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 .

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