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  "body": "\n## Statement\n\nLet $(M, g)$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] with negative [[sectional curvature|1X42JAA349]].\nThen every nontrivial abelian subgroup of the [[fundamental group|S1XK3HXHFX]] $\\pi_1(M)$ is infinite cyclic.\n\n## Remarks\n\nThe idea of the proof: by the [[Cartan–Hadamard theorem|428R02EYWS]], the universal cover $\\tilde M$ is diffeomorphic to $\\R^n$ and $\\pi_1(M)$ acts on it by deck isometries.\nEach nontrivial element has infinite order and, by compactness, moves points along a unique invariant [[geodesic|NSYMQMHPDN]], its axis.\nNegative curvature forces two commuting elements to have the same axis.\nSo an abelian subgroup acts by translations on one geodesic line, and a discrete group of translations of a line is infinite cyclic.\n\nExample: let $M_1$ and $M_2$ be compact manifolds whose fundamental groups contain elements of infinite order, such as $M_1 = M_2 = S^1$.\nThen $\\pi_1(M_1 \\times M_2)$ contains $\\Z^2$, so $M_1 \\times M_2$ carries no metric of negative sectional curvature.\n\nNegative curvature is needed: the flat torus $\\R^2 / \\Z^2$ has curvature $0$ and fundamental group $\\Z^2$.\n\n## Explanations\n\n### L1 human\n\nIn a negatively curved compact space, two loops that commute must go \"in the same direction\": both are powers of a single loop.\nSo the fundamental group of such a space contains no copy of $\\Z^2$, the group of loops on a torus.\n\n### L2 human\n\nPreissmann's theorem is a group-theoretic obstruction to negative curvature.\nIt rules out, for example, every product of two compact manifolds whose fundamental groups have elements of infinite order.\nIt is an early instance of the link between negative curvature and the algebra of the fundamental group that later led to Gromov's hyperbolic groups.\n",
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