Skip to main content
SpringerLink
Log in

Baumslag–Solitar Groups

  • Chapter
  • First Online:
Book cover A Sampling of Remarkable Groups

Part of the book series: Compact Textbooks in Mathematics ((CTM))

  • 1061 Accesses

Abstract

We introduce and explore the Baumslag–Solitar groups which is the famous collection of groups first described by G. Baumslag and D. Solitar. This collection, also referred to as BS groups, includes the first examples of finitely generated, one-relator non-hopfian groups. BS groups are widely known and provide many counter-examples and test-cases in combinatorial and geometric group theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anshel, M., Stebe, P.: The solvability of the conjugacy problem for certain HNN groups. Bull. Am. Math. Soc. 80, 266–270 (1974)

    Article  MathSciNet  Google Scholar 

  2. Baer, R.: Groups without proper isomorphic quotients. Bull. Am. Math. Soc. 50(4), 267–278 (1944)

    Article  MathSciNet  Google Scholar 

  3. Bartholdi, L., Šunić, Z.: Some solvable automaton groups mn. Contemp. Math. 394, 11–30 (2006)

    Article  Google Scholar 

  4. Baumslag, G.: A non-Hopfian group. Bull. Am. Math. Soc. 68(3), 196–198 (1962)

    Article  MathSciNet  Google Scholar 

  5. Baumslag, G., Solitar, D.: Some two generator one-relator non-Hopfian groups. Bull. Am. Math. Soc. 68(9), 199–201 (1962)

    Article  MathSciNet  Google Scholar 

  6. Belk, J.: One sheet of the Cayley graph of the Baumslag–Solitar group BS(1,2). Wikimedia Commons (2013)

    Google Scholar 

  7. Belk, J.: Picture showing how the sheets of the Cayley graph of the Baumslag-Solitar group BS(1,2) fit together. Wikimedia Commons (2013)

    Google Scholar 

  8. Brunner, A.M., Sidki, S.: The generation of GL(n, Z) by finite state automata. Int. J. Alg. Comput. 8(1), 127–139 (1998)

    Article  MathSciNet  Google Scholar 

  9. Chandler, B., Magnus, W.: A History of Combinatorial Group Theory: A Case Study in the History of Ideas. Springer, New York (1982)

    Book  Google Scholar 

  10. de la Harpe, P.: Topics in Geometric Group Theory, pp. 191–196. University of Chicago Press, Chicago (2000)

    Google Scholar 

  11. Higman, G.: A finitely related group with an isomorphic proper factor group. J. London Math. Soc. 26, 59–61 (1951)

    Article  MathSciNet  Google Scholar 

  12. Magnus, W.: Das identitätsproblem fur̈ Gruppen mit einer definierenden relation. Math. Ann. 106, 295–307 (1932)

    Article  MathSciNet  Google Scholar 

  13. Magnus, W.: Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring. Math. Ann. 111, 259–280 (1935)

    Article  MathSciNet  Google Scholar 

  14. Moldavanskii, D.I.: Isomorphism of the Baumslag-Solitar Groups. Ukrainskii Matematicheskii Zhurnal 43(12), 1684–1686 (1991)

    MathSciNet  Google Scholar 

  15. Neumann, B.H.: A two-generator group isomorphic to a proper factor group. J. Lond. Math. Soc. s1-25(4), 247–248 (1950)

    Article  MathSciNet  Google Scholar 

  16. Neumann, B.H.: An essay on free products of groups with amalgamations. Philos. Trans. R. Soc. Lond. Ser. A. 246, 503–554 (1954)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Math and Computer Science, New York City College of Technology, The City University of New York, Brooklyn, NY, USA

    Marianna C. Bonanome

  2. Department of Mathematics, Borough of Manhattan Community College, The City University of New York, New York, NY, USA

    Margaret H. Dean

  3. Department of Mathematics, Monroe Community College, Rochester, NY, USA

    Judith Putnam Dean

Authors
  1. Marianna C. Bonanome
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Margaret H. Dean
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Judith Putnam Dean
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bonanome, M.C., Dean, M.H., Putnam Dean, J. (2018). Baumslag–Solitar Groups. In: A Sampling of Remarkable Groups. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01978-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation