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  "body": "\n## Summary\n\nA solvmanifold is a homogeneous space $G/H$ of a connected [[solvable Lie group|9CNSVFQ801]] $G$ by a closed subgroup $H$.\nTori, nilmanifolds and many mapping tori of torus maps are solvmanifolds.\nThe compact ones of the form $G/\\Gamma$, with $\\Gamma$ a lattice, are natural test spaces in geometry and dynamics.\nFor example, they give examples of symplectic manifolds without Kähler structure.\n\n## Explanations\n\n### L1 human\n\nA solvable group is one built from commutative groups in finitely many steps.\nA solvmanifold is a space on which such a group acts, so that any point can be moved to any other point.\nThe line is an example: the maps $x \\mapsto ax + b$ already move any point of the line to any other, and their group is solvable.\nThe torus $\\R^n / \\Z^n$ is a compact example, with the group $\\R^n$ acting by translations.\n",
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