Inventiones mathematicae

, Volume 126, Issue 1, pp 85–109 | Cite as

Rational and transcendental growth series for the higher Heisenberg groups

  • Michael Stoll

Abstract.

This paper considers growth series of 2-step nilpotent groups with infinite cyclic derived subgroup. Every such group G has a subgroup of finite index of the form Hn×ℤm, where Hn is the discrete Heisenberg group of length 2n+1. We call n the Heisenberg rank of G.

We show that every group of this type has some finite generating set such that the corresponding growth series is rational. On the other hand, we prove that if G has Heisenberg rank n ≧ 2, then G possesses a finite generating set such that the corresponding growth series is a transcendental power series.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Michael Stoll
    • 1
  1. 1.Mathematisches Institut der Universität, Beringstr. 1, D–53 115 Bonn, Germany;¶:e mail: michael−stoll@ math.uni-bonn.deDE

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