Arbitrarily large Galois orbits of non-homeomorphic surfaces

TL;DR

This paper constructs explicit large Galois orbits of surfaces with non-isomorphic fundamental groups, demonstrating the existence of unbounded Galois orbits in the moduli space of surfaces.

Contribution

It introduces a method to produce explicit Galois orbits of arbitrary size containing non-homeomorphic surfaces with specific Beauville groups.

Findings

Constructed Galois orbits of unbounded size

Surfaces have mutually non-isomorphic fundamental groups

Uses Beauville surfaces with PGL_2(p) groups

Abstract

We construct orbits of the absolute Galois group, of explicit unbounded size, consisting of surfaces with mutually non-isomorphic fundamental groups. These are Beauville surfaces with Beauville group PGL_2(p).