Home Galois action on Fuchsian surface groups and their solenoids
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Galois action on Fuchsian surface groups and their solenoids

  • Amir Džambić EMAIL logo and Gabino González-Diez
Published/Copyright: December 12, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 2

Abstract

Let C be a complex algebraic curve uniformized by a Fuchsian group Γ. In the first part of this paper we identify the automorphism group of the solenoid associated with Γ with the Belyaev completion of its commensurator Comm⁢(Γ) and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of Gal⁢(ℂ/ℚ) on algebraic curves. In turn, this fact yields a proof of the Galois invariance of the arithmeticity of Γ independent of Kazhhdan’s. In the second part we focus on the case in which Γ is arithmetic. The list of further Galois invariants we find includes: (i) the periods of Comm⁢(Γ), (ii) the solvability of the equations X2+sin2⁡2⁢π2⁢k+1 in the invariant quaternion algebra of Γ and (iii) the property of Γ being a congruence subgroup.

Keywords: Algebraic curves; arithmetic Fuchsian groups; commensurators; Galois invariants; group completions; Riemann surface solenoids
MSC 2010: 14H30; 20H10; 30F10; 11F06; 22D99

Communicated by Jan Bruinier

Funding statement: Second author partially supported by Spanish Government Research Project MTM2016-79497-P.

Acknowledgements

The authors would like to thank Adrián Ubis for the explicit choice of the constant bp in Section 2.1.1, Jürgen Wolfart for many valuable suggestions and Andrei Jaikin-Zapirain for his helpful comments at an early stage of this paper. Finally, the authors would like to thank the referee for his/her careful reading of the paper and the valuable comments and suggestions.

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Received: 2020-06-08
Revised: 2020-11-03
Published Online: 2020-12-12
Published in Print: 2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
https://doi.org/10.1515/forum-2020-0148
Keywords for this article
Algebraic curves; arithmetic Fuchsian groups; commensurators; Galois invariants; group completions; Riemann surface solenoids

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
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