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Here is an argument that "most" points in the $SL(2, {\mathbb C})$-character variety $X(F)$ of the surface $F$ do not correspond to representations extendible to 3-manifold groups (as in the question) …

Suppose that $M$ is a compact irreducible 3-manifold. Assume that $M$ is neither a Nil nor a Sol-manifold. Then $G=\pi_1(M)$ is automatic, which implies that $G$ has quadratic Dehn function and the w …

This is an answer to questions 7 and 8 (I have to say, having 8 questions in one post is way too much for my taste): Suppose that $M$ is a finite-volume quotient of $H^3$ or a compact quotient of $H^ …

It all depends on your definition of a "geometric manifold." One definition would require the existence of a complete finite volume locally homogeneous Riemannian metric (from Thurston's list of eigh …

The question stems from a misinterpretation of Theorem 1.1 in the paper by Boileau and Zieschang. Theorem 1.1 excludes a fair number of cases, in particular, it does not apply to (totally oriented) c …

The situation is similar to the one of closed manifolds. One defines "boundary-prime" manifolds as those that cannot be decomposed nontrivially in a boundary-connected sum. Note that if $M$ is connect …

Thurston never finished his project, hence, we cannot know for sure what exactly did he have in mind in this part of the diagram. Here is what we know: Fundamental groups of good compact 3-dimensiona …

Ok, to convert my comment to an answer. Let $S$ be a closed orientable triangulated surface of genus $\ge 1$. Let $M$ be the cone over $S$. Then $M$ has a natural orientable pseudomanifold structure. …