It is known that if a finite group $G$ admits a faithful *topological* action on the 3-sphere $S^3$, then $G$ admits a faithful action on $S^3$ by *isometries*. (Pardon proved that a topological action implies a smooth action, and Dinkelbach & Leeb proved that a smooth action implies an isometric one.) I wonder if this extends to infinite groups acting on $R^3$:


**Question:** Let $G$ be a finitely generated group that admits a faithful, co-compact, topological action on $R^3$, such that no orbit has an accumulation point. Must $G$ admit an action by isometries on one of Thurston’s geometries, preserving the above properties (i.e. faithful, co-compact, accumulation-free)?


**Update:** The comments below suggest that the answer is negative in this generality (an "official" answer with references and explanation would be welcome). **What if G is assumed to be Gromov-hyperbolic?** I'm most interested in the 1-ended case, anticipating an isometric action on $\mathbb{H}^3$. (1-endedness excludes $\mathbb{S}^2 \times \mathbb{R}$.) This is partly motivated by Cannon's conjecture.


By topological action I mean an action by homeomorphisms.


**Update:** instead of just assuming that no orbit has an accumulation point, I'm happy with stronger discreteness conditions such as proper discontinuity



From topological actions on $\mathbb{R}^3$ to isometric actions
I believe (though have not checked carefully) that the argument in my paper proves:



> 
> If $\Gamma$ (discrete) acts continuously and properly discontinuously on a smooth three-manifold $M$, then that action can be uniformly approximated by a smooth action.
> 
> 
> 


The point is simply that each step in the argument is local on the quotient space $M/\Gamma$ (which is a reasonable topological space given proper discontinuity).


Here is a (sketched) better argument, which proves the indented statement above as a consequence of my paper. Fix $x\in M$, and consider the stabilizer $\Gamma\_x\leq\Gamma$, which is finite. Choose coset representatives $g\_i\in\Gamma/\Gamma\_x$, so $\Gamma x=\{g\_ix\}\_i$. Fix a $\Gamma\_x$-invariant open neighborhood $U$ of $x$ whose translates $g\_iU$ are all disjoint (should exist by proper discontinuity). Now smooth the action of $\Gamma\_x$ on $U$ using my paper, and smooth the homeomorphisms $g\_i:U\to g\_iU$ using Bing--Moise. This determines a smoothing of the action of $\Gamma$ on $\Gamma U\subseteq M$. By making the approximations sufficiently $C^0$-close, we ensure that this smoothed action of $\Gamma$ on $\Gamma U\subseteq M$ splices together with the original action of $\Gamma$ on $M\setminus\Gamma U$ to define a new action of $\Gamma$ on $M$, which is now smooth over $\Gamma U$. Now iterate a (locally) finite number of times to cover all of $M$.


