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We show that, in many situations, a homeomorphism $f$ of a manifold $M$ maybe recovered from the (marked) isomorphism class of a finitely generated groupof homeomorphisms containing $f$. As an application, we relate the notions of{\em critical regularity} and of {\em differentiable rigidity}, give examplesof 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 andKoberda that there exist finitely generated groups of $C^\alpha$diffeomorphisms of a 1-manifold $M$, not embeddable into$\mathrm{Diff}^\beta(M)$ for any $\beta > \alpha > 1$.

Reconstructing maps out of groups