Recall that our goal is to determine, for a given map of graphs of spaces such as the one shown below, whether the map \Phi can be extended to a covering map \widehat{\Phi}.

Figure 1

Figure 1

Let \mathcal{X}, \mathcal{X}' be graphs of spaces equipped with maps \Xi' \to \Xi, \phi_{v'} : X_{v'} \to X_v and \phi_{e'} : X_{e'} \to X_e as before.  Recall that in Stallings’ proof of Hall’s Theorem, we completed an immersion to a covering map by gluing in edges.  We will aim to do the same thing with graphs of spaces.

Definition: Let \mathcal{X} be a graph of spaces, and let \eta : X_{\mathcal{X}} \to \Xi be the map to the underlying graph.  If \Delta \subseteq \Xi is a subgraph, then \eta^{-1}(\Delta) \subseteq X_{\mathcal{X}} has a graph-of-spaces structure \mathcal{Y} with underlying graph \Delta.  Call \mathcal{Y} a subgraph of spaces of \mathcal{X}.

We’re seeking a condition on \mathcal{X}' such that \mathcal{X}' is realized as a subgraph of spaces of some \widehat{\mathcal{X}} with a covering map \widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}} such that the following diagram commutes:

diagram1Definition: For each edge map \partial_e^{\pm} : X_e \to X_v of \mathcal{X}, and each v' \mapsto v a vertex of \mathcal{X}', let

\mathcal{E}^{\pm}(e) = \bigcup_{v' \mapsto v} \mathcal{E}^{\pm}(e, v').

For each possible degree \mathcal{D}, let \mathcal{E}_{\mathcal{D}}^{\pm}(e) \subseteq \mathcal{E}^{\pm}(e) be the set of elevations of degree \mathcal{D}.  We will say \mathcal{X}' satisfies Stallings’ condition if and only if the following two things hold:

(a) Every edge map of \mathcal{X}' is an elevation of the appropriate edge map of \mathcal{X}.
(b) For each e \in E(\Xi) and \mathcal{D}, there is a bijection \mathcal{E}_{\mathcal{D}}^+(e) \leftrightarrow \mathcal{E}_{\mathcal{D}}^-(e).

So in Figure 1, the graph of spaces \mathcal{X}' is something you might be able to turn into a covering.  In the picture, \mathcal{E}_{\mathcal{D}}^+(e) is represented by the blue circles, and \mathcal{E}_{\mathcal{D}}^-(e) is represented by the green circles.  Observe that the blue circles are in bijection with the green circles.

Corollary: \mathcal{X}' satisfies Stallings’ condition if and only if \mathcal{X}' can be realized as a subgraph of spaces of some \widehat{\mathcal{X}} such that

(a) V(\Xi ') = V(\widehat{\Xi}), and
(b) there is a covering map \widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}} such that the following diagram commutes:

diagram1

Proof of Corollary. First we’ll show that if \Phi can be extended to a covering map as described above, then \mathcal{X}' satisfies Stallings’ condition.  By Theorem 17, every edge map of \widehat{\mathcal{X}} is an elevation.  So there are inclusions

diagram2

Furthermore, these maps are surjective, and clearly degree-preserving.

Now assume that \mathcal{X}' satisfies Stallings’ condition.  Then we build \widehat{\mathcal{X}} as follows.  Let V(\widehat{\Xi}) = V(\Xi ').  As above, we have degree-preserving inclusions (this time, not surjections)

diagram3

Extend these inclusions to bijections \mathcal{E}_{\mathcal{D}}^+(e) \leftrightarrow \mathcal{E}_{\mathcal{D}}^-(e).  Now we set E(\widehat{\Xi}) = \bigcup_{e \in E(\Xi)} \mathcal{E}^+(e).  Each of these \widehat{e} is an elevation

diagram4

This defines an edge space X_{\widehat{e}} and an edge map \partial_{\widehat{e}}^+.  Consider the corresponding elevation in \bigcup_{e \in E(\Xi)} \mathcal{E}^-(e):

diagram5

Because \partial_{\widehat{e}}^+ and \partial_{\widehat{e}}^- are of the same degree, we have a covering transformation X_{\widehat{e}} ' \to X_{\widehat{e}}.  So we can identify them, and use \partial_e^- as the other edge map.  By construction, \widehat{\mathcal{X}} satisfies the conditions of Theorem 17, so there is a suitable covering map \widehat{\Phi} : X_{\widehat{\mathcal{X}}} \to X_{\mathcal{X}}.

Exercise 25: (This will be easier later, but we have the tools necessary to do this now.)  Prove that if G_1 and G_2 are LERF groups, then so is G_1 * G_2.

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