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