TITLE: Residual Finiteness of Fundamental Groups of Seifert Fibered Spaces
QUESTION [10 upvotes]: I'm trying to understand why, if $S$ is a Seifert fibered space, then $\pi_1(S)$ is residually finite.  From theorems 12.2 and 11.10 in Hempel's "3-manifolds", we can work with a finite-sheeted covering $M$ such that $M$ is an $S^1$-bundle over a closed surface $T$.  According to a 1987 paper by Hempel, there exists a finite-sheeted covering $\widehat{M}$ of $M$, such that $\widehat{M}$ is of the type $S^1\times F$, $F$ a closed surface.  Of course, the result follows from this, but my question is:
Why does $\widehat{M}$, with such a homeomorphism type, exist?
I'm not actually sure if the existence of $\widehat{M}$ is necessary to deduce the r.f. of $\pi_1(M)$, but I don't think knowing only that it fits into the sequence $1\rightarrow\mathbb{Z}\rightarrow\pi_1(M)\rightarrow\pi_1(T)\rightarrow 1$ is enough.

REPLY [6 votes]: The proof of residual finiteness of the central cyclic extensions of surface groups is not hard: It suffices to consider the extension group with the presentation
$G=<a_1,b_1,...,a_g, b_g, t| [a_1,b_1]...[a_g,b_g]=t, [a_i,t]=1, [b_i,t]=1>$.
Now, kill all $a_i, b_i$ except for $a_1, b_1$. The quotient is the integer 
Heisenberg group 
$H=<a,b, t | [a,b]=t>$. 
The latter group is isomorphic to the group of integer 3x3 upper triangular matrices with 1's on the diagonal. We thus get a homomorphism
$h: G\to H$ that is injective on the subgroup $<t>$. We also have the 
epimorphism $f: G\to F$, where $F$ is the surface group  which kills the generator $t$. Now, take the product homomorphism $f\times h: G\to F\times H$ which is clearly injective. Since both $F$ and $H$ are residually finite, it follows that $G$ is residually finite as well. This argument also shows that $G$ is linear. It is an open problem (due to W.Thurston in 1980's) if the fundamental group of every compact 3-manifold $M$ is linear. (The hard case is when the JSJ-decomposition of $M$ is nontrivial, otherwise it follows from Perelman.)