= Free isometric realization of a finitely generated group
{title2=$T\cong F_r/\ker\pi$}
Every finitely generated group acts freely and properly discontinuously by <Riemannian isometries> on a connected manifold in any dimension $d\geq3$. Choose a surjection $F_r\to T$ and the <regular covering> of $\#_r(S^1\times S^{d-1})$ associated with its kernel. Pulling back a metric makes the <deck transformation group> isometric. Uniqueness of lifts makes its action free. The cover can be noncompact and need not be <simply connected>; no claim about realizing every group as a closed three-manifold <fundamental group> is involved.
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