As before, are \chi' and \chi are graphs of spaces equipped with maps \Phi \colon X_{\chi'} \to X_{\chi}, \varXi' \to \varXi, \phi_{v'} \colon X_{v'} \to X_v, and \phi_{e'} \colon X_{e'} \to X_e such that



Lemma 28: Suppose that every edge map of \chi' is an elevation.  Then the map \Phi is \pi_1-injective.

Proof: The idea is to add extra vertex spaces to \chi' so that \chi' satisfies Stalling’s condition.  As before, we have inclusions:fig_28_2

If \chi' does not satisfy Stalling’s condition then one of these maps is not surjective.  Without loss of generality, suppose thatfig_28_31

does not arise as an edge map of \chi'.  Suppose \partial_{e}^{-} \colon X_e \to X_u.  Then (\partial_{e}^{-} \circ \phi_{e'})_{*}(\pi_1(X_{e'})) is a subgroup of \pi_1(X_u).  Let X_{u'} be the corresponding covering space of X_u.  Since \pi_1(X_{u'}) contains (\partial_{e}^{-} \circ \phi_{e'})_{*}(\pi_1(X_{e'})), there is a lift \partial_{e'} \colon X_{e'} \to X_{u'} of \partial_{e'} \circ \phi_{e'}.  Now replace \chi' by \chi' \cup X_{u'} and repeat.  After infinitely many repetitions, the result \hat{\chi} satisfies the hypothesis of Stalling’s condition, and \chi' is contained in a subgraph of spaces.

Theorem 18: If G_{+} and G_{-} are residually finite groups then G_{+} * G_{-} is also residually finite.

Proof: Let X_{\pm} = K(G_{\pm}, 1), let e = * be a point, and fix maps \partial^{\pm} \colon e \to X_{\pm}.  This defines a graph of spaces \chi.  By the Seifert-van Kampen Theorem, \pi_1(X_{\chi}) \cong G_{+} * G_{-}.  Let \tilde{\chi} be the graph of spaces structure on the universal cover of X_{\chi}, and let \Delta \subseteq X_{\tilde{\chi}} be a compact subset.  We may assume that \Delta is connected; we may also assume that if \tilde{e} \in \tilde{\varXi} (Bass-Serre tree) and \Delta \cap \tilde{e} \ne \emptyset, then \tilde{e} \subseteq \Delta.  Let \eta \colon X_{\tilde{\chi}} \to \tilde{\varXi} be the map to the underlying map of the Bass-Serre tree.  Then \eta(\Delta) is a finite connected subgraph, \varXi'.  For each v' \subseteq v(\varXi'), let \Delta_{v'} = \Delta \cap X_{v'}, a compact subspace of X_{v'}.  Because G_{\pm} are residually finite, we have a diagramfig_28_6

where X_{\hat{v}} \to X_{\pm} is a finite-sheeted covering map and \Delta_{v'} embeds in X_{\hat{v}}.  Let \hat{\chi} be defined as follows.  Set \hat{\varXi} = \varXi'; for a vertex \hat{v} \in \hat{\varXi}, the vertex space is the X_{\hat{v}} corresponding to v'.  For each \hat{e} \in E(\hat{\varXi}) corresponding to e' \in E(\varXi'), define \partial_{\hat{e}}^{\pm} so that the diagramfig_28_81

commutes.  Now sum.

Exercise 26: If G_1 and G_2 are residually finite and H is finite, prove that G_1 *_H G_2 is residually finite.

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