In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...

, a length function is a function that assigns a number to each element of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

Definition

A length function ''L'' : ''G'' → R⁺ on a

''G'' is a function satisfying: :

\beginL(e) &= 0,\\
L(g^) &= L(g)\\
L(g_1 g_2) &\leq L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G.
\end

Compare with the

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s for a

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

and a filtered algebra.

Word metric

An important example of a length is the

word metric In group theory, a word metric on a discrete group G is a way to measure distance between any two elements of G . As the name suggests, the word metric is a metric on G , assigning to any two elements g , h of G a distance d(g,h) that me ...

: given a

presentation of a group In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...

by generators and relations, the length of an element is the length of the shortest word expressing it.

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

s (including the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...

) have combinatorially important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: length of a Weyl group element. A

longest element of a Coxeter group In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by ''w''0. Properties * A C ...

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 := \

do not form

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

s 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

{{PlanetMath attribution, id=4365, title=Length function Group theory Geometric group theory