Word metric

can be expressed as a word whose letters come from a generating set for the group.

The word metric on G is very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G. A generating set for

While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in geometric group theory.

The integer -3 can be expressed as -1-1-1+1-1, a word of length 5 in these generators.

More generally, the distance between two integers m and n in the word metric is equal to |m-n|, because the shortest word representing the difference m-n has length equal to |m-n|.

can be thought of as vectors in the Cartesian plane with integer coefficients.

It can be pictured in the plane as an infinite square grid of city streets, where each horizontal and vertical line with integer coordinates is a street, and each point of

lies at the intersection of a horizontal and a vertical street.

Each horizontal segment between two vertices represents the generating vector

can make the trip by many different routes.

But no matter what route is taken, the car must travel at least |1 - (-2)| = 3 horizontal blocks and at least |2 - 4| = 2 vertical blocks, for a total trip distance of at least 3 + 2 = 5.

If the car goes out of its way the trip may be longer, but the minimal distance travelled by the car, equal in value to the word metric between

Let G be a group, let S be a generating set for G, and suppose that S is closed under the inverse operation on G. A word over the set S is just a finite sequence

are elements of S. The integer L is called the length of the word

has length zero, and its evaluation is the identity element of G. Given an element g of G, its word norm |g| with respect to the generating set S is defined to be the shortest length of a word

is equal to g. Given two elements g,h in G, the distance d(g,h) in the word metric with respect to S is defined to be

Equivalently, d(g,h) is the shortest length of a word w over S such that

The proof of the symmetry axiom d(g,h) = d(h,g) for a metric uses the assumption that the generating set S is closed under inverse.

The word metric has an equivalent definition formulated in more geometric terms using the Cayley graph of G with respect to the generating set S. When each edge of the Cayley graph is assigned a metric of length 1, the distance between two group elements g,h in G is equal to the shortest length of a path in the Cayley graph from the vertex g to the vertex h. The word metric on G can also be defined without assuming that the generating set S is closed under inverse.

To do this, first symmetrize S, replacing it by a larger generating set consisting of each

Then define the word metric with respect to S to be the word metric with respect to the symmetrization of S. Suppose that F is the free group on the two element set

This can be visualized in terms of the Cayley graph, where the shortest path between b and a has length 2.

The group G acts on itself by left multiplication: the action of each

In general, the word metric on a group G is not unique, because different symmetric generating sets give different word metrics.

However, finitely generated word metrics are unique up to bilipschitz equivalence: if

are two symmetric, finite generating sets for G with corresponding word metrics

This proof is also easy: any word over S can be converted by substitution into a word over T, expanding the length of the word by a factor of at most K, and similarly for converting words over T into words over S. The bilipschitz equivalence of word metrics implies in turn that the growth rate of a finitely generated group is a well-defined isomorphism invariant of the group, independent of the choice of a finite generating set.

This topic is discussed further in the article on the growth rate of a group.

A principle that generalizes the bilipschitz invariance of word metrics says that any finitely generated word metric on G is quasi-isometric to any proper, geodesic metric space on which G acts, properly discontinuously and cocompactly.

Metric spaces on which G acts in this manner are called model spaces for G. It follows in turn that any quasi-isometrically invariant property satisfied by the word metric of G or by any model space of G is an isomorphism invariant of G. Modern geometric group theory is in large part the study of quasi-isometry invariants.