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Our goal is to understand the structure of subgroups of graphs of groups. This will enable us to prove things like: is LERF.
Exercise 22. A group is coherent if every fg subgroup is fp. Prove that of a finite graph of groups with coherent vertex groups and cyclic edge group is coherent.
Recall, that if is fg then H has an induced graph of groups structure. where T is the Bass-Serre tree of .
Topologically, defines a covering space , which inherits a graph of spaces structure with underlying graph .
is a finite connected subgraph and the image of in is a connected subcomplex such that . has the structure of a graph of space with underlying graph
There is an explicit algebraic description of , which follows immediately from Lemmas 16,17,18.
Lemma 23. (a) The vertices of are in bijection with .
The vertex is labelled by: , well-defined up to conjugation in .
(b) The edges of are in bijection with . The vertex is labelled with .
(c) The edges of adjoining the vertex corresponding to are in bijection with .
Now we will try to understand this topologically. In particular, we will understand the edge maps of a covering space of a graph of spaces.
Let be a continuous map and a covering map. If makes the diagram commutes, then is a lift of .
Lemma 24. Fix basepoint , , and mapping to . There is a lift with if and only if . Furthermore, if this lift exists it is unique.
It may be impossible to lift at . But it is possible if we pass to a covering space. Eg., if is the universal cover. Intuitively, an elevation is a minimal lift.
Definition: Let be as above. A based connected covering space together with a based map such that
commutes is an elevation (of at ) if whenever the diagram
commutes and is a covering map of degree larger than 1 then the composition does not lift to at .
The unbased covering map or equivalently the conjugacy class of the subgroup is called the degree of the elevation .
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.