Definition 2

Let G be a group. A real filtration $(G_\alpha)$ on G is called central if $G = \bigcup_{\alpha > 0} G_\alpha$ and the commutator $(x, y) = x^{-1}y^{-1}xy$ of an element x of $G_\alpha$ and an element y of $G_\beta$ belongs to $G_{\alpha + \beta}$.

In terms of the order function v, the above definition translates into the relations

We deduce that $v((x, y)) > v(x)$ if $v(x) \neq +\infty$; if we write $x^y = y^{-1}xy$ (cf. Algebra, Chapter I, § 6, no. 2), then $x^y = x.(x, y)$, whence This relation expresses the fact that each of the subgroups $G_\alpha$ of G is normal. The $G_\alpha$ form a fundamental system of neighbourhoods of e for a topology compatible with the group structure on G (General Topology, Chapter III, § 1, no. 2, Example) said to be defined by the filtration $(G_\alpha)$.

In the rest of this no., G denotes a group with a central filtration $(G_\alpha)$. For all $\alpha \in \mathbf{R}$, we define the subgroup $G_\alpha^+$ of G by

In particular $G_\alpha^+ = G_\alpha = G$ for $\alpha \leq 0$. Recall that if A and B are two subgroups of G, (A, B) denotes the subgroup of G generated by the commutators $(a, b)$ with $a \in A$ and $b \in B$. With this notation we have the formulae By (14), $G_\alpha^+$ is a normal subgroup of $G_\alpha$ for all $\alpha \in \mathbf{R}$ and the quotient group $\mathrm{gr}_\alpha(G) = G_\alpha / G_\alpha^+$ is commutative. We write $\mathrm{gr}(G) = \bigoplus_{\alpha \in \mathbf{R}} \mathrm{gr}_\alpha(G)$ and give this group the graduation of type $\mathbf{R}$ in which $\mathrm{gr}_\alpha(G)$ consists of elements of degree $\alpha$. Then $\mathrm{gr}_\alpha(G) = \{0\}$ for $\alpha \leq 0$.