# The fundamental group of the mapping torus is doubly degenerate

Consider an hyperbolic compact surface $S$ (hence with genus $>1$) and a Pseudo-Anosov diffeomorphism $\varphi\colon S\to S$. We call "mapping torus" the 3-manifold $$M=M(\varphi)=S\times\left[0,1\right]/(x,0)\sim (\varphi(x),1)$$ Such a manifold can be given an hyperbolic structure and fibres over the circle, with fibre $S$. I would like to prove that the fundamental group of $M$ is doubly degenerate, that is its limit sets in $S^2_\infty$ coincides with the whole sphere $S^2_\infty$. The only I know is that the immersion $S\to M$ induces an injection $\pi_1(S)\to\pi_1(M)$; moreover, the fundamental group of $S$ is non-abelian (and I suppose virtually non-abelian either), thus non-elementary, therefore its limit sets contains infinitely many points, and consequently the same holds also for $\pi_1(M)$, but I'm still very far from what I'd like to show. Could you help me with that? Thank you.

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