Coy L. May

Groups of small symmetric genus

Group actions on compact surfaces have received considerable attention during the past century. The surface has often carried an analytic structure and been considered a Riemann surface or, equivalently, a complex algebraic curve.

The symmetric crosscap number of a group

Let G be a finite group. The symmetric crosscap number \tilde \sigma (G) is the minimum topological genus of any compact non-orientable surface (with empty boundary) on which G acts effectively. We first survey some of the basic facts about the symmetric crosscap number; this includes relationships between this parameter and others. We obtain formulas for the symmetric crosscap number for three families of groups, the dicyclic groups, the abelian groups with most factors in the canonical form isomorphic to Z_(2) , and the hamiltonian groups with no odd order part. We also determine \tilde \sigma (G) for each group G with order less than 16. The groups with symmetric crosscap numbers 1 and 2 have been classified. We show here that there are no groups with \tilde\sigma=3 ; this affirms a conjecture of Tucker.

THE REAL GENUS OF 2-GROUPS

Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we consider 2-groups acting on bordered Klein surfaces. The main focus is determining the real genus of each of the 51 groups of order 32. We also obtain some general results about the partial presentations that 2-groups acting on bordered surfaces must have. In addition, we obtain genus formulas for some families of 2-groups and show that if G is a 2-group with positive real genus, then ρ(G) ≡ 1 mod 4.

Complex doubles of bordered Klein surfaces with maximal symmetry

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A ( X ) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.

THE SYMMETRIC GENUS OF 2-GROUPS

Let G be a finite group. The symmetric genus σ( G ) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2 m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2 m −3 and a dihedral subgroup of index 4 or else the exponent of G is 2 m −2 . We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G . A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

Finite metacyclic groups acting on bordered surfaces

A group is called metacyclic in case both its commutator subgroup and commutator quotient group are cyclic. Thus a metacyclic group is a cyclic extension of a cyclic group, and metacyclic groups are among the best understood of the nonabelian groups. Many interesting groups are metacyclic. For instance, the dihedral groups and the “odd” dicyclic groups are metacyclic; see [4, pp. 9–11] for more examples. Here we shall consider the actions of these groups on bordered Klein surfaces.

The symmetric genus of metacyclic groups

Abstract Let G be a finite group. The symmetric genus of G is the minimum genus of any Riemann surface on which G acts faithfully. Here we determine a useful lower bound for the symmetric genus of a finite group with a cyclic quotient group. The lower bound is attained for the family of K -metacyclic groups, and we determine the symmetric genus of each nonabelian subgroup of a K -metacyclic group. We also provide some examples of other groups for which the lower bound is attained. We use the standard representation of a finite group as a quotient of a noneuclidean crystallographic (NEC) group by a Fuchsian surface group.

THE 2-GROUPS OF ODD STRONG SYMMETRIC GENUS

Let G be a finite group. The strong symmetric genusσ0(G) is the minimum genus of any Riemann surface on which G acts preserving orientation. We show that a 2-group G has strong symmetric genus congruent to 3 (mod 4) if and only if G is in one of 14 families of groups. A consequence of this classification is that almost all positive integers that are the genus of a 2-group are congruent to 1 (mod 4).

GROUPS OF EVEN REAL GENUS

Let G be a finite group. The real genusρ (G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we develop some constructions of groups of even real genus, first using the notion of a semidirect product. As a consequence, we are able to show that for each integer g in certain congruence classes, there is at least one group of genus g. Next we consider the direct product Zn × G, in which one factor is cyclic and the other is a group of odd order that is generated by two elements. By placing a restriction on the genus action of G, we find the real genus of the direct product, in case n is relatively prime to |G|. We give some applications of this result, in particular to O*-groups, the odd order groups of maximum possible order. Finally we apply our results to the problem of determining whether there is a group of real genus g for each value of g. We prove that the set of integers for which there is a group has lower density greater than 5/6.