Skip to main content
SpringerLink
Log in

Growth functions of amalgams

  • Conference paper
Arboreal Group Theory

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 19))

Abstract

Let G be a finitely generated group with a finite generating set S. Then S determines a length function ℓ on G (see §1 below) which is used to define the growth series of G:

$$G(z) = \sum\limits_{g \in G} {{z^{\ell (g)}}} $$

recently there has been a lot of interest in the study of these series.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones, Math. Ann. 255 (1977), 69–76.

    Article  MathSciNet  Google Scholar 

  2. M. Benson, Growth series of finite extensions of Zn are rational, Invent. Math. 73 (1983), 251–269.

    MathSciNet  MATH  Google Scholar 

  3. M. Benson, On the rational growth of virtually nilpotent groups, in “Combinatorial group theory and topology”, S.M. Gersten and J.R. Stallings (eds), Annals of Mathematics Studies 111, 185–196.

    Google Scholar 

  4. N. Bourbaki, “Groupes et algèbres de Lie. Chapitres 4, 5 et 6.,” Hermann, Paris, 1968.

    Google Scholar 

  5. K.S. Brown, “Cohomology of Groups,” Springer-Verlag, New York Heidelberg Berlin, 1982.

    Google Scholar 

  6. J.W. Cannon, The growth of the closed surface groups and the compact hyperbolic Coxeter groups, (preprint).

    Google Scholar 

  7. W.J. Floyd and S.P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), 1–29.

    MathSciNet  MATH  Google Scholar 

  8. W.J. Floyd and S.P. Plotnick, Symmetries of planar growth functions, Invent. Math. 93 (1988), 501–543.

    MathSciNet  MATH  Google Scholar 

  9. M.A. Grayson, Geometry and growth in three dimensions Ph.D. Thesis. Princeton University (1983), 1–119.

    Google Scholar 

  10. J-M. Lemaire, “Algèbres Connexes et Homologie des Espaces de Lacets,” LNM 422 Springer-Verlag, New York Heidelberg Berlin, 1974.

    Google Scholar 

  11. J. Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2 (1968), 1–7.

    MathSciNet  MATH  Google Scholar 

  12. W. Parry, Counterexamples involving growth series and Euler characteristics, Proc. AMS 102 (1988), 49–51.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Scott and C.T.C. Wall, Topological methods in group theory, in “Homological group theory”, C.T.C. Wall (ed), London Math. Soc. L.N.S. 36, 137–203.

    Google Scholar 

  14. J-P. Serre, Cohomologie des groupes discrets, Ann. of Math. Studies 70 (1971), 77–169.

    Google Scholar 

  15. J-P. Serre, “Trees,” Springer-Verlag, New York Heidelberg Berlin, 1980.

    Google Scholar 

  16. N. Smythe, Growth functions and Euler series, Invent. Math. 77 (1984), 517–531.

    MathSciNet  MATH  Google Scholar 

  17. M. Shapiro, A geometric approach to the almost convexity and growth of some nilpotent groups, (preprint).

    Google Scholar 

  18. C. Soulé, The cohomology of SL 3(Z), Topology 17 (1978), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Wagreich, The growth function of a discrete group, Proc. Conference on Algebraic Varieties with Group Actions, LNM 956 (1982), 125–144, Springer-Verlag, New York Heidelberg Berlin.

    Google Scholar 

Download references

Author information

Author notes

  1. Juan M. Alonso

    Present address: Department of Mathematics, University of Stockholm, P.O.Box 6701, S-113 85, Stockholm, Sweden

Authors and Affiliations

  1. Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California, 94720, USA

    Juan M. Alonso

Authors
  1. Juan M. Alonso
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, San Jose State University, San Jose, 95192, California, USA

    Roger C. Alperin

  2. Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA, 94720, USA

    Roger C. Alperin

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag New York, Inc.

About this paper

Cite this paper

Alonso, J.M. (1991). Growth functions of amalgams. In: Alperin, R.C. (eds) Arboreal Group Theory. Mathematical Sciences Research Institute Publications, vol 19. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3142-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-3142-4_1

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7811-5

  • Online ISBN: 978-1-4612-3142-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation