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This volume contains seventeen of the best papers delivered at the SIGMAP Workshop 2014, representing the most recent advances in the field of symmetries of discrete objects and structures, with a particular emphasis on connections between maps, Riemann surfaces and dessins d’enfant.Providing the global community of researchers in the field with the opportunity to gather, converse and present their newest findings and advances, the Symmetries In Graphs, Maps, and Polytopes Workshop 2014 was the fifth in a series of workshops. The initial workshop, organized by Steve Wilson in Flagstaff, Arizona, in 1998, was followed in 2002 and 2006 by two meetings held in Aveiro, Portugal, organized by Antonio Breda d’Azevedo, and a fourth workshop held in Oaxaca, Mexico, organized by Isabel Hubard in 2010.This book should appeal to both specialists and those seeking a broad overview of what is happening in the area of symmetries of discrete objects and structures.iv>



Powers of Skew-Morphisms

Skew-morphisms have important applications in the classification of regular Cayley maps, and have also been shown to be fundamental in the study of complementary products of finite groups AB with B cyclic and \(A\cap B = \{1\}\). As natural generalizations of group automorphisms, they share many of their properties but proved much harder to classify. Unlike automorphisms, not all powers of skew-morphisms are skew-morphisms again. We study and classify the powers of skew-morphisms that are either skew-morphisms or group automorphisms and consider reconstruction of skew-morphisms from such powers. We also introduce a new class of skew-morphisms that generalize the widely studied t-balanced skew-morphisms and which we call coset-preserving skew-morphisms. We show that, in certain cases, all skew-morphisms have powers that belong to this class and can therefore be reconstructed from these.
Martin Bachratý, Robert Jajcay

Census of Quadrangle Groups Inclusions

In a classical result of 1972 Singerman classifies the inclusions between triangle groups. We extend the classification to a broader family of triangle and quadrangle groups forming a particular subfamily of Fuchsian groups. With two exceptions, each inclusion determines a finite bipartite map (hypermap) on a 2-dimensional spherical orbifold that encodes the complete information and gives a graphical visualisation of the inclusion. A complete description of all the inclusions is contained in the attached tables.
António Breda d’Azevedo, Domenico A. Catalano, Ján Karabáš, Roman Nedela

Some Unexpected Consequences of Symmetry Computations

This paper gives some instances of experimental computations involving the action of groups on graphs and maps with a high degree of symmetry, that have led to unexpected theoretical discoveries. These include new presentations for 3-dimensional special linear groups, a closed-form definition for the binary reflected Gray codes, a new theorem on groups expressible as a product of an abelian group and a cyclic group, and some revealing observations about the genus spectrum of particular classes of regular maps on surfaces.
Marston D. E. Conder

A 3D Spinorial View of 4D Exceptional Phenomena

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via ‘sandwiching’. This extends to a description of orthogonal transformations in general by means of ‘sandwiching’ with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflections by the Cartan-Dieudonné theorem. We begin by viewing the largest non-crystallographic reflection/Coxeter group \(H_4\) as a group of rotations in two different ways—firstly via a folding from the largest exceptional group \(E_8\), and secondly by induction from the icosahedral group \(H_3\) via Clifford spinors. We then generalise the second way by presenting a construction of a 4D root system from any given 3D one. This affords a new, spinorial, perspective on 4D phenomena, in particular as the induced root systems are precisely the exceptional ones in 4D, and their unusual automorphism groups are easily explained in the spinorial picture; we discuss the wider context of Platonic solids, Arnold’s trinities and the McKay correspondence. The multivector groups can be used to perform concrete group-theoretic calculations, e.g. those for \(H_3\) and \(E_8\), and we discuss how various representations can also be constructed in this Clifford framework; in particular, representations of quaternionic type arise very naturally.
Pierre-Philippe Dechant

Möbius Inversion in Suzuki Groups and Enumeration of Regular Objects

We compute the Möbius function for the subgroup lattice of the simple Suzuki group Sz(q), and use it to enumerate regular objects such as maps, hypermaps, dessins d’enfants and surface coverings with automorphism groups isomorphic to Sz(q).
Martin Downs, Gareth A. Jones

More on Strongly Real Beauville Groups

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. A particularly interesting subclass are the ‘strongly real’ Beauville surfaces that have an analogue of complex conjugation defined on them. In this survey we discuss these objects and in particular the groups that may be used to define them. En route we discuss several open problems, questions and conjectures and in places make some progress made on addressing these.
Ben Fairbairn

On Pentagonal Geometries with Block Size 3, 4 or 5

Let PLS(kr) be a partial linear space which is both uniform, i.e. every line has the same cardinality \(k \ge 2\), and regular, i.e. every point is incident with the same number \(r \ge 1\) of lines. In a recent paper (J. Combin. Des. 21 (2013), 163–179), Ball, Bamberg, Devillers & Stokes introduced the concept of a pentagonal geometry PENT(kr) as a PLS(kr) in which all the points not collinear with any given point are themselves collinear. They also determined the existence spectrum for \(k=1\) or 2 and \(r=k\) or \(k+1\). In this paper we prove that the existence spectrum for PENT(3, r) is \(r \equiv \) 0 or 1 (mod 3) except \(r=4\) or 6. We also prove that there exists a PENT(4, r) for \(r \equiv \) 1 (mod 8) and a PENT(5, r) for \(r \equiv \) 1 (mod 5), \(r \ne 6\), apart from nine possible exceptions. Further we construct an infinite class of pentagonal geometries PENT(\(2^m, 2^{m+1}+1\)), \(m \ge 1\), and a PENT(6, 13).
Terry S. Griggs, Klara Stokes

