28. Separability properties of amalgams

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

fig_28_11

commutes.

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 that fig_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 diagram fig_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 diagram fig_28_81