 Solution

The surviving conclusion is a quasi-isometry from a word metric for a possibly infinite generating set of a group. Finite generation cannot be asserted. Choose $x\in X$ and $R\ge 0$ so that $Gx$ is $R$-dense; the cobounded group action hypothesis provides such a choice. PutThis set is symmetric because $d_X(x,g^{-1}x)=d_X(gx,x)$. The metric geodesic subdivision argument in the next part givesIn particular $S$ generates $G$. Apply the same estimates to $g^{-1}h$ to obtain the two-sided bounds for the orbit map; its image is $R$-dense. Thus a cobounded isometric action on a metric geodesic space admits an orbit quasi-isometry for a suitable, possibly infinite, generating set. This is the cobounded orbit map lemma.

Infinite stabilizers cause no obstruction to these bounds: all their nonidentity elements belong to $S$ and have length one, although their orbit displacement is zero. The additive constant permits precisely this collapse. For example, every group acts coboundedly on a point; its complete Cayley graph for $S=G\setminus \{1\}$ is bounded, but the group need not be finitely generated.