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 .