# Reconstructing maps out of groups

## TL;DR

This paper demonstrates how a homeomorphism of a manifold can be reconstructed from the isomorphism class of a finitely generated group of homeomorphisms, linking critical regularity and differentiable rigidity with applications to 1-manifolds.

## Contribution

It introduces methods to recover manifold homeomorphisms from group isomorphism classes and connects concepts of regularity and rigidity, providing new insights and proofs.

## Key findings

- Homeomorphism can be reconstructed from group isomorphism class.
- Existence of groups of diffeomorphisms with strong differential rigidity.
- Finiteness of groups of $C^eta$ diffeomorphisms not embeddable into higher regularity groups.

## Abstract

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 $\beta > \alpha > 1$.