieck's theory of dessins d'enfants), we refer the reader to [4,10,11].A regular hypermap may admit a symmetry that induces a reversal of some local orientation of the supporting surface. At the group theory level, this is equivalent to the existence of an automorphism ϑ of a (k, l, m)-group G presented as above, such that ϑ inverts two of the three generators. Such regular hypermaps are called reflexible. If ϑ is actually given by conjugation of some element of order two in G, then the corresponding (k, l, m)-generating triple for G gives rise to two distinct reflexible hypermaps: one ...

Nonorientable regular maps over linear fractional groups

“…Firstly, there is a much more general version of Theorem 2.2 in which the prime p is not necessarily coprime to k and m, in particular, covering the case when both k and m are equal to p and G ∼ = P SL(2, p); see Propositions 3.1, 3.2, 4.6 and 6.1 of [3]. In order to avoid a rather long re-statement of these facts we invite the reader to check that part (2) of Proposition 6.1 combined with Proposition 3.1 of [3] imply: Theorem 4.1. If p is a prime congruent to 1 mod 4, then there exists a nonorientable regular map of type (p, p) with automorphism group isomorphic to P SL(2, p).…”

“…We begin by recalling the characterisation of such automorphism groups from [8]; for a much more detailed proof we refer to [3]. Proposition 2.1.…”

“…Necessary and sufficient conditions for G(ξ, η) to give rise to a nonorientable regular map were given in [3]. Here we present an excerpt sufficient for our purposes.…”

“…Our approach was based on developing some results obtained in [3] in the course of analysing regular maps over linear fractional groups. The scope of [3] is, however, broader and covers regular hypermaps. For a general theory of hypermaps and their surface representations we recommend [5].…”

“…Facts collected in [8,3] imply that for any hyperbolic triple (k, m, l), that is, such that 1/k + 1/m + 1/l < 1, there exist infinitely many regular hypermaps of type (k, m, l) on orientable surfaces, with automorphism group isomorphic to a linear fractional group over a finite field. By the theory developed in [3] combined with the findings in this paper, to establish a nonorientable analogue of this result requires analysing conditions under which the quantity A = 4+(α+α −1 )(β+β −1 )(γ+γ −1 )−(α+α −1 ) 2 −(β+β −1 ) 2 −(γ+γ −1 ) 2 , where α, β and γ are primitive 2k-th, 2m-th, and 2l-th roots of unity in C, projects onto a non-zero square in a quotient field of the ring of algebraic integers of Q[α, β, γ] generated by a suitable prime ideal; note that for l = 2 we have γ 2 = 1 and γ + γ −1 = 0 and then A reduces to the quantity A introduced earlier. In fact, methods of Section 3 can be adapted in an obvious way to construct, for any hyperbolic triple (k, m, l) and for suitable triples (α, β, γ) of primitive roots of unity as above, an infinite number of nonorientable regular hypermaps of type (k, l, m) over linear fractional groups.…”