Regular maps with primitive automorphism groups

Classification of reflexible Cayley maps for dihedral groups

A regular map $${\mathcal{M}}$$ is an embedding of a graph into a compact surface S such that its automorphism group $${{\rm Aut}^+(\mathcal{M}) \subseteq {\rm Aut}^+(S)}$$ acts transitively on the directed edges. A Petrie polygon of $${\mathcal{M}}$$ is a zig-zag circuit in which every two consecutive edges but no three belong to the same face. It is known that the number of sides of every Petrie polygon of $${\mathcal{M}}$$ is the same and this number is called the Petrie length of $${\mathcal{M}}$$ . In this paper our main concern is the Petrie lengths of toroidal regular maps. There are mainly two types of such regular maps up to duality, and for each type we prove a theorem that enables us to determine the Petrie length of the corresponding map. We also find the Petrie lengths of Accola–Maclachlan, Wiman and Fermat maps, which are well-known families of regular maps. Finally, we consider regular maps with Petrie length 2, and prove that only Accola–Maclachlan and Wiman surfaces underlie such regular maps. In this way, we obtain new characterizations of Accola–Maclachlan and Wiman surfaces.

This paper provides a complete determination of which of the alternating groups An and the symmetric groups Sn occur as the automorphism group of some regular or chiral map on an orientable surface, and which of them occur as the automorphism group of a regular map on a non-orientable surface. The situation for some given types (m,k) is also considered, where k is the valency and m is the face-size, with special focus on types with m=3, and more particularly with (m,k)=(3,7) or (3,8), or their duals. Some observations are made also about what happens for regular and orientably-regular maps with given valency, and for regular and chiral polyhedra. Much but certainly not all of what is presented follows from theorems in previous papers by the author and others, and this one brings them and some new observations together into a single reference.

Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic [formula omitted

Regular maps are embeddings of graphs or multigraphs on closed surfaces (which may be orientable or non-orientable), in which the automorphism group of the embedding acts regularly on flags. Such maps may admit external symmetries that are not automorphisms of the embedding, but correspond to combinations of well known operators that may transform the map into an isomorphic copy: duality, Petrie duality, and the ‘hole operators’, also known as ‘taking exponents’. The group generated by the external symmetries admitted by a regular map is the external symmetry group of the map. We will be interested in external symmetry groups of regular maps in the case when the map admits both the above dualities (that is, if it has trinity symmetry) and all feasible hole operators (that is, if it is kaleidoscopic). Existence of finite kaleidoscopic regular maps was conjectured for every even valency by Wilson, and proved by Archdeacon, Conder and Širáň (2010).It is well known that regular maps may be identified with quotients of extended triangle groups. In other words, these groups may be regarded as ‘universal’ for constructions of regular maps. It is therefore interesting to ask if similar ‘universal’ groups exist for kaleidoscopic regular maps with trinity symmetry. A satisfactory answer, however, is likely to be very complex, if indeed feasible at all. We demonstrate this (and other things) by a construction of an infinite family of finite kaleidoscopic regular maps with trinity symmetry, all of valency 8, such that the orders of their external symmetry groups are unbounded. Also we explicitly determine the external symmetry groups for the family of kaleidoscopic regular maps of even valency mentioned above.

