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Let $(M,g)$ be a closed orientable connected riemannian $3$-manifold (closed means compact and without boundary).

Is the following affirmation true?

Afirmação: If $(M,g)$ is flat, then there is a closed orientable connected surface $\Sigma$ embedded in $M$ such that $\Sigma$ has positive genus and the induced homomorphism in the fundamental groups by inclusion $i:\Sigma\rightarrow M$ is injective.

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  • $\begingroup$ Thurston's geometrization tells you that $M$ should be mapping torus of finite automorphism group of $T^2$. So you can always find an incompressible torus there which has positive genus $\endgroup$ – Anubhav Mukherjee Nov 6 '18 at 18:09
  • $\begingroup$ @AnubhavMukherjee Can you point me any reference? $\endgroup$ – Michael Nov 7 '18 at 12:26

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