Galois action on Fuchsian surface groups and their solenoids

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.