Length function

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

A length function L : G → R⁺ on a group G is a function satisfying:^[1]^[2]^[3]

{\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

Properties

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

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

References

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