  We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may
be recovered from the (marked) isomorphism class of a finitely generated group
of homeomorphisms containing $f$. As an application, we relate the notions of
{\em critical regularity} and of {\em differentiable rigidity}, give examples
of groups of diffeomorphisms of 1-manifolds with strong differential rigidity,
and in so doing give an independent, short proof of a recent result of Kim and
Koberda that there exist finitely generated groups of $C^\alpha$
diffeomorphisms of a 1-manifold $M$, not embeddable into
$\mathrm{Diff}^\beta(M)$ for any $\beta > \alpha > 1$.
