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The R ∞ property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups

  • Wagner C. Sgobbi , Dalton C. Silva and Daniel Vendrúscolo EMAIL logo
Published/Copyright: January 31, 2023
Journal of Group Theory
From the journal Journal of Group Theory Volume 26 Issue 4

Abstract

We say a group 𝐺 has property R ∞ if the number R ⁢ ( φ ) of twisted conjugacy classes is infinite for every automorphism 𝜑 of 𝐺. For such groups, the R ∞ -nilpotency degree is the least integer 𝑐 such that G / γ c + 1 ⁢ ( G ) has property R ∞ . In this work, we compute the R ∞ -nilpotency degree of all Generalized Solvable Baumslag–Solitar groups Γ n . Moreover, we compute the lower central series of Γ n , write the nilpotent quotients Γ n , c = Γ n / γ c + 1 ⁢ ( Γ n ) as semidirect products of finitely generated abelian groups and classify which invertible integer matrices can be extended to automorphisms of Γ n , c .

Funding source: Fundação de Amparo à Pesquisa do Estado de São Paulo

Award Identifier / Grant number: 2016/24707-4

Award Identifier / Grant number: 2017/21208-0

Award Identifier / Grant number: 2019/03150-0

Funding statement: Wagner C. Sgobbi was supported by grants 2017/21208-0 and 2019/03150-0, São Paulo Research Foundation (FAPESP). Daniel Vendrúscolo was partially supported by grant 2016/24707-4 São Paulo Research Foundation (FAPESP)

Acknowledgements

The authors would like to thank the referee, whose valuable comments and advice improved much of the paper.

  1. Communicated by: Dessislava Kochloukova

References

[1] M. Calvez and I. Soroko, Property R ∞ for some spherical and affine Artin–Tits groups, J. Group Theory 25 (2022), no. 6, 1045–1054. 10.1515/jgth-2022-0010Search in Google Scholar

[2] K. Dekimpe and D. L. Gonçalves, The R ∞ property for free groups, free nilpotent groups and free solvable groups, Bull. Lond. Math. Soc. 46 (2014), no. 4, 737–746. 10.1112/blms/bdu029Search in Google Scholar

[3] K. Dekimpe and D. L. Gonçalves, The R ∞ property for nilpotent quotients of surface groups, Trans. London Math. Soc. 3 (2016), no. 1, 28–46. 10.1112/tlms/tlw002Search in Google Scholar

[4] K. Dekimpe and D. L. Gonçalves, The R ∞ -property for nilpotent quotients of Baumslag-Solitar groups, J. Group Theory 23 (2020), no. 3, 545–562. 10.1515/jgth-2018-0182Search in Google Scholar

[5] K. Dekimpe, D. L. Gonçalves and O. Ocampo, The R ∞ property for pure Artin braid groups, Monatsh. Math. 195 (2021), no. 1, 15–33. 10.1007/s00605-020-01484-7Search in Google Scholar

[6] A. Fel’shtyn and D. L. Gonçalves, The Reidemeister number of any automorphism of a Baumslag–Solitar group is infinite, Geometry and Dynamics of Groups and Spaces, Progr. Math. 265, Birkhäuser, Basel (2008), 399–414. 10.1007/978-3-7643-8608-5_9Search in Google Scholar

[7] D. Gonçalves and P. Wong, Twisted conjugacy classes in nilpotent groups, J. Reine Angew. Math. 633 (2009), 11–27. 10.1515/CRELLE.2009.058Search in Google Scholar

[8] B. J. Jiang, Lectures on Nielsen Fixed Point Theory, Contemp. Math. 14, American Mathematical Society, Providence, 1983. 10.1090/conm/014Search in Google Scholar

[9] G. Levitt and M. Lustig, Most automorphisms of a hyperbolic group have very simple dynamics, Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), no. 4, 507–517. 10.1016/S0012-9593(00)00120-8Search in Google Scholar

[10] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Dover, New York, 1976. Search in Google Scholar

[11] R. Mikhailov and I. B. S. Passi, Lower Central and Dimension Series of Groups, Lecture Notes in Math. 1952, Springer, Berlin, 2009. Search in Google Scholar

[12] K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936), no. 1, 586–593. 10.1007/BF01565432Search in Google Scholar

[13] V. Roman’kov, Twisted conjugacy classes in nilpotent groups, J. Pure Appl. Algebra 215 (2011), no. 4, 664–671. 10.1016/j.jpaa.2010.06.015Search in Google Scholar

[14] J. Taback and P. Wong, Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag–Solitar groups, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 705–717. 10.1112/jlms/jdm024Search in Google Scholar
Received: 2022-08-03
Revised: 2022-11-20
Published Online: 2023-01-31
Published in Print: 2023-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Curvature distribution and hyperbolicity
  3. On Whitehead’s cut vertex lemma
  4. Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups
  5. On endomorphisms of the direct product of two free groups
  6. The R ∞ property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups
  7. A note on torsion length and torsion subgroups
  8. On the separability of subgroups of nilpotent groups by root classes of groups
  9. The Addition Theorem for two-step nilpotent torsion groups
  10. A bound for the orders of centralizers of irreducible subgroups of algebraic groups
  11. The average character degree of finite groups and Gluck’s conjecture
  12. Character degrees of normally monomial 𝑝-groups of maximal class
  13. Finite groups with an affine map of large order
  14. Classification of non-solvable groups whose power graph is a cograph
https://doi.org/10.1515/jgth-2022-0129

Articles in the same Issue

  1. Frontmatter
  2. Curvature distribution and hyperbolicity
  3. On Whitehead’s cut vertex lemma
  4. Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups
  5. On endomorphisms of the direct product of two free groups
  6. The R ∞ property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups
  7. A note on torsion length and torsion subgroups
  8. On the separability of subgroups of nilpotent groups by root classes of groups
  9. The Addition Theorem for two-step nilpotent torsion groups
  10. A bound for the orders of centralizers of irreducible subgroups of algebraic groups
  11. The average character degree of finite groups and Gluck’s conjecture
  12. Character degrees of normally monomial 𝑝-groups of maximal class
  13. Finite groups with an affine map of large order
  14. Classification of non-solvable groups whose power graph is a cograph
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