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  "body": "\n## Summary\n\nA one-relator group is a [[group|G5S1M9]] with a [[presentation|0JGEQXQ4AC]] that has a single defining relator.\nExamples are $\\Z^2$, the fundamental groups of closed surfaces, and the [[Baumslag–Solitar groups|429JJ5ZNBY]].\nThey are the natural next class after [[free groups|0KTW26C80J]], and they have a rich general theory.\nMagnus proved the [[Freiheitssatz|B2TRERFWEH]] for them and showed that their [[word problem|PBGTTAVMMT]] is solvable.\n\n## Explanations\n\n### L1 human\n\nTake a free group, where no rule holds between the generators, and impose exactly one rule.\nFor example, $\\langle a, b \\mid a b a^{-1} b^{-1} \\rangle$ imposes $ab = ba$, and the result is the group $\\Z^2$ of pairs of integers.\nOne rule is little, so these groups stay close to free groups.\nStill they include many important groups, such as the fundamental groups of closed surfaces.\n",
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