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ADDED (29 May, 2013) As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaki …

One can also see it using the theory of ends. If $\pi_1M$ were freely decomposable, then it would follow from the easy direction of Stallings' Ends Theorem that $\pi_1M$ had two or infinitely many en …

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it' …

I would motivate the simple loop conjecture as follows. (I'm fairly idiosyncratic about this; I fear I'm going to turn off many 3-manifold topologists.) As well as understanding spaces, we want to u …

Apologies for the shameless self-promotion, but as you ask for necessary conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'. Aschenbrenner, …

It does hold - Wise proved that finite-volume non-compact hyperbolic 3-manifolds are virtually special, hence virtually RFRS and so virtually fibred by one of Agol's results. Details and references a …

Agol's original paper on RFRS gives a nice short proof that the fundamental group of any manifold which also happens to be a finite-index subgroup of your favourite right-angled reflection group is RF …

Theorem(Long and Niblo): If $M$ is a 3-manifold and $S$ is an incompressible component of $\partial M$ then $\pi_1 S$ is separable in $\pi_1 M$ (pick a base point in $S$ to make sense of this). This …

This example is neither particularly easy nor explicit, but it is at least a definite family of examples. It follows from a paper of Bergeron--Haglund--Wise and work of Agol that any 'standard' arith …

Gabai proved that homotopy hyperbolic 3-manifolds are virtually hyperbolic, in the paper of that name: Gabai, David, Homotopy hyperbolic 3-manifolds are virtually hyperbolic. J. Amer. Math. Soc. 7 (1 …

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 …

The answer is 'yes'. Furthermore, this is more or less equivalent to a group-theoretic fact, which applies in much greater geneality, called Shenitzer's Lemma. First, note that we may assume that $X …

The trouble is that computing the outer automorphism group is a very similar to the isomorphism problem, but actually a little harder. The only algorithm that I know of to compute the outer automorph …

I think about it like this. For convenience, I'll assume $M$ is closed. Given a homomorphism $\phi:\pi_1M\to\mathbb{Z}$, Stallings explains how to find an essential surface $S\subset M$ with $\pi_1S\ …

This doesn't actually answer the question, but concerns Tim's comment:- "The core of the question - I think! - is whether group theory, plus a bit of extra topological input, recognizes the geometric …