Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi\_1(M)$ is residually finite. The outline of the proof is basically:


1. Reduce to the case $M$ is closed and irreducible;
2. Use the JSJ decomposition to split $M$ along tori, getting pieces which are either 


	* Seifert fibered spaces or
	* hyperbolic 3-manifolds, via Thurston's hyperbolization theorem.
3. Realize $\pi\_1(M)$ as the fundamental group of a graph of groups, where the vertex groups come from the Seifert fibered spaces and hyperbolic 3-manifolds, and the edge groups come from the tori we split along.
4. Show all these vertex groups are essentially linear, and the $\mathbb{Z}\times\mathbb{Z}$ edge groups correspond to maximal unipotent subgroups.
5. Conclude, using the main theorem of the paper, that $\pi\_1(M)$ is residually finite.


I was wondering if, via the geometrization conjecture, this basic outline now works for **any** compact 3-manifold $M$. I guess my question is really two (related) questions:



> 
> Does Hempel's argument essentially work in the general case? Are there significant changes necessary?
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and



> 
> If this is correct, are there different arguments? (I actually really like Hempel's proof, but maybe it's not "in" right now.)
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> 
> 


Thanks, Steve



Residual Finiteness of Fundamental Group of Compact 3-Manifold
For the first question, the answer is yes. Geometrization implies that the only non-Haken manifolds irreducible manifolds are compact hyperbolic manifolds (with no cusps), and there again $\pi\_1$ is residually finite.


For the second part, I'm not aware of other arguments, and I think this is still the standard way to prove this. Geometrization just says that the argument works in all cases.


