Residual finiteness for graph manifold groups Is there a simple proof that 3-dimensional graph manifolds 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. 
 A: As far as I'm aware, every proof of this fact is essentially the same as Hempel's original proof.  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 (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, 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.
