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  "body": "\n## Summary\n\nA graph of groups is a connected graph with a group attached to each vertex and each edge.\nEach edge group comes with injective homomorphisms into the groups of the end vertices of the edge.\nIts fundamental group glues the vertex groups along the edge groups.\nIt is the central object of [[Bass–Serre theory|6M731X66RD]]: groups acting on trees are exactly fundamental groups of graphs of groups.\nThe simplest cases give the amalgamated free product and the [[HNN extension|YEYERRY37K]].\n\n## Explanations\n\n### L1 human\n\nThink of each vertex group as a building block, and each edge as an instruction to glue two blocks along a common piece, the edge group.\nWith two vertices and one edge one gets an amalgam $A *_C B$: two groups glued along a shared subgroup.\nWith one vertex and a loop one glues a group to itself, which gives an HNN extension.\nIf all the groups are trivial, the fundamental group is just the fundamental group of the graph, a free group.\n",
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