The quotient graph of groups attached to a tree action

Let a group $G$ act without inversions on a simplicial tree $T$, and let $X=G\mathrm{\backslash }T$ be the quotient graph from The quotient graph of an action without inversions. Choose one lift $\tilde{v}$ of each quotient vertex $v$. For each geometric quotient edge, choose one orientation $e$ and a lift $\tilde{e}$ with origin $\widetilde{o(e)}$, and choose $g_e\in G$ satisfying $t(\tilde{e})=g_e\widetilde{t(e)}$. For the opposite orientation set $\widetilde{\stackrel{ˉ}{e}}:=g_e^{-1}\overline{\stackrel{~}{e}},g_{\bar{e}}:=g_e^{-1}\mathrm{.}$ Then $o(\widetilde{\stackrel{ˉ}{e}})=\widetilde{t(e)}$ and $t(\widetilde{\stackrel{ˉ}{e}})=g_{\bar{e}}\widetilde{o(e)}$, so both orientations start at the chosen lift of their origin.

The resulting quotient graph of groups has:

• vertex group $G_v={\mathrm{Stab}}_G(\tilde{v})$,
• for each chosen orientation $e$, edge group $G_e=G_{\bar{e}}:={\mathrm{Stab}}_G(\tilde{e})$,
• boundary map ${\alpha }_e:G_e\hookrightarrow G_{o(e)}$ by inclusion,
• boundary map ${\alpha }_{\bar{e}}:G_e\to G_{t(e)}$ given by $h\mapsto g_e^{-1}h{g}_e$.

Different choices of lifts change this data by conjugation.