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  "body": "\n## Statement\n\nLet $G$ be a group.\nThen $G$ has [[cohomological dimension|EF4ZWACHD8]] at most $1$ if and only if $G$ is a [[free group|0KTW26C80J]].\n\n## Remarks\n\nStallings proved the theorem for finitely generated groups, and Swan extended it to all groups.\n\nOne direction is direct: if $G$ is free on a set $X$, then the augmentation ideal of the group ring $\\Z[G]$ is a free $\\Z[G]$-module with basis $\\set{x - 1 : x \\in X}$.\nThis gives a free resolution $0 \\to \\bigoplus_{x \\in X} \\Z[G] \\to \\Z[G] \\to \\Z \\to 0$ of length $1$.\nFor the converse with $G$ finitely generated and nontrivial, a group of dimension $1$ is torsion-free and has more than one end.\nBy [[Stallings theorem about ends of groups|GWZN3EKEWD]], it splits as a free product or HNN extension over a finite subgroup, which must be trivial.\nInduction on the number of generators finishes the proof.\n\nExample: every subgroup $H$ of a free group $F$ satisfies $\\mathrm{cd}(H) \\le \\mathrm{cd}(F) \\le 1$, so $H$ is free.\nThis recovers the [[Nielsen–Schreier theorem|W97X7WX5G8]].\n\nExample: $\\mathrm{cd}(\\Z^2) = 2$, so $\\Z^2$ is not free.\n\n## Explanations\n\n### L1 human\n\nFree groups are the groups with no relations among their generators.\nThe theorem says they are also exactly the groups whose cohomology dies above degree one.\nSo an algebraic condition on cohomology detects a combinatorial property of the group.\n\n### L2 human\n\nSerre proved that a torsion-free group has the same cohomological dimension as its subgroups of finite index.\nWith this, the theorem shows that a torsion-free group with a free subgroup of finite index is itself free.\nIn topology, it says that every group of cohomological dimension at most $1$ has a graph as a [[classifying space|WA796ZCNRX]].\n",
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