As before, are and are graphs of spaces equipped with maps , , , and such that
Lemma 28: Suppose that every edge map of is an elevation. Then the map is -injective.
Proof: The idea is to add extra vertex spaces to so that satisfies Stalling’s condition. As before, we have inclusions:
If does not satisfy Stalling’s condition then one of these maps is not surjective. Without loss of generality, suppose that
does not arise as an edge map of . Suppose . Then is a subgroup of . Let be the corresponding covering space of . Since contains , there is a lift of . Now replace by and repeat. After infinitely many repetitions, the result satisfies the hypothesis of Stalling’s condition, and is contained in a subgraph of spaces.
Theorem 18: If and are residually finite groups then is also residually finite.
Proof: Let , let be a point, and fix maps . This defines a graph of spaces . By the Seifert-van Kampen Theorem, . Let be the graph of spaces structure on the universal cover of , and let be a compact subset. We may assume that is connected; we may also assume that if (Bass-Serre tree) and , then . Let be the map to the underlying map of the Bass-Serre tree. Then is a finite connected subgraph, . For each , let , a compact subspace of . Because are residually finite, we have a diagram
where is a finite-sheeted covering map and embeds in . Let be defined as follows. Set ; for a vertex , the vertex space is the corresponding to . For each corresponding to , define so that the diagram
commutes. Now sum.
Exercise 26: If and are residually finite and is finite, prove that is residually finite.