Rational and transcendental growth series for the higher Heisenberg groups

Title data

Stoll, Michael:
Rational and transcendental growth series for the higher Heisenberg groups.
In: Inventiones Mathematicae. Vol. 126 (1996) Issue 1 . - pp. 85-109.
ISSN 1432-1297
DOI: https://doi.org/10.1007/s002220050090

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Abstract in another language

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 H n ×ℤm , where H_n 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.