Non-compact 3-manifold with incompressible boundary

Question

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times [0,\infty)$. I know that if $M$ is compact then it is forced to be $\Sigma_2\times I$, is a similar result with $\Sigma_2\times [0,1)$ true?

Answer 1

While this isn't a duplicate of the question, the question is answered by the user studiosus here: A 3-manifold with fundamental group isomorphic to a surface group.

I'll mention that the existence of a compact 3-manifold $S\subset M$ (with boundary) with the inclusion a homotopy equivalence is a consequence of Scott's Core theorem.

G.P. Scott ,"Compact submanifolds of 3-manifolds", J. London Math. Soc. (2) 7 (1973) 246-250

Answer 2

As a counter example was already presented, I'll focus on the last part of your question, and conclude with a (a priori large) class of manifolds which will not be able to serve you with counter examples.

Claim: The result holds if and only if $M$ is tame.

One direction is immediate. For the other direction, let $M$ be tame and hence properly embeds $M\hookrightarrow M'$, where $M'$ is compact and the image of $M$ contains the interior of $M'$. By compactness we have $M'=\Sigma_g\times I$ and hence $M=\Sigma_g\times [0,\infty)$.

In particular, as the tameness conjecture was proven by Agol a little more then 10 year ago, we know that we have:

Theorem: Let $M$ be any hyperbolic 3-manifold with $\partial M =\Sigma_g$ and $\pi_1M=\pi_1\Sigma_g$. Then $M\cong \Sigma_g\times [0,\infty)$.