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Gareth A. Jones

Publications and source records attributed to Gareth A. Jones.

At least 19 recordsLinked to original sources
arXiv2025

Regularity properties of Macbeath-Hurwitz and related maps and surfaces

The Macbeath-Hurwitz maps $M$ of type $\{3,7\}$, obtained from the Hurwitz groups $G={\rm PSL}_2(q)$ found by Macbeath, are fully regular by a result of Singerman, with automorphism group $G\times{\rm C}_2$ or ${\rm PGL}_2(q)$. Hall's criterion determines which of these two properties, called inner and outer regularity, $M$ has. Inner (but not outer) regular maps $M$ yield non-orientable regular maps $M/{\rm C}_2$ of the same type with automorphism group $G$. If $q=p^3$ for a prime $p\equiv\pm 2$ or $\pm 3$ mod~$(7)$ the unique map $M$ is inner regular if and only if $p\equiv 1$ mod~$(4)$. If $q=p$ for a prime $p\equiv\pm 1$ mod~$(7)$ there are three maps $M$; we use the density theorems of Frobenius and Chebotarev to show that in this case the sets of such primes $p$ for which $0, 1, 2$ or $3$ of them are inner regular have relative densities $1/8$, $3/8$, $3/8$ and $1/8$ respectively. Hall's criterion and its consequences are extended to the analogous Macbeath maps of type $\{3,n\}$ obtained from ${\rm PSL}_2(q)$ for all $n\ge 7$; theoretical predictions on their number and properties are supported by evidence from the map databases of Conder and Poto\v cnik.

math.CO↗
arXiv2025

Non-orientable regular hypermaps of arbitrary hyperbolic type

One of the consequences of residual finiteness of triangle groups is that for any given hyperbolic triple $(\ell,m,n)$ there exist infinitely many regular hypermaps of type $(\ell,m,n)$ on compact orientable surfaces. The same conclusion also follows from a classification of those finite quotients of hyperbolic triangle groups that are isomorphic to linear fractional groups over finite fields. A non-orientable analogue of this, that is, existence of regular hypermaps of a given hyperbolic type on {\em non-orientable} compact surfaces, appears to have been proved only for {\em maps}, which arise when one of the parameters $\ell,m,n$ is equal to $2$. In this paper we establish a non-orientable version of the above statement in full generality by proving the following much stronger assertion: for every hyperbolic triple $(\ell,m,n)$ there exists an infinite set of primes $p$ of positive Dirichlet density, such that (i) there exists a regular hypermap $\mathcal{H}$ of type $(\ell,m,n)$ on a compact non-orientable surface such that the automorphism group of $\mathcal{H}$ is isomorphic to $\PSL(2,p)$, and, moreover, (ii) the carrier compact surface of {\em every} regular hypermap of type $(\ell,m,n)$ with rotation group isomorphic to $\PSL(2,p)$ is necessarily non-orientable.

math.GR↗
arXiv2024

Counting conjugacy classes of subgroups of ${\rm PSL}_2(p)$

We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of self-normalising subgroups. The formulae are used to prove lower bounds $17$, $18$, $6$ and $12$ respectively satisfied by these invariants for all $p>37$. A computer search carried out for a different problem shows that these bounds are attained for over a million primes $p$; we show that if the Bateman--Horn Conjecture is true, they are attained for infinitely many primes. Also, assuming no unproved conjectures, we use a result of Heath-Brown to obtain upper bounds for these invariants, valid for an infinite set of primes $p$.

math.GR↗
arXiv2024

Permutation groups of prime power degree and $p$-complements

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime to $p$ in such representations). We deduce that every primitive permutation group of prime power degree has a regular subgroup, and that any two faithful primitive representations of a group, of the same prime power degree, are equivalent under automorphisms. In general, $p$-complements in a finite group can be inequivalent under automorphisms, or even non-isomorphic. We extend examples of such phenomena due to Buturlakin, Revin and Nesterov by showing that the number of inequivalent classes of complements can be arbitrarily large. Questions concerning the existence of prime power representations and $p$-complements in groups with socle ${\rm PSL}_d(q)$ are related to some difficult open problems in Number Theory.

math.GR↗
arXiv2024

A number-theoretic problem concerning pseudo-real Riemann surfaces

Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes $p$ such that $p+2$ has all its prime factors $q\equiv -1$ mod~$(4)$. We use theorems of Landau and Raikov to prove that the number of integers $n\le x$ with only such prime factors $q$ is asymptotic to $cx/\sqrt{\ln x}$ for a specific constant $c=0.4865\ldots$. Heuristic arguments, following Hardy and Littlewood, then yield a conjecture that the number of such primes $p\le x$ is asymptotic to $c'\int_2^x(\ln t)^{-3/2}dt$ for a constant $c'=0.8981\ldots$. The theorem, the conjecture and a similar conjecture applying the Bateman--Horn Conjecture to other pseudo-real Riemann surfaces are supported by evidence from extensive computer searches.

