Lower-central generators, residue coordinates and weighted length

Let $G$ be finitely generated nilpotent with ${\gamma }_{c+1}=1$. Choose, in each finitely generated abelian factor ${\gamma }_i\mathrm{/}{\gamma }_{i+1}$, an ordered cyclic decomposition, and lift its generators to elements $u_{ij}\in {\gamma }_i$. Infinite cyclic factors use exponents $a_{ij}\in \mathbb{Z}$; a finite cyclic factor of order $d_{ij}>1$ uses $a_{ij}\in \{0,\dots ,d_{ij}-1\}$. All factors exist by Finite generation of lower-central factors and Integer abelian structure and rank by finite reduction. Order products by increasing $i$, then increasing $j$. Assign $u_{ij}$ weight $i$.

For a normalized tuple $a$, its integer weighted coordinate length is the least integer $R\ge 0$ such that $\mathrm{\mid }{a}_{ij}\mathrm{\mid }\le R^i$ on every infinite factor and $R\ge 1$ whenever a residue coordinate is nonzero. The zero tuple has length zero. For real $R\ge 1$ let $Q(R)$ consist of all ordered products with $\mathrm{\mid }{a}_{ij}\mathrm{\mid }\le \lfloor R^i\rfloor$ on infinite factors and every allowed finite residue. The number of infinite coordinates in layer $i$ is $r_i$ of Bass–Guivarc’h dimension and nilpotent Hirsch length. Unique parametrization of group elements is justified by Finite lower-central coordinate systems with torsion accounted for ↗.

For any finite labelled alphabet whose weight-$i$ letters represent elements of ${\gamma }_i$, a word has weighted word counts $(N_1,\dots ,N_c)$ if it has $N_i$ letters of assigned weight $i$; inverse letters retain that weight. These are counts before normalization, distinct from coordinate exponents and from the minimal ordinary word length of Word length of a group element with respect to a generating set. Identity letters may be deleted. An lcs generating alphabet means its letters of weights at least $i$ generate ${\gamma }_i$.

Integral coordinates from a central cyclic refinement and mixed lower-central coordinates are different constructions. Even in a torsion-free group, finite cyclic lower-central factors must keep their residue coordinates. For a group with torsion, use $G\mathrm{/}T(G)$ before invoking a torsion-free integral model; keep finite kernel representatives when lifting back.

Druţu–Kapovich, Geometric Group Theory (837-page edition), Definitions 14.18–14.19 and Proposition 14.25, printed pp.504–505,510–511. Revised Definitions 14.18–14.19 provide lcs alphabets and weighted word counts. The mixed tuple conventions match Proposition 14.25.