Bass-Serre structure theorem

Choose the lift data from [L1] successively along the maximal subtree $T_0$ so that every tree edge has connecting element $g_e=1$. Send each chosen vertex stabilizer element to itself in $G$ and each oriented edge symbol to the corresponding $g_e$. The identity $g_{\bar{e}}=g_e^{-1}$ from [L1] respects the reversal relation, and the definition of the boundary map respects the conjugacy relation of [L2], so this defines a homomorphism from the path group to $G$. Because every tree edge maps to $1$, [L3] yields a homomorphism $\mathrm{\Phi }:{\pi }_1(\mathcal{G},T_0)⟶G\mathrm{.}$

Fix a chosen lift ${\tilde{v}}_0$ of some quotient vertex $v_0$. Let $g\in G$. The unique path in the tree $T$ from ${\tilde{v}}_0$ to $g{\tilde{v}}_0$ projects to an edge path $e_1,\dots ,e_n$ in the quotient graph. Inductively along that path, choose stabilizer elements $h_0,\dots ,h_n$ at the intermediate chosen lifts so that the $j$th lifted edge is $\mathrm{\Phi }(h_0{e}_1{h}_1\cdots e_j){\tilde{e}}_j$ and its terminal vertex is $\mathrm{\Phi }(h_0{e}_1{h}_1\cdots e_j{h}_j)\widetilde{t(e_j)}$. At the end one obtains $g=\mathrm{\Phi }(h_0{e}_1{h}_1\cdots e_n{h}_n)$. Hence $\mathrm{\Phi }$ is surjective.

Let $\tilde{X}$ be the Bass-Serre tree of $(\mathcal{G},T_0)$. Using [L4], define $\mathrm{\Psi }(\gamma G_v):=\mathrm{\Phi }(\gamma )\tilde{v},\mathrm{\Psi }(\gamma G_e):=\mathrm{\Phi }(\gamma )\tilde{e}\mathrm{.}$ This is well defined because the vertex and edge groups in [L1] are actual stabilizer subgroups of the chosen lifts, and the incidence formulas of [L4] match the connecting elements $g_e$ by construction.

The map $\mathrm{\Psi }$ is $\mathrm{\Phi }$-equivariant by definition. At a chosen vertex coset $G_v$, the incident edges of $\tilde{X}$ above an oriented quotient edge $e$ are the cosets $h{G}_e$ with $h\in G_v$, and $\mathrm{\Psi }$ sends them bijectively to the incident edges $h\tilde{e}$ at the chosen lift $\tilde{v}$. By equivariance the same holds at every vertex. Since [L6] says $\tilde{X}$ is a tree and step 2.1 shows that every translate of every chosen lift lies in the image, $\mathrm{\Psi }$ is a covering map from a connected tree onto the tree $T$, hence an isomorphism of graphs.

Let $\gamma \in {\pi }_1(\mathcal{G},T_0)$ satisfy $\mathrm{\Phi }(\gamma )=1$. By [L5], choose a reduced closed graph-of-groups word representing $\gamma$. Under the map $\mathrm{\Psi }$ of step 2.2, its closed edge path begins and ends over the same quotient vertex, and the endpoint is the $\mathrm{\Phi }(\gamma )$-translate of the starting point. Thus it traces a closed path in $T$. If the word had positive edge length, this would be a nontrivial reduced closed path in the tree $T$, impossible. Therefore the reduced representative has edge length $0$, so by the length-$0$ clause of [L5] it is just one coefficient from a chosen vertex stabilizer. But $\mathrm{\Phi }$ restricts on each vertex group to the actual inclusion into $G$, so $\mathrm{\Phi }(\gamma )=1$ forces that coefficient to be the identity. Hence $\gamma =1$, and $\mathrm{\Phi }$ is injective.

Steps 2.1 and 3.2 show that $\mathrm{\Phi }$ is an isomorphism, and step 3.1 gives the $G$-equivariant identification of the Bass-Serre tree with the original tree. This proves both claims.