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A torus bundle is labeled by an element $M$ of $SL(2,\mathbb{Z})$ -- the mapping class group of a torus. How to compute the fundamental group of a torus bundle from the 2-by-2 matrix $M \in SL(2,\mathbb{Z})$? Any references?

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A quick google search on "fundamental group of a mapping torus" gives many hits - eg:

https://math.stackexchange.com/questions/39589/fundamental-group-of-mapping-torus?rq=1

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Take a look at the section on monodromy in my old preprint (Rigidity of Fibering, arxiv.org 1106.4595.)

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