A [torus bundle](https://en.wikipedia.org/wiki/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?



The fundamental group of torus bundle
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>


