= Cobounded orbit map lemma
A <cobounded group action> by isometries on a nonempty <geodesic metric space> admits an <orbit map> that is a <quasi-isometry> for a possibly infinite <generating set of a group>. If $Gx$ is $R$-dense, take $S=\{g\ne1:d(x,gx)\leq2R+1\}$. Subdivide a <metric geodesic> into segments of length at most one and approximate subdivision vertices by orbit points, keeping the endpoints exact. The resulting group increments generate and give
$$
|g|_S\leq d(x,gx)+1,\qquad d(x,gx)\leq(2R+1)|g|_S.
$$
The first estimate includes zero displacement: a nontrivial stabilizer element has length one. No finiteness assertion about $S$ survives without properness.
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