Galois action on Fuchsian surface groups and their solenoids

TL;DR

This paper explores the Galois action on Fuchsian surface groups and their solenoids, establishing invariants related to automorphisms, arithmeticity, and quaternion algebra properties, advancing understanding of algebraic curves and their symmetries.

Contribution

It identifies the automorphism group of solenoids with the Belyaev completion of the commensurator and proves Galois invariance of arithmeticity, introducing new Galois invariants for arithmetic Fuchsian groups.

Findings

Automorphism group of solenoid equals Belyaev completion of commensurator

Galois invariance of the arithmeticity of Fuchsian groups

New Galois invariants including periods and quaternion algebra properties

Abstract

Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $\Gamma$. In the first part of this paper we identify the automorphism group of the solenoid associated with $\Gamma$ with the Belyaev completion of its commensurator $\mathrm{Comm}(\Gamma)$ and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of $\mathrm{Gal}(\mathbb C/\mathbb Q)$ on algebraic curves. In turn this fact yields a proof of the Galois invariance of the arithmeticity of $\Gamma$ independent of Kazhhdan's. In the second part we focus on the case in which $\Gamma$ is arithmetic. The list of further Galois invariants we find includes: i) the periods of $\mathrm{Comm}(\Gamma)$, ii) the solvability of the equations $X^2+\sin^2 \frac{2\pi}{2k+1}$ in the invariant quaternion algebra of $\Gamma$ and iii) the property of $\Gamma$ being a congruence subgroup.