math.NT↗
arXiv2023

Regular dessins with primitive automorphism groups

We classify the dessins $\mathcal D$ for which the automorphism group $G$ acts primitively and faithfully on the points over one of the three critical values (without loss of generality the black vertices in the usual bipartite map representation). We show that they are all generalised Paley dessins, in which the black vertices are the elements of a finite field ${\mathbb F}_q$, and $G$ is a subgroup of the affine group ${\rm AGL}_1(q)$. Using earlier results obtained with Streit and Wolfart, we determine the orbits of the absolute Galois group on these dessins, we show that they are all defined over certain cyclotomic fields, and we obtain defining equations in some special cases. Relaxing the condition of a faithful action allows only cyclic regular coverings of these dessins.

math.NT↗
arXiv2023

Regular maps with primitive automorphism groups

We classify the regular maps $\mathcal M$ which have automorphism groups $G$ acting faithfully and primitively on their vertices. As a permutation group $G$ must be of almost simple or affine type, with dihedral point stabilisers. We show that all such almost simple groups, namely all but a few groups ${\rm PSL}_2(q)$, ${\rm PGL}_2(q)$ and ${\rm Sz}(q)$, arise from regular maps, which are always non-orientable. In the affine case, the maps $\mathcal M$ occur in orientable and non-orientable Petrie dual pairs. We give the number of maps associated with each group, together with their genus and extended type. Some of this builds on earlier work of the first author on generalised Paley maps, and on recent work of Jajcay, Li, \vSirá\vn and Wang on maps with quasiprimitive automorphism groups. There are tables of data for the maps in appendices to this paper.

math.GR↗
arXiv2022

Orders of simple groups and the Bateman--Horn Conjecture

We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and Hölder in the 1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ has just six prime factors. The conjecture suggests that there are infinitely many such primes, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, and the construction of highly symmetric block designs.

math.GR↗
arXiv2022

Klein's ten planar dessins of degree 11, and beyond

We reinterpret ideas in Klein's paper on transformations of degree $11$ from the modern point of view of dessins d'enfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman-Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for some prime power $q$, in which case infinitely many groups ${\rm PSL}_n(q)$ arise as permutation groups and monodromy groups of degree $p$ (an open problem in group theory).

math.GR↗
arXiv2021

Hole operations on Hurwitz maps

For a given group $G$ the orientably regular maps with orientation-preserving automorphism group $G$ are used as the vertices of a graph $Ø(G)$, with undirected and directed edges showing the effect of duality and hole operations on these maps. Some examples of these graphs are given, including several for small Hurwitz groups. For some $G$, such as the affine groups ${\rm AGL}_1(2^e)$, the graph $Ø(G)$ is connected, whereas for some other infinite families, such as the alternating and symmetric groups, the number of connected components is unbounded.

math.GR↗
arXiv2021

Finite simple automorphism groups of edge-transitive maps

Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.

math.GR↗
arXiv2021

Groups of prime degree and the Bateman-Horn Conjecture

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We present heuristic arguments and computational evidence based on the Bateman-Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\rm PSL}_n(q)$, ${\rm PSU}_n(q)$ and ${\rm PSp}_{2n}(q)$ which arise in the work of Dixon and Zalesskii on linear groups of prime degree.

math.GR↗
arXiv2021

Block designs and prime values of polynomials

A recent construction by Amarra, Devillers and Praeger of block designs with specific parameters depends on certain quadratic polynomials, with integer coefficients, taking prime power values. The Bunyakovsky Conjecture, if true, would imply that each of them takes infinitely many prime values, giving an infinite family of block designs with the required parameters. We have found large numbers of prime values of these polynomials, and the numbers found agree very closely with the estimates for them provided by Li's recent modification of the Bateman-Horn Conjecture. While this does not prove that these polynomials take infinitely many prime values, it provides strong evidence for this, and it also adds extra support for the validity of the Bunyakovsky and Bateman-Horn Conjectures.

math.NT↗
arXiv2020

Hurwitz groups as monodromy groups of dessins: several examples

We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.

math.AG↗
arXiv2020

Primes in geometric series and finite permutation groups

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$.

math.NT↗
arXiv2020

Groups of automorphisms of Riemann surfaces and maps of genus $p+1$ where $p$ is prime

We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $ρ(g-1)$ for some integer $ρ\ge 1$, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for $ρ>6$, and of the first and third authors for $ρ=3, 4, 5$ and $6$. As a corollary we classify the orientably regular hypermaps (including maps) of genus $p+1$, together with the non-orientable regular hypermaps of characteristic $-p$, with automorphism group of order divisible by the prime $p$; this extends results of Conder, \v Sirá\v n and Tucker for maps.

math.AG↗
arXiv2019

A short proof of Greenberg's Theorem

Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated groups $A$.

math.GR↗
arXiv2019

Infinite Paley graphs

Infinite analogues of the Paley graphs are constructed, based on uncountably many infinite but locally finite fields. Weil's estimate for character sums shows that they are all isomorphic to the random or universal graph of Erd\H os, Rényi and Rado. Automorphism groups and connections with model theory are considered.

math.CO↗