Let be a finitely generated group. It has a surjection for S finite. Elements of the kernel are called relations and the elements of S are called genereators. Suppose the kernel is generated as a normal subgroup by a subgroup . Then we write to mean . If R is finite, the is said to be finitely presentable.
Let’s develop a topological point of view.
Let be the standard rose for . Each relation corresponds to a (homotopy class of a) map . We construct a 2-complex X gluing on a 2-cell using the map for each relation .
Proof: This is a simple application of the Seifert-van Kampen Theorem.
We call X a presentation complex for , and we deduce that every finitely generated group is of a 2-complex.
Exercise 8: Every finitely presented group is of a closed 4-manifold.
Therefore, acts freely and properly discontinuously on some 2-complex , the universal cover of .
Definition: Let S be a finite generating set for as above. Then is the Cayley graph of with respect to S.
To see that this only depends on S, let’s give the more standard definition. The vertices of are just the elements of . For each generator , two vertices are joined by an edge iff
The group acts by left translation.
- The tree for .
Definition: The Cayley graph induces a natural length metric on , denoted and called the word metric. Note that , the word length of .
The action of on is by isometries.
Given a metric space , a geodesic is just an isometric embedding of a compact interval into .
A metric space is geodesic if any pair of point is joined by a geodesic. Note that is a geodesic metric space.
Definition: Let . A -quasi isometric embedding is a map of metric space such that .
If for some , we also have that for every , there is such that , then is a quasi-isometry.
Exercise 9: Quasi-isometry is an equivalent relation (use the Axiom of Choice).
Exercise 10: Let and be two finite generating sets for . Then is a quasi-isometry.
Example: is a quasi-isometry.