Infinite families of automorphism groups of Riemann surfaces

Summary

Introduction

Let a, b be two rational numbers, with a > 0. Consider the sequence N_a,b : g ↦ ag + b, for g = 2,3,…. We say that this sequence is admissible if for infinitely many values of g the number ag + b is the order of an automorphism group G of a compact Riemann surface X_g of genus g. When this occurs, the pair (X_g,G), or simply X_g or G, is said to belong to N_a,b.

A cocompact Fuchsian group gives rise to an admissible sequence of the form a(g – 1), where a is a positive rational number which depends only on the group. For example the {2,3, 7}–triangle group gives rise to the sequence 84(g – 1); see Macbeath [Me] for an early thorough discussion of these aspects. Conversely, it was shown in [K] that every admissible sequence of the form a(g – 1) arises from a fixed finite number of cocompact Fuchsian groups.

There are a few known admissible sequences N_a,b where a + b ≠ 0. Wiman showed that 4g + 2 is the largest order of a cyclic automorphism group of a compact Riemann surface of genus g, and he also exhibited such surfaces for every g; see [W], [H] Accola and independently Maclachlan showed that for infinitely many values of g the largest order of an automorphism group of a compact Riemann surface of genus g is 8g + 8.