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.

中文翻译：

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对 Fuchsian 表面群及其螺线管的伽罗瓦作用

令 $C$ 是由 Fuchsian 群 $\Gamma$ 统一化的复杂代数曲线。在本文的第一部分中，我们确定了与 $\Gamma$ 相关联的螺线管的自同构群以及它的公称量 $\mathrm{Comm}(\Gamma)$ 的 Belyaev 完成，我们使用这个识别来证明同构类这种完成是 $\mathrm{Gal}(\mathbb C/\mathbb Q)$ 在代数曲线上的自然伽罗瓦作用的不变量。反过来，这个事实产生了独立于 Kazhhdan 的 $\Gamma$ 算术性的伽罗瓦不变性的证明。在第二部分，我们关注 $\Gamma$ 是算术的情况。我们发现的进一步伽罗瓦不变量列表包括： i) $\mathrm{Comm}(\Gamma)$ 的周期，