Structure Theorem
Let Γ be the fundamental group corresponding to the spanning tree T. For every vertex x and edge y, Gx and Gy can be identified with their images in Γ. It is possible to define a graph with vertices and edges the disjoint union of all coset spaces Γ/Gx and Γ/Gy respectively. This graph is a tree, called the universal covering tree, on which Γ acts. It admits the graph Y as fundamental domain. The graph of groups given by the stabiliser subgroups on the fundamental domain corresponds to the original graph of groups.
Read more about this topic: Graph Of Groups
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