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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 …

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 …

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\ …

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 …

This fact doesn’t need $M$ to be hyperbolic. It just needs one general theorem about 3-manifold topology, namely the Scott core theorem. Let $N\to M$ be the covering space corresponding to the subgrou …

The paper of Scott and Swarup mentioned in the question cites Kropholler and Roller's 'Splittings of Poincaré duality groups'. On page 35 they write: 'Let $K$ be a Poincaré duality group of dimen …

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 …

See pp. 449--457 of Peter Scott's article The geometries of 3-manifolds for a complete description of all 3-manifolds with finite fundamental group. The article is available on his website. There don …

The second question is very similar to this question of Igor Belegradek. Hempel's argument is still the only known one. It's the 'obvious' way to prove residual finiteness using geometrisation. To …

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 …

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 …

The Virtual Fibring theorem provides the topological classification of closed 3-manifolds. Very roughly, after passing to finite covers, we can build 3-manifolds by gluing together simpler pieces that …

This is a very nice question, which I don't know how to answer. But I hope the following (shameless self-promotion) will be of interest. You seem to be looking for some sort of local condition that c …