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''length metric''

A [[Metric|metric]] on a [[Finitely-generated group|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|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|Polynomial and exponential growth in groups and algebras]]; [[#References|[a1]]], [[#References|[a2]]]; see also [[#References|[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|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 [[#References|[a1]]], [[#References|[a4]]]). Hyperbolic groups are always finitely presented (cf. also [[Finitely-presented group|Finitely-presented group]]), and as such realizable as the [[Fundamental group|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.A. Ufnarovskii,   "Combinatorial and asymptotic methods in algebra"  A.I. Kostrikin (ed.)  I.R. Shafarevich (ed.) , ''Algebra'' , '''VI''' , Springer  (1995)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Grigorchuk,   T. Nagnibeda,   "Operator growth functions of discrete groups"  ''Invent. Math.''  (to appear)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.Yu. Ol'shanskii,   "Geometry of defining relations in groups" , Kluwer Acad. Publ.  (1991)  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Gromov,   "Hyperboloic groups" , ''Essays in Group Theory'' , Springer  (1987)  pp. 75–263</TD></TR></table>