Bass–Guivarc’h dimension and nilpotent Hirsch length

For a finitely generated nilpotent group $G$ of class $c$, let $r_i$ be the number of infinite cyclic summands in ${\gamma }_i(G)\mathrm{/}{\gamma }_{i+1}(G)$. These factors are finitely generated abelian by Finite generation of lower-central factors, and the number is intrinsic by Integer abelian structure and rank by finite reduction. Equivalently $r_i={\dim }_{\mathbb{Q}}\mathrm{Hom}({\gamma }_i\mathrm{/}{\gamma }_{i+1},(\mathbb{Q},+))$.

Define the Bass–Guivarc'h dimension and the nilpotent Hirsch length by $D(G)={\sum }_{i=1}^{c}i{r}_i,h(G)={\sum }_{i=1}^c{r}_i\mathrm{.}$ For $G=1$, use $c=0$ and empty sums equal to zero. Inserting trailing trivial factors does not change either sum. These definitions concern nilpotent groups only.

Druţu–Kapovich, Geometric Group Theory (837-page edition), Definition 13.46, printed p.474. Revised Definition 13.46 supplies the weighted and unweighted sums; the local integer lemma supplies well-defined ranks.