A Schottky description of a Theorem of Conder-Maclahlan-Vasiljevic-Wilson

Abstract

In a recently paper of Conder-Maclachlan-Vasiljevic-Wilson [7] it has been proved that for every positive integer \( g \geq 2 \) there exists a closed non-orientable surface of algebraic genus \( g \) with at least \( 4(g + 1) \) automorphisms if \( g \) is even, or at least \( 8(g - 1) \) automorphisms if \( g \) is odd. The main purpose of this note is to provide explicitly such kind of situations in terms of Schottky groups. We also provide a construction of closed non-orientable surfaces of algebraic genus \( g \), for infinite many values of integers \( g \geq 2 \), so that they admit a group of automorphisms of order \( 12(g - 1) \) which can be reflected by Schottky groups.