Length function

In mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.

Definition

Let $G$ be a group. A length function on $G$ is a function $L\colon G\to \mathbb {R} ^{+}$ satisfying:

{\begin{aligned}L(e)&=0,\\L(g^{-1})&=L(g)\\L(g_{1}g_{2})&\leq L(g_{1})+L(g_{2}),\quad \forall g_{1},g_{2}\in G.\end{aligned}}

Compare with the axioms for a metric and a filtered algebra.

Word metric

An important example of a length is the word metric: given a presentation of a group by generators and relations, the length of an element is the length of the shortest word expressing it.

Coxeter groups (including the symmetric group) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1).

A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).

Properties

A group with a length function does not form a filtered group, meaning that the sublevel sets $S_{i}:=\{g\mid \ell (g)\leq i\}$ do not form subgroups in general.

However, the group algebra of a group with a length functions forms a filtered algebra: the axiom $\ell (gh)\leq \ell (g)+\ell (h)$ corresponds to the filtration axiom.

Length function at PlanetMath.