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Galois action on Fuchsian surface groups and their solenoids. (English) Zbl 07509454

Summary: Let \(C\) be a complex algebraic curve uniformized 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 \(\text{Comm}(\Gamma)\) and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of \(\text{Gal}(\mathbb{C}/\mathbb{Q})\) 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 \(\text{Comm}(\Gamma)\), (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.

MSC:

14H30 Coverings of curves, fundamental group
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F10 Compact Riemann surfaces and uniformization
11F06 Structure of modular groups and generalizations; arithmetic groups
22D99 Locally compact groups and their algebras
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