Our goal is to understand the structure of subgroups of graphs of groups. This will enable us to prove things like: F \ast_{z} F is LERF.

Exercise 22. A group is coherent if every fg subgroup is fp. Prove that \pi_1 of a finite graph of groups with coherent vertex groups and cyclic edge group is coherent.

Recall, that if H \subseteq \pi_1 \mathcal{G}=G is fg then H has an induced graph of groups structure. \mathcal{H} =H \backslash T_H \subseteq H \backslash T where T is the Bass-Serre tree of \mathcal{G}.

Topologically, H defines a covering space X^H \rightarrow X_g, which inherits a graph of spaces structure with underlying graph H\backslash T.

\Gamma' =H \backslash T_H \subseteq H \backslash T is a finite connected subgraph and the image of \Gamma' in X^H is a connected subcomplex X' such that \pi_1 X' = \pi_1 X^H =H. X' has the structure of a graph of space with underlying graph \Gamma'

There is an explicit algebraic description of H\backslash T, which follows immediately from Lemmas 16,17,18.

Lemma 23. (a) The vertices of H\backslash T are in bijection with \coprod_{v \in V(\Gamma)} H \backslash G/G_v=\coprod_{v \in V(\Gamma)}\{HgG_v\mid g \in G\}.

The vertex HgG_v is labelled by: H \cap gG_vg^{-1}, well-defined up to conjugation in H.

(b) The edges of H\backslash T are in bijection with \coprod_{e \in E(\Gamma)} H \backslash G/G_e. The vertex HgG_e is labelled with H \cap g G_e g ^{-1}.

(c) The edges of H\backslash T adjoining the vertex corresponding to HgG_v  are in bijection with \coprod_{e~\mathrm{adjoining}~v} (g^{-1} H g \cap G_v ) \backslash G_v/G_e.

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 f:X \rightarrow Y be a continuous map and Y' \rightarrow Y a covering map. If f':X \rightarrow Y' makes the diagram commutes, then f' is a lift of f.

Lemma 24. Fix basepoint x \in X, y=f(x) \in Y, and y' \in Y' mapping to y. There is a lift f':X \rightarrow Y' with f'(x)=y' if and only if f_{\ast} \pi_1(X,x) \subseteq \pi_1(Y',y').  Furthermore, if this lift exists it is unique.

It may be impossible to lift at y'. But it is possible if we pass to a covering space. Eg., if \tilde{X} \rightarrow X is the universal cover.  Intuitively, an elevation is a minimal lift.

Definition: Let X,Y,Y', x,y,y' be as above. A based connected covering space (X',x') \rightarrow (X,x) together with a based map f':(X',x')\to (Y',y') such that

elevation1

commutes is an elevation (of f at y') if whenever the diagram

elevation

commutes and X' \rightarrow \bar{X} is a covering map of degree larger than 1 then the composition (\bar{X},\bar{x}) \rightarrow (X,x) \rightarrow (Y,y) does not lift to Y' at y'.

The unbased covering map X' \rightarrow X or equivalently the conjugacy class of the subgroup \pi_1(X') \subseteq \pi_1(X) is called the degree of the elevation f'.

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