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Counterexamples are easily constructed using the Thurston norm. In fact, any example of a fibered, oriented, closed 3-manifold $M$, with a fiber of genus $\ge 2$ and with pseudo-Anosov monodromy, and …

Cannon's Conjecture: Every finitely generated word hyperbolic group with Gromov boundary $S^2$ has a finite normal subgroup whose quotient is the fundamental group of a closed hyperbolic 3-orbifold.

To answer your question about allowing $M$ to have infinite volume, there exist such examples on $M=S \times \mathbb{R}$ itself. These were originally constructed by Bers, his "singly degenerate" grou …

Read Chapter 4 of Thurston's notes http://library.msri.org/books/gt3m/. He produces infinitely many closed hyperbolic 3-manifold of different volumes, and hence non-homeomorphic, just by doing Dehn su …

Points in the Teichmuller geodesic $\sigma$ are given quite concretely in terms of the pseudo-Anosov data, namely the stable and unstable measured foliations, by using the standard method one uses to …

Sutured manifolds and sutured manifold hierarchies were defined for the very purpose of studying surfaces of minimal genus within a homology class. See the original papers of Gabai on this topic, star …

Yes, $Y$ is still irreducible, and this holds by a simple connectivity argument. Suppose that $\Sigma \subset Y \subset X$ is a smoothly embedded 2-sphere. Since $X$ is irreducible, $\Sigma$ bounds …

I thought perhaps I should write up my comments to Agol's answer as a separate answer itself. Proving asphericity of a graph manifold $M$ with $\pi_1$-injective tori can be done from the point of vie …

You are asking for free actions of finite, nonsolvable groups on $Y$. If you truly don't care that $Y$ is a rational homology sphere, for any finite group $G$ there exists a closed, connected, orienta …