Abstract
A generalized Baumslag–Solitar group is a finitely generated group that acts on a tree with infinite cyclic edge and vertex stabilizers. A group G is residually a finite \(\pi \)-group, for a set of primes \(\pi \), if every non-trivial element of G has non-trivial image in a quotient of G that is a finite \(\pi \)-group. We provide a criterion for generalized Baumslag–Solitar groups to be residually a finite \(\pi \)-group.
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References
Azarov, D.N.: A criterion for the \(\cal{F}_\pi \)-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks. Sib. Math. J. 57(3), 377–384 (2016)
Baumslag, G., Solitar, D.: Some two-generator one-relator non-Hopfian groups. Bull. AMS 68(3), 199–201 (1962)
Brazil, M.: Growth functions for some nonautomatic Baumslag–Solitar groups. Trans. Am. Math. Soc. 342, 137–154 (1994)
Clay, M., Forester, M.: On the isomorphism problem for generalized Baumslag-Solitar groups. Algebr. Geom. Topol. 8, 2289–2322 (2008)
Delgado, A.L., Robinson, D.J.S., Timm, M.: Cyclic normal subgroups of generalized Baumslag–Solitar groups. Comm. Algebra 45(4), 1808–1818 (2017)
Dudkin, F.A.: Computation of the centralizer dimension of generalized Baumslag–Solitar groups. Sib. Elect. Math. Rep. 15, 1823–1841 (2018)
Edjvet, M., Johnson, D.L.: The growth of certain amalgamated free products and HNN-extensions. J. Austral. Math. Soc. 52, 285–298 (1992)
Forester, M.: Deformation and rigidity of simplicial group actions on trees. Geom. Topol. 6, 219–267 (2002)
Ivanova, O.A., Moldavanskii, D.I.: \(\cal{F}_\pi \)-residuality of some one related groups (Russian). Nauch. Trudi IvGU, Math. 6, 51–58 (2008)
Kargapolov, M.I., Merzljakov, J.I.: Fundamentals of the Theory of Groups. Springer, Berlin (1979)
Kropholler, P.H.: Baumslag–Solitar groups and some other groups of cohomological dimension two. Comment. Math. Helv. J. 65, 547–558 (1990)
Levitt, G.: On the automorphism group of generalized Baumslag–Solitar groups. Geom. Topol. 11, 473–515 (2007)
Levitt, G.: Quotients and subgroups of Baumslag–Solitar groups. J. Group Theory 18(1), 1–43 (2015)
Moldavanskii, D.I.: On the residuality of Baumslag–Solitar groups (Russian). Cebysevskij sb. 13(1), 110–114 (2012)
Robinson, D.J.S.: Generalized Baumslag–Solitar Groups: A Survey of Recent Progress. In: Groups St Andrews 2013, LMS, Lecture Note Series, vol. 422, pp. 457–469, Cambridge (2016)
Serre, J.P.: Trees. Springer, Berlin (1980)
Shalen, P.B.: Three-manifolds and Baumslag–Solitar groups. Topol. Appl. 110(1), 113–118 (2001)
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The work was supported by Russian Science Foundation (project 19-11-00039).
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Dudkin, F. \(\mathcal {F}_\pi \)-residuality of generalized Baumslag–Solitar groups. Arch. Math. 114, 129–134 (2020). https://doi.org/10.1007/s00013-019-01404-8
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DOI: https://doi.org/10.1007/s00013-019-01404-8
Keywords
- Residual \(\pi \)-finiteness
- Residual finiteness
- Generalized Baumslag–Solitar group
- Baumslag–Solitar group