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We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus $M=M(\varphi)$ of $S$, where $\varphi$ is some automorphism of $S$. I want to show that if $\varphi$ is periodic or reducible, then $M$ cannot be given any hyperbolic structure, thus $\varphi$ must be a pseudo-Anosov automorphism. I know that also the converse holds, and it is a complicated theorem of Thurston, but I'd like to give a simple proof at least of this easier implication. Could you help with that, or at least give me some hints?

Than you very much! bye

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Well, if $\varphi$ is periodic, $M$ will be Seifert-fibred. – user641 Oct 20 '11 at 18:44
Thank you Steve, but could you be a bit more precise please? – fatoddsun Oct 25 '11 at 12:09
For a periodic map, after finally many times going around, a point comes back to itself. There's your fiber. – user641 Oct 25 '11 at 16:30

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