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
Best Answer
I believe (though have not checked carefully) that the argument in my paper proves:
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 $M$.