The Grothendieck-Teichmüller Group of a Finite Group and $${\varvec{G}}$$ G -Dessins d’enfants

For each finite group G, we define the Grothendieck-Teichmüller group of G, denoted \(\mathrm{{GT}}(G)\), and explore its properties. The theory of dessins d’enfants shows that the inverse limit of \(\mathrm{{GT}}(G)\) as G varies can be identified with a group defined by Drinfeld and containing \(\mathrm{{Gal}}(\overline{{\mathbb {Q}}}/ {\mathbb {Q}})\). We give, in particular, an identification of \(\mathrm{{GT}}(G)\), in the case when G is simple and non-abelian, with a certain very explicit group of permutations that can be analyzed easily. With the help of a computer, we obtain precise information for \(G= PSL_2({\mathbb {F}}_q)\) when \(q \in \{4, 7, 8, 9, 11, 13, 16, 17, 19\}\), and we treat \(A_7\), \(PSL_3({\mathbb {F}}_3)\) and \(M_{11}\). In the rest of the paper we give a conceptual explanation for the technique which we use in our calculations. It turns out that the classical action of the Grothendieck-Teichmüller group on dessins d’enfants can be refined to an action on “G-dessins”, which we define, and this elucidates much of the first part.
Pierre Guillot

Discrete Groups and Surface Automorphisms: A Theorem of A.M. Macbeath

This short article re-examines the interaction between group actions in hyperbolic geometry and low-dimensional topology, focussing in particular on some contributions of Murray Macbeath to the study of Riemann surface automorphisms. A brief account is included of a potential extension to hyperbolic 3-manifolds.
W. J. Harvey

Isometric Point-Circle Configurations on Surfaces from Uniform Maps

We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with a maximum number of automorphisms for their genus, and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle \(v_4\) configurations on p-gonal surfaces with two p-gonal morphisms is given. The image of these configurations on the sphere under the two p-gonal morphisms is also described.
Milagros Izquierdo, Klara Stokes

Dessins, Their Delta-Matroids and Partial Duals

Given a map \(\mathcal M\) on a connected and closed orientable surface, the delta-matroid of \(\mathcal M\) is a combinatorial object associated to \(\mathcal M\) which captures some topological information of the embedding. We explore how delta-matroids associated to dessins behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.
Goran Malić

Faithful Embeddings of Planar Graphs on Orientable Closed Surfaces

A graph G is said to be faithfully embeddable on a closed surface \(F^2\) if G can be embedded on \(F^2\) in such a way that any automorphism of G extends to an auto-homeomorphism of \(F^2\). It has been known that every 3-connected planar graph is faithfully embeddable on the sphere. We shall show that every 3-connected planar graph is faithfully embeddable on a suitable orientable closed surface other than the sphere unless it is one of seven exceptions.
Seiya Negami

The Higher Dimensional Hemicuboctahedron

The paper describes the first known infinite sequence of 2-orbit d-polytopes in \(\mathbb {R}^d\) with \(d \ge 3\). The sequence has the remarkable property that its d-dimensional member has vertex-figures isomorphic to the \((d-1)\)-dimensional member.
Daniel Pellicer

Groups of Order at Most 6,000 Generated by Two Elements, One of Which Is an Involution, and Related Structures

A (2,*)-group is a group that can be generated by two elements, one of which is an involution. We describe the method we have used to produce a census of all (2,*)-groups of order at most 6,000. Various well-known combinatorial structures are closely related to (2,*)-groups and we also obtain censuses of these as a corollary.
Primož Potočnik, Pablo Spiga, Gabriel Verret

Even-Integer Continued Fractions and the Farey Tree

Singerman introduced to the theory of maps on surfaces an object that is a universal cover for any map. This object is a tessellation of the hyperbolic plane together with a certain subset of the ideal boundary. The 1-skeleton of this tessellation comprises the edges of an infinite tree whose vertices belong to the ideal boundary. Here we show how this tree can be used to give a beautiful geometric representation of even-integer continued fractions. We use this representation to prove some of the fundamental theorems on even-integer continued fractions that are already known, and we also prove some new theorems with this technique, which have familiar counterparts in the theory of regular continued fractions.
Ian Short, Mairi Walker

Triangle Groups and Maps

We develop a Belyi type theory that applies to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular we extend Belyi’s famous theorem from Riemann surfaces to Klein surfaces.
David Singerman

Nilpotent Symmetric Dessins of Class Two

A dessin is a cellular embedding of a connected bipartite graph into an orientable closed surface with a fixed colouring of vertices and prescribed global orientation. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the set of edges, and a regular dessin is symmetric if it admits an external symmetry transposing the vertex colours. The symmetric dessins whose automorphism groups are nilpotent of class two are classified.
Na-Er Wang, Roman Nedela, Kan Hu
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