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Lifting subgroups of PSL2 to SL2 over local fields

  • Naomi Andrew , Matthew J. Conder , Ari Markowitz and Jeroen Schillewaert ORCID logo EMAIL logo
Published/Copyright: February 27, 2025
Journal of Group Theory
From the journal Journal of Group Theory Volume 28 Issue 4

Abstract

Let 𝐾 be a non-archimedean local field. We show that discrete subgroups without 2-torsion in PSL 2 ⁢ ( K ) can always be lifted to SL 2 ⁢ ( K ) , and provide examples (when char ⁡ ( K ) ≠ 2 ) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into PSL 2 ⁢ ( K ) to SL 2 ⁢ ( K ) .

Funding source: H2020 European Research Council

Award Identifier / Grant number: 850930

Funding source: Marsden Fund

Award Identifier / Grant number: 3723563

Award Identifier / Grant number: 3725880

Funding source: Royal Society Te Apārangi

Award Identifier / Grant number: 3726956

Funding statement: This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 850930). It has also received funding from the New Zealand Marsden Fund (grant numbers 3723563 and 3725880) and the Rutherford Foundation (grant number 3726956).

Acknowledgements

We thank Sam Hughes for cohomological advice. The first author would like to thank the University of Auckland for its hospitality during a visit in September 2023.

  1. Communicated by: Michael Giudici

References

[1] J. Bellaïche and G. Chenevier, Sous-groupes de GL 2 et arbres, J. Algebra 410 (2014), 501–525. 10.1016/j.jalgebra.2014.02.005Search in Google Scholar

[2] J. O. Button, Lifting Möbius groups, New York J. Math. 8 (2002), 181–188. Search in Google Scholar

[3] J. O. Button, Lifting Möbius groups: Addendum, New York J. Math. 8 (2002), 257–258. Search in Google Scholar

[4] J. W. S. Cassels, Local Fields, London Math. Soc. Stud. Texts 3, Cambridge University, Cambridge, 1986. Search in Google Scholar

[5] M. Culler, Lifting representations to covering groups, Adv. Math. 59 (1986), no. 1, 64–70. 10.1016/0001-8708(86)90037-XSearch in Google Scholar

[6] V. Guirardel and G. Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1 (2007), no. 2, 135–181. 10.4171/ggd/8Search in Google Scholar

[7] A. Hatcher, Algebraic Topology, Cambridge University, Cambridge, 2002. Search in Google Scholar

[8] F. Kato, Non-Archimedean orbifolds covered by Mumford curves, J. Algebraic Geom. 14 (2005), no. 1, 1–34. 10.1090/S1056-3911-04-00384-4Search in Google Scholar

[9] W. Magnus, Noneuclidean Tesselations and Their Groups, Pure Appl. Math. 61, Academic Press, New York, 1974. Search in Google Scholar

[10] J. W. Morgan and P. B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2) 120 (1984), no. 3, 401–476. 10.2307/1971082Search in Google Scholar

[11] J. Neukirch, Algebraic Number Theory, Grundlehren Math. Wiss. 322, Springer, Berlin, 1999. 10.1007/978-3-662-03983-0Search in Google Scholar

[12] J.-P. Serre, Trees, Springer, Berlin, 1980. 10.1007/978-3-642-61856-7Search in Google Scholar

[13] R. C. Valentini and M. L. Madan, A hauptsatz of L. E. Dickson and Artin–Schreier extensions, J. Reine Angew. Math. 318 (1980), 156–177. 10.1515/crll.1980.318.156Search in Google Scholar

[14] M. van der Put and H. H. Voskuil, Discontinuous subgroups of PGL 2 ⁢ ( K ) , J. Algebra 271 (2004), no. 1, 234–280. 10.1016/S0021-8693(03)00398-3Search in Google Scholar

[15] R. A. Wilson, The Finite Simple Groups, Grad. Texts in Math. 251, Springer, London, 2009. 10.1007/978-1-84800-988-2Search in Google Scholar
Received: 2024-03-16
Revised: 2024-12-03
Published Online: 2025-02-27
Published in Print: 2025-07-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the kernel of actions on asymptotic cones
  3. CAT(0) cube complexes and asymptotically rigid mapping class groups
  4. Iwip endomorphisms of free groups and fixed points of graph selfmaps
  5. Space of orders with finite Cantor–Bendixson rank
  6. Lifting subgroups of PSL2 to SL2 over local fields
  7. Regular 3-polytopes of order 2𝑛𝑝
  8. On the number of tuples of group elements satisfying a first-order formula
  9. Exponent-critical groups
  10. On soluble groups in which commutators have prime power order
  11. A character theoretic criterion for Fitting height
  12. Hilbert divisors and degrees of irreducible Brauer characters
  13. Character triples and relative defect zero characters
https://doi.org/10.1515/jgth-2024-0064

Articles in the same Issue

  1. Frontmatter
  2. On the kernel of actions on asymptotic cones
  3. CAT(0) cube complexes and asymptotically rigid mapping class groups
  4. Iwip endomorphisms of free groups and fixed points of graph selfmaps
  5. Space of orders with finite Cantor–Bendixson rank
  6. Lifting subgroups of PSL2 to SL2 over local fields
  7. Regular 3-polytopes of order 2𝑛𝑝
  8. On the number of tuples of group elements satisfying a first-order formula
  9. Exponent-critical groups
  10. On soluble groups in which commutators have prime power order
  11. A character theoretic criterion for Fitting height
  12. Hilbert divisors and degrees of irreducible Brauer characters
  13. Character triples and relative defect zero characters
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