Is there a simple proof that 3-dimensional [graph manifolds](http://en.wikipedia.org/wiki/Graph_manifold) have residually finite fundamental groups? 


By "simple" I mean the proof that does not use any hard 3d topology. I care because I wish to generalize this to higher-dimensional analogs of graph manifolds. 



Residual finiteness for graph manifold groups
As far as I'm aware, every proof of this fact is essentially the same as [Hempel's original proof](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=AUCN&pg7=ALLF&pg8=ET&review_format=html&s4=hempel&s5=residual&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=895623). I don't know whether it's "simple" enough for you! The key point is that the fundamental group G of a Seifert-fibred piece has the following property.


**Property.** There exists an integer K such that for any positive integer n there is a finite-index normal subgroup Gn of G such that any peripheral subgroup P intersects Gn in KnP.


It's not too hard to prove. There's a nice account in [a paper by Emily Hamilton](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=629944&vfpref=html&r=5&mx-pid=1851085) (which generalizes Hempel's result).


The other important fact is that peripheral subgroups in Seifert-fibred manifold groups are separable (ie closed in the profinite topology, for any non-experts out there).


Using these two pieces of information, you can piece together finite quotients of Seifert-fibred pieces into a virtually free quotient of π1 of the graph manifold in which your favourite element doesn't die.


**Note on separability of peripheral subgroups.** Of course, Scott proved that Seifert-fibred manifold groups are LERF. But, by a pretty argument of [Long and Niblo](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=AUCN&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=long&s5=niblo&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq), a subgroup is separable if and only if the double along it is residually finite. In particular, you can deduce peripheral separability from the easier fact that Seifert-fibred manifold groups are residually finite.


