 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 $S=\{g\ne 1:d(x,gx)\le 2R+1\}$. 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 giveThe first estimate includes zero displacement: a nontrivial stabilizer element has length one. No finiteness assertion about $S$ survives without properness.