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A Sampling of Remarkable Groups pp 1–36Cite as

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Preliminaries

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Abstract

We outline basic group theory terms and concepts while providing greater depth for the fundamentals of group presentations, free groups, and trees. Groundwork is set for topics for further exploration, such as dead-end elements, revisited in the exploration of groups in the later chapters.

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References

  1. Bencsath, K., Bonanome, M., Dean, M., Zyman, M.: Lectures on Finitely Generated Solvable Groups. Springer Briefs. Birkhauser, Basel (2013)

    Book  Google Scholar 

  2. Bogopol’skii, O.V.: Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent. Algebra Logic 36(3), 155–163 (1997)

    Article  MathSciNet  Google Scholar 

  3. Dehn, M.: Uber unendliche diskontinuierliche Gruppen. Math. Ann. 71 (1), 116–144 (1911)

    Article  MathSciNet  Google Scholar 

  4. Grigorchuk, R.I.: Degrees of growth of finitely generated groups and the theory of invariant means. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 48(5), 939–985 (1984)

    MathSciNet  Google Scholar 

  5. Grigorchuk, R.I., Šunić, Z.: Self-similarity and branching in group theory. London Math. Soc. Lecture Note Ser. 339, 36–95 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Gromov, M.: With an appendix by J. Tits. Groups of polynomial growth and expanding maps. Inst. HautesÉtudes Sci. Publ. Math. 53, 53–73 (1981). MR 623534

    Article  MathSciNet  Google Scholar 

  7. Gromov, M.: Hyperbolic Groups, Essays in Group Theory (S.M. Gersten, ed.). MSRI Publications, vol. 8. Springer, Berlin (1987)

    Google Scholar 

  8. Lehnert, J.: Some remarks on depth of dead ends in group. Int. J. Algebra Comput. 19(4), 585–594 (2009)

    Article  MathSciNet  Google Scholar 

  9. Lim, C.H.: Growth rate of groups (2009). http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Lim.pdf

  10. Lyndon, R., Schupp, P.E. : Combinatorial Group Theory. Springer, Berlin (1977). MR 0577064; reprinted in ‘Classics in Mathematics’ series, Springer-Verlag, Berlin, (2001). MR 1812024

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Magnus, W., Karass, A., & Solitar, D.: Combinatorial Group Theory, Presentations of Groups in Terms of Generators and Relations, 2nd edn. Dover Publications, New York (2014)

    Google Scholar 

  13. Miller, C. F.: On Group-Theoretic Decision Problems and Their Classification. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  14. Milnor, J.: Growth of finitely generated solvable groups. J. Differ. Geom. 2, 447–449 (1968). MR 0244899 (39 6212)

    Article  MathSciNet  Google Scholar 

  15. Nielsen, J.: Om regning med ikke-kommutative faktorer og dens anvendelse i gruppeteorien (in Danish). Math. Tidsskrift B 78–94 (1921)

    Google Scholar 

  16. Olshanskii, A.J.: On a geometric method in the combinatorial group theory. Proc. I.C.M. Warsaw, 1, 415–424 (1984)

    Google Scholar 

  17. Riley, T., Warshall, A.D.: The unbounded dead-end depth property is not a group invariant. Int. J. Alg. Comput. 16(5), 969–984 (2006)

    Article  MathSciNet  Google Scholar 

  18. Schreier, O.: Die Untergruppen der freien Gruppe. Habilitation Thesis, Hamburg University (1926)

    MATH  Google Scholar 

  19. Scott, N.: Growth of finitely generated groups, University of Melbourne, Honours Thesis, supervised by Dr. Lawrence Reeves. http://www.ms.unimelb.edu.au/publications/ScottNick.pdf

  20. Šunić, Z.: Frobenius problems and dead ends in integers. J. Number Theory 128(5), 1211–1223 (2008)

    Article  MathSciNet  Google Scholar 

  21. Tao, T.: Some notes on amenability. https://terrytao.wordpress.com/2009/04/14/some-notes-on-amenability/

  22. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  Google Scholar 

  23. von Neumann, J.: Zur allgemeinen Theorie des Masses. Fund. Math. 13, 73–116 (1929)

    Article  Google Scholar 

  24. Warshall, A.: A group with deep pockets for all finite generating sets. Israel J. Math. 185, 317–342 (2011)

    Article  MathSciNet  Google Scholar 

  25. Wolf, J.A.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differ. Geom. 2, 421–446 (1968). MR 0248688 (401939)

    Article  MathSciNet  Google Scholar 

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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
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  2. Margaret H. Dean
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  3. Judith Putnam Dean
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Bonanome, M.C., Dean, M.H., Putnam Dean, J. (2018). Preliminaries. In: A Sampling of Remarkable Groups. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01978-5_1

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