arXiv subjects
Publications and source records attributed to Gareth A. Jones.
Building on earlier work of Biggs, James, Wilson and the author, and using the Graver-Watkins description of the 14 classes of edge-transitive maps, we complete the classification of the edge-transitive embeddings of complete graphs.
For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Extending earlier results of Siran, Tucker and Watkins, these are used to determine which symmetric groups $S_n$ can arise in this way for each class. Similar results are obtained for all finite simple groups, building on work of Leemans and Liebeck, Nuzhin and others on generating sets for such groups. It is also shown that each edge-transitive class realises finite groups of every sufficiently large nilpotence class or derived length, and also realises uncountably many non-isomorphic infinite groups. Edge-transitive embeddings of complete graphs are classified, and there is a detailed discussion of edge-transitive maps with boundary.
It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, every countable group $A$ is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if $A$ is finite. In particular, this applies to dessins d'enfants, regarded as finite oriented hypermaps. The proof, involving maximal subgroups of various triangle groups, yields a simple construction of a regular map whose automorphism group contains an isomorphic copy of every finite group.
An operation of joining coset diagrams for a given group, introduced by Higman and developed by Conder in connection with Hurwitz groups, is reinterpreted and generalised as a connected sum operation on dessins d'enfants of a given type. A number of examples are given.
A map which is non-orientable or has non-empty boundary has a canonical double cover which is orientable and has empty boundary. The map is called stable if every automorphism of this cover is a lift of an automorphism of the map. This note describes several infinite families of unstable maps, and relates them to similar phenomena for graphs, hypermaps and Klein surfaces.
In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong C_3*C_2$ are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers $p\ge 3$, $q\ge 2$ the triangle group $Γ=Δ(p,q,\infty)\cong C_p*C_q$ has uncountably many conjugacy classes of nonparabolic maximal subgroups. We also extend results of Tretkoff and of Brenner and Lyndon for the modular group by constructing uncountably many conjugacy classes of such subgroups of $Γ$ which do not arise from Neumann's original method. These maximal subgroups are all generated by elliptic elements, of finite order, but a similar construction yields uncountably many conjugacy classes of torsion-free maximal subgroups of the Hecke groups $C_p*C_2$ for odd $p\ge 3$. Finally, an adaptation of work of Conder yields uncountably many conjugacy classes of maximal subgroups of $Δ(2,3,r)$ for all $r\ge 7$.
A detailed proof is given of a theorem describing the centraliser of a transitive permutation group, with applications to automorphism groups of objects in various categories of maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism group of an object is the centraliser of its monodromy group. An alternative form of the theorem, valid for finite objects, is discussed, with counterexamples based on Baumslag--Solitar groups to show how it fails more generally. The automorphism groups of objects with primitive monodromy groups are described, as are those of non-connected objects.
If $\mathcal S$ is a Beauville surface $({\mathcal C}_1\times{\mathcal C}_2)/G$, then the Hurwitz bound implies that $|G|\le 1764\,χ({\mathcal S})$, with equality if and only if the Beauville group $G$ acts as a Hurwitz group on both curves ${\mathcal C}_i$. Equivalently, $G$ has two generating triples of type $(2,3,7)$, such that no generator in one triple is conjugate to a power of a generator in the other. We show that this property is satisfied by alternating groups $A_n$, their double covers $2.A_n$, and special linear groups $SL_n(q)$ if $n$ is sufficiently large, but by no sporadic simple groups or simple groups $L_n(q)$ ($n\le 7$), ${}^2G_2(3^e)$, ${}^2F_4(2^e)$, ${}^2F_4(2)'$, $G_2(q)$ or ${}^3D_4(q)$ of small Lie rank.
This paper discusses some aspects of the history of the Paley graphs and their automorphism groups.
Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order $2$. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with vertex-stabilisers of order $2$.
This paper considers how many conjugacy classes of reflections a map can have, under various transitivity conditions. It is shown that for vertex- and for face-transitive maps there is no restriction on their number or size, whereas edge-transitive maps can have at most four classes of reflections. Examples are constructed, using topology, covering spaces and group theory, to show that various distributions of reflections can be achieved. Connections with real forms of algebraic curves are also discussed.
We compute the Möbius function for the subgroup lattice of the simple Suzuki group $Sz(q)$; this is used to enumerate normal subgroups of certain Hecke groups with quotients isomorphic to $Sz(q)$.
We state and prove a corrected version of a theorem of Singerman, which relates the existence of symmetries (anticonformal involutions) of a quasiplatonic Riemann surface $\mathcal S$ (one uniformised by a normal subgroup $N$ of finite index in a cocompact triangle group $Δ$) to the properties of the group $G=Δ/N$. We give examples to illustrate the revised necessary and sufficient conditions for the existence of symmetries, and we relate them to properties of the associated dessins d'enfants, or hypermaps.
Generalising a conjecture of Singerman, it is shown that there exist orientably regular chiral hypermaps of every non-spherical type. The proof uses the representation theory of automorphism groups acting on homology and on various spaces of differentials. Some examples are given.
The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group Γ, with quotient group isomorphic to Γ/N. It is shown how to enumerate such objects with a given finite automorphism group G, how to represent them all as quotients of a single regular object U(G), and how they are acted on by the outer automorphism group of Γ. Examples constructed include kaleidoscopic maps with trinity symmetry.
We use the Möbius function of the simple Suzuki group Sz(q) to enumerate regular objects such as maps, hypermaps, dessins d'enfants and surface coverings with automorphism groups isomorphic to Sz(q).
Dessins d'enfants are combinatorial structures on compact Riemann surfaces defined over algebraic number fields, and regular dessins are the most symmetric of them. If G is a finite group, there are only finitely many regular dessins with automorphism group G. It is shown how to enumerate them, how to represent them all as quotients of a single regular dessin U(G), and how certain hypermap operations act on them. For example, if G is a cyclic group of order n then U(G) is a map on the Fermat curve of degree n and genus (n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus 274218830047232000000000000000001. For other non-abelian finite simple groups, the genus is much larger.
A Beauville surface (of unmixed type) is a complex algebraic surface which is the quotient of the product of two curves of genus at least 2 by a finite group G acting freely on the product, where G preserves the two curves and their quotients by G are isomorphic to the projective line, ramified over three points. Such a group G is called a Beauville group. We show that if a characteristically simple group G is a cartesian power of a finite simple alternating group, then G is a Beauville group if and only if it has two generators and is not isomorphic to A_5.