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  "body": "\n## Summary\n\nBass–Serre theory describes the groups that act on [[trees|062D9KXJB1]] by building them from simpler groups.\nIts central object is a [[graph of groups|DC1CQMD3WK]]: a graph with a group at each vertex and each edge.\nThe structure theorem says that a group acting on a tree without inversions is the fundamental group of the quotient graph of groups.\nThe vertex and edge groups of this graph of groups are stabilisers.\nConversely, the fundamental group of a graph of groups acts on a tree, the Bass–Serre tree.\nIt unifies [[free products|TM7QS7P7X9]], amalgamated free products and [[HNN extensions|YEYERRY37K]].\nIt gives short proofs of subgroup theorems such as the [[Nielsen–Schreier theorem|W97X7WX5G8]] and the [[Kurosh subgroup theorem|RY8K5RQJSR]].\n\n## Explanations\n\n### L1 human\n\nA tree is a graph with no cycles, so there is exactly one path between any two points.\nWhen a group moves a tree around by symmetries, the quotient picture is a graph, and each vertex and edge of it remembers the group of symmetries fixing it.\nBass–Serre theory says one can rebuild the whole group from this decorated graph.\nFor instance, $\\mathrm{SL}_2(\\Z)$ acts on a tree with quotient a single edge.\nThis shows that it is an amalgam of a cyclic group of order $4$ and a cyclic group of order $6$ over a common subgroup of order $2$.\nA subgroup of the group still acts on the same tree, which is why the theory proves so much about subgroups.\n",
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