A 2-cell decomposition of a closed orientable surface is called a regular map if its automorphism group acts transitively on the set of all its darts (or arcs). It is well known that the group $$G = \mathrm{Aut}^+(\mathcal {M})$$G=Aut+(M) of all orientation-preserving automorphisms of such a map $$\mathcal {M}$$M is a finite quotient of the free product $$\Gamma = \mathbb {Z}* C_2$$Γ=ZźC2. In this paper we investigate the situation where G is nilpotent and the underlying graph of the map is simple (with no multiple edges). By applying a theorem of Labute (Proc Amer Math Soc 66:197---201, 1977) on the ranks of the factors of the lower central series of $$\Gamma $$Γ (via the associated Lie algebra), we prove that the number of vertices of any such map is bounded by a function of the nilpotency class of the group G. Moreover, we show that for a fixed nilpotency class c there is exactly one such simple regular map $$\mathcal {M}_c$$Mc attaining the bound, and that this map is universal, in the sense that every simple regular map $$\mathcal {M}$$M for which $$\mathrm{Aut}^+(\mathcal {M})$$Aut+(M) is nilpotent of class at most c is a quotient of $$\mathcal {M}_c$$Mc. In particular, there are finitely many such quotients for any given value of c, and every regular map $$\mathcal {M}$$M, whether simple or non-simple, for which $$\mathrm{Aut}^+(\mathcal {M})$$Aut+(M) is nilpotent of class at most c, is a cyclic cover of exactly one of them.

This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g −1 , where g is the genus, all orientably-regular maps of genus p+1 for p prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Širáň classification of non-orientable regular maps of Euler characteristic −p where p is prime.

Möbius regular maps

Regular combinatorial maps

Regular and orientably-regular maps are central to the part of topological graph theory concerned with highly symmetric graph embeddings. Classification of such maps often relies on factoring out a normal subgroup of automorphisms acting intransitively on the set of the vertices of the map. Maps whose automorphism groups act quasiprimitively on their vertices do not allow for such factorization. Instead, we rely on classification of quasiprimitive group actions which divides such actions into eight types, and we show that four of these types, HS, HC, SD, and CD, do not occur as the automorphism groups of regular or orientably-regular maps. We classify regular and orientably-regular maps with automorphism groups of the HA type, and construct new families of regular as well as both chiral and reflexible orientably-regular maps with automorphism groups of the TW and PA types. We provide a brief summary of the known results concerning the AS type, which has been extensively studied before.

Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle G,f\rangle$ contains a constant map, and that $G$ is a \emph{synchronizing group} if $G$ synchronizes \emph{every} non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it has previously been conjectured that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks $5$ and $6$. In addition we produce graphs whose automorphism groups have approximately $\sqrt{n}$ \emph{non-synchronizing ranks}, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree $n$ synchronizes every non-uniform transformation of rank $n-1$ and $n-2$, and here this is extended to $n-3$ and $n-4$. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

In 2017 a first selfintersection-free polyhedral realization of Hurwitz’s regular map {3, 7}18 of genus 7 was found by Michael Cuntz and the first author. For any regular map which had previously been realized as a polyhedron without self-intersections in 3-space, it was also possible to find such a polyhedron with nontrivial geometric symmetries. So it is natural to ask of whether we can find for the above-mentioned regular map a corresponding version with some non-trivial geometric symmetry. The orientation-preserving combinatorial automorphism group of this Hurwitz map is the projective special linear group PSL(2,8) of order 504 = 23 ⋅ 32 ⋅ 7. All non-trivial subgroups of PSL(2,8) are candidates for such a geometric symmetry. Using the GAP software for exploring the subgroup structure, we found that it is sufficient to consider only four cyclic subgroups whose order is 9, 7, 3, and 2, respectively. We prove that there are obstructions for selfintersection-free polyhedral realizations of the Hurwitz map {3, 7}18 of genus 7 with geometric rotational symmetries of order 9 or 3. We provide new small integer coordinates within the realization space known from 2017, which are also suitable for making a 3D-printed model. We present Kepler–Poinsot type realizations, both with 7-fold and with 3-fold rotational symmetry, the latter with integer coordinates.

Regular maps whose groups do not act faithfully on vertices, edges, or faces

This is the first in a series of papers whose results imply the validity of a strengthened version of the Sims conjecture on finite primitive permutation groups from the authors’ article “Stabilizers of graph’s vertices and a strengthened version of the Sims conjecture”, Dokl. Math. 59 (1), 113–115 (1999). In this paper, the case of not almost simple primitive groups and the case of primitive groups with alternating socle are considered.

Characterization and construction of Cayley graphs admitting regular Cayley maps