Cyclic groups of automorphisms of schottky type

Rubén A Hidalgo

doi:10.22199/s07160917.1999.0001.00002

Abstract

Given an abstract group of orden n, we call its Schottky genus to the minimum genus ? ? 2 on which it acts as group of conformal automorphisms of Schottky type. In this note, we compute the Schottky genus for both cyclic and dihedral groups. In particular, we obtain that the Schottky genus of the dihedral group of order 2n is the same as for the cyclic group of order n. Since every dihedral group is of Schottky type, we have that the Schottky genus of a dihedral group of order 2n is also its minimum genus.

Highlights

Given an abstmrt gmup of or·de1· n, we call its Schottl.:y genus to the minimum gcnus g :::0: 2 on which it acts as group of conforma/ automor'phisms of Schottl.:y type
We say that the three-tuple (0, G, P : O-+ S) is a Schottky uniformization of S
The importance of Schottky uniformizations respect to the non-universal uniformizations of Riemann surfaces is given by the easy understanding of their geometry

Summary

Sorne Types of Schottky Groups

For each n > 2 we construct finite normal extensions K of a Schottky group G of genus S(n) 2': 2 so that K/G is isomorphic toa dihedral group of arder.

Main Construction

Computation of the Schottky genus for Cyclic groups

Since li