Book contents
- Frontmatter
- Contents
- Introduction
- 1 Finite group schemes
- 2 Algorithms for polycyclic groups
- 3 The spread of finite and infinite groups
- 4 Discrete subgroups of semisimple Lie groups, beyond lattices
- 5 Complete reducibility and subgroups of exceptional algebraic groups
- 6 Axial algebras of Jordan and Monster type
- 7 An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams
- 8 Finite groups and the class-size prime graph revisited
- 9 Character bounds for finite simple groups and applications
- 10 Generalized Baumslag-Solitar groups: a topological approach
- References
10 - Generalized Baumslag-Solitar groups: a topological approach
Published online by Cambridge University Press: 21 November 2024
Book contents
- Frontmatter
- Contents
- Introduction
- 1 Finite group schemes
- 2 Algorithms for polycyclic groups
- 3 The spread of finite and infinite groups
- 4 Discrete subgroups of semisimple Lie groups, beyond lattices
- 5 Complete reducibility and subgroups of exceptional algebraic groups
- 6 Axial algebras of Jordan and Monster type
- 7 An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams
- 8 Finite groups and the class-size prime graph revisited
- 9 Character bounds for finite simple groups and applications
- 10 Generalized Baumslag-Solitar groups: a topological approach
- References
Summary
Generalized Baumslag-Solitar groups are a class of combinatorially interesting groups. Their group theory is also closely associated to a the topology of a class of 2-dimensional spaces. These 2-dimensional spaces are Seifert fibred. We develop the basic topology of these fibrations and derive some of the most immediate group theoretic consequences of this topology.
- Type
- Chapter
- Information
- Groups St Andrews 2022 in Newcastle , pp. 407 - 440Publisher: Cambridge University PressPrint publication year: 2024
References
[1] Baumslag, G., and Solitar, D. 1962. Some two-generator one-relator nonHopfian groups. Bull. Amer. Math. Soc., 68, 199–201.CrossRefGoogle Scholar
[2] Bedenikovic, T., Delgado, A., and Timm, M. 2006. A classification of 2-complexes with nontrivial self-covers. Topology and Its Applications, 153, 2073–2091.CrossRefGoogle Scholar
[3] Delgado, A., D.J.S., Robinson, and Timm, M. 2011. Generalized Baumslag-Solitar groups and geometric homomorphisms. J. Pure and Applied Algebra, 215, 398–410.Google Scholar
[4] Delgado, A., Robinson, D.J.S, and Timm, M. 2014. Generalised Baumslag-Solitar Graphs with Soluble Fundamental Groups. Algebra Colloquium, 21, 53–58.CrossRefGoogle Scholar
[5] Delgado, A., Robinson, D.J.S., and Timm, M. 2017. Cyclic normal subgroups of generalized Baumslag-Solitar groups. Communications in Algebra, 1801–1818.Google Scholar
[6] Delgado, A., Robinson, D.J.S., and Timm, M. 2018. 3 manifolds and generalized Baumslag-Solitar groups. Communications in Analysis and Geometry, 26, 571–584.CrossRefGoogle Scholar
[7] Forester, M. 2003. On uniqueness of JSJ decompositions of finitely generated groups. Comment. Math. Helv., 78, 740–751.CrossRefGoogle Scholar
[8] Heil, W.H. 1975. Some finetely presented non-3-manifold groups. PAMS, 53, 497–500.CrossRefGoogle Scholar
[9] Hempel, J. 1976. 3-Manifolds. Ann. of Math. Studies, vol. 86. Princeton, New Jersey: Princeton University Press.Google Scholar
[11] P.H., Kropholler 1990. Baumslag-Solitar groups and some other groups of cohomological dimension 2. Commentarii Hatematicic Helvetici, 65, 445–461.Google Scholar
[12] Levit, G. 2015. Generalized Baumslag-Solitar groups: rank and finite index subgroups. Ann. Inst. Fourier (Grenoble), 65, 725–762.CrossRefGoogle Scholar
[13] Levitt, G. 2005. Characteriszing rigid simplicial actions on trees. Pages 27–33 of: Geometric Methods in Group Theory. Contemporay Mathematics, vol. 372. Providence, Rhode Island: Amer. Math. Soc.CrossRefGoogle Scholar
[14] Levitt, G. 2007. On the automorphism group of generalized Baumslag-Solitar groups. Geom. Top., 11, 473–515.CrossRefGoogle Scholar
[15] Levitt, G. 2015. Quotients and subgroups of Baumslag-Solitar groups. J. Group Theory, 18, 1–43.CrossRefGoogle Scholar
[17] Margolis, A. 2022. Model geometries of finitely generated groups. arXiv:2207.10509v1Google Scholar
[18] Robinson, D.J.S. 2011. The Schur multiplier of a generalized Baumslag-Solitar group. Rend. Se. Mat. Univ. Padova, 125, 207–215.CrossRefGoogle Scholar
[19] Robinson, D.J.S. 2015. Generalized Baumslag-Solitar groups: a survey of recent progress. Page 457–468 of: Campbell, C. M., Quick, M. R., Robertson, E. F., and Roney-Dougal, C. M. (eds), Groups St Andrews 2013. London Mathematical Society Lecture Note Series. Cambridge University Press.Google Scholar
[20] Serre, J.P. 2003. Trees. Monographs on Mathematics. Berlin: Springer-Verlag. Translated from the French original by John Stillwell. Corrected 2nd printing of the 1980 English translation.Google Scholar
[21] Sokolov, E.V. 2021. Certain residual properties of the generalized Bausmslag-Solitar groups. Journal of Algebra, 582(September), 1–25. https://doi.org/10.1016/j.jalgebra.2021.05.001.CrossRefGoogle Scholar
[22] Timm, M. 2022. A topologist’s interactions with Derek J. S. Robinson and his mathematics. pre-print.Google Scholar
[23] Whyte, K. 2001. The large scale geometry of the higher Baumslag-Solitar groups. Geometric and Functional Analysis, 11, 1327–1343. doi 10.1007/s00039-001-8232-6.CrossRefGoogle Scholar
[24] Yoshikawa, K. 1988. A ribbon knot group which has no free base. Proceedings of the American Mathematical Society, 102, 1065–1070.CrossRefGoogle Scholar