David Singerman

Regular hypermaps

An algebraic map is a pair (G, fl), where G is a group generated by x, y, with x2 = I, acting transitively on a set n. It is regular if its automorphism group is transitive on n.Out of an algebraic map we can construct a topological map which is a two-cell decomposition of an orientable surface. There has been a lot of work on regular maps; on the sphere, for example, they are the Platonic solids. In this paper we show how we can construct a topological hypermap out of an algebraic hypermap. We put particular emphasis on the regular ones. On the sphere they are Archimedean solids and we also describe all examples on a torus and on a surface of genus 2.

PSL(2, q) AS AN IMAGE OF THE EXTENDED MODULAR GROUP WITH APPLICATIONS TO GROUP ACTIONS ON SURFACES

The modular group PSH2, Z), which is isomorphic to a free product of a cyclic groupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed q) tha is at PSUl,n image of the modular group ifq =£ 9. (Here, as usual, q is a prime power.) The extended modular group PGU2, Z)contains PSL{2, Z) with index 2. It has a presentationthe subgroup PSL(2, Z) being generated by UV and VW.A simple group which is an image of PGL(2, Z) is also a Z)n. Fo imagr e of PSU2,many reasons connected with PSU2, q) actions on surfaces (which we discuss in Section4) it is important to know when PSU2, q) is also an image of PGL(2 Z). W, e will prove

Symmetries and pseudo-symmetries of hyperelliptic surfaces

by DAVID SINGERMAN(Received 29 September, 1978)1. Let X be a closed Riemann surface of g genu > 2 ans d let Aut X denote the groupof automorphisms of X where, in this paper, an automorphism means a conformal oranticonformal self-homeomorphism. X is called hyperelliptic if it admits a conformalautomorphism J of order 2 such that XIH has genu =s (J) 0, i whers thee grou H p oforder 2 generated by J. Thus X is a two-sheeted covering of the sphere which is branchedover 2g + 2 points and J is the sheet-interchange map. J is the unique conformalautomorphism of order 2 such that X/(J) has genus 0 and it follows that if U e Aut X, thenUJIT

Hypermaps on Surfaces with Boundary

We introduce a theory of hypermaps on surfaces with boundary. A topological hypermap can be associated to an algebraic hypermap which is a quintuple (G, ?, c1, c2, c3 ,), where G is a group, generated by three involutions c1, c2, and c3 , that acts transitively on the set ?. Conversely, the topological hypermap can be reconstructed from the algebraic hypermap. This theory is based on the ideas of Cori and Machi, and generalizes the papers of Jones and Singerman and Corn and Singerman.

Regular maps and principal congruence subgroups of Hecke groups

Regular q-valent maps correspond to normal subgroups of the triangle group (2, q, ∞). This group has a representation as the Hecke group Hq which is generated by z → -1/z and z → -1/z+λq, where λq := 2 cos π/q. We investigate the regular maps corresponding to the principal congruence subgroups of Hq. Those of low index give many interesting regular maps.

BELYI˘ UNIFORMIZATION OF ELLIPTIC CURVES

Belyis Theorem implies that a Riemann surface X represents a curve defined over a number field if and only if it can be expressed as U/?, where U is simply-connected and ? is a subgroup of finite index in a triangle group. We consider the case when X has genus 1, and ask for which curves and number fields ? can be chosen to be a lattice. As an application, we give examples of Galois actions on Grothendieck dessins.

Reflections of regular maps and Riemann surfaces

A compact Riemann surface of genus g is called an M-surface if it admits an anti-conformal involution that fixes g +1 simple closed curves, the maximum number by Harnack’s Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack’s theorem still apply. Here we show that for each genus g> 1 there is a unique M-surface of genus g that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix g curves.

ORIENTABLE AND NON-ORIENTABLE KLEIN SURFACES WITH MAXIMAL SYMMETRY

by DAVID SINGERMAN(Received 8 September, 1983)1. Let X be a bordered Klein surface, by which we mean a Klein surface withnon-empty boundary. X is characterized topologically by its orientability, the number k ofits boundary components and the genus p of the closed surface obtained by filling in allthe holes. Th algebraice genus g of X is defined byf2p + fc — 1 if X is orientableg = <lp + fc-1 ifXis non-orientable.If gs=2 it is known that if G is a group of automorphisms of X then |G|^12(g-l) andthat the upper bound is attained for infinitely many values of g ([4], [5]). A borderedKlein surface for which this upper bound is attained is said to have maximal symmetry. Agroup of 12(g-l) automorphisms of a bordered Klein surface of algebraic genus g iscalled an M*-group and it is known that a finite group G is an M*-group if and only if itis generated by 3 non-trivial element

Unicellular dessins and a uniqueness theorem for Klein's Riemann surface of genus 3

If we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, we have a unifacial dessin whose automorphism group is transitive on the edges but not on the directed edges of the dessin. We show that Kleins surface is the unique platonic surface with this property.

Weierstrass Points and Regular Maps

It was shown by G. A. Jones and the first author in [8] that underlying any map on a compact orientable surface S there is a natural complex structure making S into a Riemann surface. In this paper we consider regular maps and enquire about the Weierstrass points on the underlying Riemann surface. We are particularly interested to know when these are geometric, i.e. whether they lie at vertices, face-centres or edge-centres of the map.