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Gareth A. Jones

Publications and source records attributed to Gareth A. Jones.

53 records · Page 3Linked to original sources
arXiv2013

Characteristically simple Beauville groups, II: low rank and sporadic groups

A Beauville surface is a rigid complex surface of general type, isogenous to a higher product by the free action of a finite group, called a Beauville group. Here we consider which characteristically simple groups can be Beauville groups. We show that if G is a cartesian power of a simple group L_2(q), L_3(q), U_3(q), Sz(2^e), R(3^e), or of a sporadic simple group, then G is a Beauville group if and only if it has two generators and is not isomorphic to A_5.

math.GR↗
arXiv2012

Primitive permutation groups containing a cycle

The primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. The contributions of Jordan and Marggraff to this topic are briefly discussed.

math.GR↗
arXiv2012

Beauville surfaces with abelian Beauville group

A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves $C_{1}$, $C_{2}$ of genera $g_{1},g_{2}\ge 2$ by the free action of a finite group $G$. In this paper we study those Beauville surfaces for which $G$ is abelian (so that $G\cong \mathbb{Z}_{n}^{2}$ with $\gcd(n,6)=1$ by a result of Catanese). For each such $n$ we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over $\mathbb{Q}$.

math.AG↗
arXiv2012

Elementary abelian regular coverings of Platonic maps, Case I: ordinary representations

We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the representations of G on certain homology groups of the sphere, punctured at the branch-points. We give a complete classification for branching over faces (or, dually, vertices) of M, and outline how the method extends to other branching patterns.

math.CO↗
arXiv2011

Classification of nonorientable regular embeddings of Hamming graphs

By a regular embedding of a graph K in a surface we mean a 2-cell embedding of K in a compact connected surface such that the automorphism group acts regularly on flags. In this paper, we classify the nonorientable regular embeddings of the Hamming graph H(d,n). We show that there exists such an embedding if and only if n=2 and d=2, or n=3 or 4 and d>0, or n=6 and d=1 or 2. We also give constructions and descriptions of these embeddings.

math.CO↗
arXiv2011

Hyperbolic manifolds and tessellations of type {3,5,3} associated with L_2(q)

We classify the normal subgroups K of the tetrahedral group Delta=[3,5,3]^+, the even subgroup of the Coxeter group Gamma=[3,5,3], with Delta/K isomorphic to a finite simple group L_2(q). We determine their normalisers N(K) in the isometry group of hyperbolic 3-space H^3, the isometry groups N(K)/K of the associated hyperbolic 3-manifolds H^3/K, and the symmetry groups N_{Gamma}(K)/K of the icosahedral tessellations of these manifolds, giving a detailed analysis of how L_2(q) acts on these tessellations.

math.GR↗
arXiv2011

Hypermaps and multiply quasiplatonic Riemann surfaces

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases with isomorphic automorphism groups.

math.CO↗
arXiv2011

Automorphism groups of Beauville surfaces

This paper shows that the automorphism group of a Beauville surface is a finite solvable group, and describes its possible structure. It relies on results of Singerman on triangle group inclusions, and of Lucchini on generators for special linear groups.

math.GR↗
arXiv2010

Constructions of Chiral Polytopes of Small Rank

An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5.

math.CO↗
arXiv2010

Characterisations and Galois conjugacy of generalised Paley maps

A generalised Paley map is a Cayley map for the additive group of a finite field F, with a subgroup S=-S of the multiplicative group as generating set, cyclically ordered by powers of a generator of S. We characterise these as the orientably regular maps with orientation-preserving automorphism group acting primitively and faithfully on the vertices; allowing a non-faithful primitive action yields certain cyclic coverings of these maps. We determine the fields of definition and the orbits of the absolute Galois group on these maps, and we show that if (q-1)/(p-1) divides |S|, where |F|=q=p^e with p prime, then these maps are the only orientably regular embeddings of their underlying graphs; in particular this applies to the Paley graphs, where |S|=(q-1)/2 is even.

math.CO↗
arXiv2010

Classification and Galois conjugacy of Hamming maps

We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an orientably regular surface embedding if and only if q is a prime power p^e. If q>2 there are up to isomorphism ϕ(q-1)/e such maps, all constructed as Cayley maps for a d-dimensional vector space over the field of order q. We show that for each such pair d, q the corresponding Belyi pairs are conjugate under the action of the absolute Galois group, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q>3.

math.CO↗
arXiv2009

Regular embeddings of complete bipartite graphs: classification and enumeration

The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in the cases where $n$ is a prime power, obtained in collaboration with Du, Kwak, Nedela and Škoviera, together with results of Itô, Hall, Huppert and Wielandt on factorisable groups and on finite solvable groups.

math.CO↗
arXiv2009

Hypermap operations of finite order

Duality and chirality are examples of operations of order 2 on hypermaps. James showed that the groups of all operations on hypermaps and on oriented hypermaps can be identified with the outer automorphism groups ${\rm Out} Δ\cong PGL_2({\bf Z})$ and ${\rm Out} Δ^+\cong GL_2({\bf Z})$ of the groups $Δ= C_2*C_2*C_2$ and $Δ^+ = F_2$. We will consider the elements of finite order in these two groups, and the operations they induce.

math.CO↗
arXiv2009

Maps admitting trialities but not dualities

We use group theory to construct infinite families of maps on surfaces which are invariant under Wilson's map operations of order 3 but not under the operations of order 2, such as duality and Petrie duality.

math.CO↗
arXiv2006

Generalised Fermat Hypermaps and Galois Orbits

We consider families of quasiplatonic Riemann surfaces characterised by the fact that -- as in the case of Fermat curves of exponent $n$ -- their underlying regular (Walsh) hypermap is the complete bipartite graph $ K_{n,n} $, where $ n $ is an odd prime power. We will show that all these surfaces, regarded as algebraic curves, are defined over abelian number fields. We will determine the orbits under the action of the absolute Galois group, their minimal fields of definition, and in some easier cases also their defining equations. The paper relies on group-- and graph--theoretic results by G. A. Jones, R. Nedela and M.Škoviera about regular embeddings of the graphs $K_{n,n}$ [JNŠ] and generalises the analogous question for maps treated in [JStW], partly using different methods.

math.AG↗