**Edited:** Due to work of [Raymond and Scott](https://doi.org/10.1007/BF01220468), there exist diffemorphisms (of certain three-dimensional nil-manifolds) whose $n$th power is diffeotopic to the identity, but which are not themselves homotopic to finite order homeomorphisms.


Do they thus provide counterexamples to the claim that that actions by homeomorphisms (on an aspherical three manifold) are conjugate (by a homeomorphism) to isometric actions?



Conjugacy of topological actions on aspherical three manifolds to isometric actions

> 
> Do they thus provide counterexamples to the claim that that actions by
> homeomorphisms (on an aspherical three manifold) are conjugate (by a
> homeomorphism) to isometric actions?
> 
> 
> 


Here is a counterexample which I find a bit simpler.


Suppose that $M$ is a closed, connected, oriented hyperbolic three-manifold. Thus $M$ is aspherical, as desired. Also, $\mathrm{Isom}(M)$ is finite. Let $f$ be a homeomorphism of $M$ which is the identity outside of a small ball $B$ and is not the identity on $B$. Then (difficult exercise) $f$ has infinite order. Now suppose that $g$ is any homeomorphism. Then (easy exercise) $gfg^{-1}$ again has infinite order. Thus $f$ is not conjugate to an isometry.


