Marston Conder

Determination of all Regular Maps of Small Genus

Complete lists are given of all reflexible orientable regular maps of genus 2 to 15, all non-orientable regular maps of genus 4 to 30, and all (orientable) rotary but chiral (irreflexible) maps of genus 2 to 15 inclusive. On each list the maps are classified according to genus and type (viz {p, q} where every face is incident with p edges and every vertex is incident with q edges). The complete lists were determined with the help of a parallel program which finds all normal subgroups of low index in a finitely-presented group.

Hurwitz groups: A brief survey

Hurwitz groups are the nontrivial finite quotients 2 3 7 of the ( 2 , 3 , 7 ) triangle group (x, y\x — y = (xy) = 1) . This paper gives a brief survey of such groups, their significance, and some of their properties, together with a description of all examples known to the author.

Automorphism groups of symmetric graphs of valency 3

The seven types of symmetric graph of valency 3 are described in a unified way in terms of generators and relations for their automorphism groups, and some relationships among the types are deduced. Also a number of new finite symmetric graphs are defined, solving the first three problems posed by Djokovic and Miller in [J. Combin. Theory Ser. B 29 (1980), 195–230].

Remarks on path-transitivity in finite graphs

This paper deals with graphs the automorphism groups of which are transitive on vertices and on undirected paths (but not necessarily on directed walks) of some fixed length. In particular, it is shown that if the automorphism groupGof a graph Γ is transitive on vertices and on undirected paths of lengthk+1in Γ, for somek≥1, thenGis also transitive onk-arcs in Γ. Further details are given for the casek=1, for the case of cubic graphs, and for the casek>4.

A Classification of 2-Arc-Transitive Circulants

A graph X is k-arc-transitive if its automorphism group acts transitively on the set of k-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants is given.

A tetravalent half-arc-transitive graph with non-abelian vertex stabilizer

A construction is given of a 4-valent ½-arc-transitive graph with vertex stabilizer isomorphic to the dihedral group D8. The graph has 10 752 vertices and is the first known example of a 4-valent ½-arc-transitive graph with non-abelian vertex stabilizer.

Some results on quotients of triangle groups

Given positive integers k,l,m, the (k,l,m) triangle group has presentation A(k,l,m) =< X,Y,Z \ X=Y= ZTM = XYZ = 1> . This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubiks cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

Regular t-balanced Cayley maps

The concept of a t-balanced Cayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Siran, M. Skoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Siran, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (1-3) (2002) 167-179], and apply our results to the specific case of regular Cayley maps of abelian groups.

The genus of compact riemann surfaces with maximal automorphism group

Any compact Riemann surface with genus g > 1 has at most 84(g–1) conformal automorphisms. In this paper it is shown that there are just 32 integers g in the range 1 < g < 11905 for which there exist compact Riemann surfaces of genus g with this maximum possible number of automorphisms. The automorphism group of each such surface is isomorphic to a quotient ΔN of the (2, 3, 7) triangle group δ =⩽ x,y| x2 = y3 =(xy)7 = 1⩾where N is one of 92 proper normal subgroups of Δ with index less than 106. These normal subgroups are found using elementary group-theoretic techniques.

On Cyclic Groups of Automorphisms of Riemann Surfaces

The question of extendability of the action of a cyclic group of automorphisms of a compact Riemann surface is considered. Particular attention is paid to those cases corresponding to Singermans list of Fuchsian groups which are not nitely-maximal, and more generally to cases involving a Fuchsian triangle group. The results provide partial answers to the question of which cyclic groups are the full automorphism group of some Riemann surface of given genus g > 1.