# Word metric

length metric

A metric on a finitely-generated group $G$, defined as follows. Let $A$ be a finite set of generators for $G$. Let $A ^ {- 1 }$ be the set of inverses of elements in $A$. If $\gamma \in G$ is not the identity element, then the length of $\gamma$ is defined as the minimal number of elements of $A \cup A ^ {- 1 }$, counted with multiplicity, such that $\gamma$ can be written as a product of these elements. The length of the identity element is defined to be zero. The word metric $d _ {A}$ on $G$ with respect to $A$ is then defined by the following formula: for all $\gamma$ and $\gamma ^ \prime$ in $G$, $d _ {A} ( \gamma, \gamma ^ \prime )$ is equal to the length of the product $\gamma ^ {\prime - 1 } \gamma$. The action of $G$ by left translations on the metric space $( G,d _ {A} )$ is an action by isometries. If $A$ and $B$ are two finite generating sets for $G$, then the identity mapping between the metric spaces $( G,d _ {A} )$ and $( G,d _ {B} )$ is a quasi-isometry.

An equivalent definition is the following: $d _ {A}$ is the maximal metric on $G$ that is invariant by the left-action of $G$ on itself, and such that the distance of any element of $A$ or $A ^ {- 1 }$ to the identity element of $G$ is equal to $1$.

The notion of word metric lies at the foundation of geometric group theory. A group $G$( equipped with a finite generating set $A$) can be canonically imbedded, as the set of vertices, in the associated Cayley graph, which is a simplicial graph. This graph has a canonical metric, and the metric induced on the vertices is the word metric.

The word metric on a group has much to do with the growth function of a finitely-generated group (cf. also Polynomial and exponential growth in groups and algebras; [a1], [a2]; see also [a3], especially Sect. 37, for other and related techniques in the study of groups).

Using the word metric (or the length of words), one defines

$$( x \cdot y ) = { \frac{1}{2} } ( \left | x \right | + \left | y \right | - \left | {x ^ {- 1 } y } \right | ) ,$$

where $| x |$ is the length of the element $x \in G$.

A group $G$ is hyperbolic (cf. also Hyperbolic group) if there is a constant $\delta \geq 0$ such that for all $x,y,z \in G$,

$$( x \cdot y ) \geq \min \{ ( x \cdot z ) , ( y \cdot z ) \} - \delta$$

(cf. also [a1], [a4]). Hyperbolic groups are always finitely presented (cf. also Finitely-presented group), and as such realizable as the fundamental group of a smooth bounded region $M$. Hyperbolicity is then equivalent to the purely geometric property that there is a constant $c$ such that for every smooth closed curve $C$ in $M$, contractible in $M$ and bounding a disc $D$, one has

$${ \mathop{\rm area} } ( D ) \leq c { \mathop{\rm length} } ( C ) .$$

This gives (further) geometric methods for studying hyperbolic groups.

#### References

 [a1] V.A. Ufnarovskii, "Combinatorial and asymptotic methods in algebra" A.I. Kostrikin (ed.) I.R. Shafarevich (ed.) , Algebra , VI , Springer (1995) (In Russian) [a2] R. Grigorchuk, T. Nagnibeda, "Operator growth functions of discrete groups" Invent. Math. (to appear) [a3] A.Yu. Ol'shanskii, "Geometry of defining relations in groups" , Kluwer Acad. Publ. (1991) (In Russian) [a4] M. Gromov, "Hyperboloic groups" , Essays in Group Theory , Springer (1987) pp. 75–263
How to Cite This Entry:
Word metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Word_metric&oldid=49235
This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article