Rational and transcendental growth series for the higher Heisenberg groups
Let \(G\) be a 2-step nilpotent group whose commutator subgroup \([G,G]\) is infinite cyclic. Every such group has a subgroup of finite index of the form \(H_n\times Z^m\), where \[ \begin{multlined} H_n=\langle x_1,\dots,x_n,y_1,\dots,y_n,z\mid[x_i,x_j]=[y_i,y_j]=1,\\ [x_i,y_j]=1\text{ for }i\neq j,\quad[x_i,y_i]=z,\quad[z,x_i]=[z,y_i]=1\rangle\end{multlined} \] is the \(n\)-th discrete Heisenberg group. We call \(n\) the Heisenberg rank of \(G\). In the paper under review the author proves that \(G\) has a finite generating set \(S\) such that \(G\) has rational growth with respect to \(S\). On the other hand, he shows that if \(G\) has Heisenberg rank at least two, then \(G\) has a finite generating set \(S\) such that \(G\) has transcendental growth with respect to \(S\). In particular, all higher discrete Heisenberg groups \(H_2,H_3,\dots\) possess transcendental growth series. Moreover the second Heisenberg group \(H_2\) has transcendental growth series with respect to its standard generating set \(\Sigma_2=\{x^{\pm1}_1,x^{\pm1}_2,y^{\pm1}_1,y^{\pm1}_2\}\).
scientific article; zbMATH DE number 931985