Abstract
A finitely generated group G that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag-Solitar group or GBS-group. Let p and q be coprime integers other than 0, 1, and −1. We prove that the Baumslag-Solitar group BS(p, q) embeds into G if and only if the equation x −1 y p x = y q is solvable in G for y ≠ 1; i.e., \(\tfrac{p} {q} \) ∈ Δ(G), where Δ is the modular homomorphism.
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References
Baumslag G. and Solitar D., “Some two-generator one-relator non-Hopfian groups,” Bull. Amer. Math. Soc., 68, No. 3, 199–201 (1962).
Dudkin F. A., “Baumslag-Solitar groups and their subgroups,” in: Itogi Nauki. Yug Rossii. Matematicheskiĭ Forum. Vol. 6: Groups and Graphs [in Russian], YuMI VNTs RAN i PSO-A, Vladikavkaz, 2012, pp. 21–28.
Serre J.-P., Trees, Springer-Verlag, Berlin, Heidelberg, and New York (1980).
Churkin V. A., “Theory of groups that act on trees,” Algebra and Logic, 22, No. 2, 159–165 (1983).
Clay M. and Forester M., “Whitehead moves for G-trees,” Bull. London Math. Soc., 41, No. 2, 205–212 (2009).
Forester M., “On uniqueness of JSJ decompositions of finitely generated groups,” Comm. Math. Helv., 78, 740–751 (2003).
Clay M., “Deformation spaces of G-trees and automorphisms of Baumslag-Solitar groups,” Groups Geom. Dyn., 3, 39–69 (2009).
Forester M., “Splittings of generalized Baumslag-Solitar groups,” Geom. Dedicata, 121, No. 1, 43–59 (2006).
Clay M. and Forester M., “On the isomorphism problem for generalized Baumslag-Solitar groups,” Algebr. Geom. Topol., 8, 2289–2322 (2008).
Levitt G., “On the automorphism group of generalized Baumslag-Solitar groups,” Geom. Topol., 11, 473–515 (2007).
Forester M., “Deformation and rigidity of simplicial group actions on trees,” Geom. Topol., 6, 219–267 (2002).
Bass H., “Covering theory for graphs of groups,” J. Pure Appl. Algebra, 89, No. 1, 3–47 (1993).
Dudkin F. A., “Subgroups of Baumslag-Solitar groups,” Algebra and Logic, 48, No. 1, 1–19 (2009).
Dudkin F. A., “Subgroups of finite index in Baumslag-Solitar groups,” Algebra and Logic, 49, No. 3, 221–232 (2010).
Levitt G., Quotients and Subgroups of Baumslag-Solitar Groups [Preprint], arXiv:1308.5122.
Lyndon R. C. and Schupp P. E., Combinatorial Group Theory, Springer-Verlag, Berlin etc. (2001).
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Original Russian Text Copyright © 2014 Dudkin F.A.
The author was supported by the Russian Foundation for Basic Research (Grants 12-01-31222 and 12-01-33102).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 90–96, January–February, 2014.
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Dudkin, F.A. Embedding of baumslag-solitar groups into the generalized Baumslag-Solitar groups. Sib Math J 55, 72–77 (2014). https://doi.org/10.1134/S0037446614010091
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DOI: https://doi.org/10.1134/S0037446614010091