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Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Generator c of $\Bbb Z$ is $c \in Aut(\Bbb Z \times \Bbb Z)=SL(2,Z)$. So c can be taken to be either of the two generators of $SL(2,Z)$.

This is towards the mapping class group calculation of a torus bundle over a circle with monodromy. So finally we are interested in $Out((\Bbb Z\times \Bbb Z) \rtimes \Bbb Z) $, if it makes it a simpler problem.

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    $\begingroup$ Note that $c^{4}$ is central in that group. $\endgroup$
    – Geoff Robinson
    Commented Aug 17, 2016 at 11:15
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    $\begingroup$ I don't see why the question was closed -- and just as I was finishing writing an answer! Maybe it can be reopened. $\endgroup$
    – Allen Hatcher
    Commented Aug 17, 2016 at 17:36
  • $\begingroup$ @AllenHatcher Agreed, I voted to reopen. $\endgroup$
    – Igor Rivin
    Commented Aug 17, 2016 at 21:21
  • $\begingroup$ It seems it was judged to be too simple. I have tried calculating it and I don't see an obvious answer. One key question was whether it would be a subrgroup of $SL(3,Z)$. $\endgroup$
    – SKShukla
    Commented Aug 17, 2016 at 22:16
  • $\begingroup$ I guess it was closed because it's an exercise... when the question was written, $c$ was an element of order 4. Of course in general there's more discussion. And no, the key question is not whether it's "a subgroup of $SL(3,Z)$" (which means nothing, or maybe whether it's isomorphic to a subgroup therein, but that would be a question coming after the answer. $\endgroup$
    – YCor
    Commented Aug 17, 2016 at 23:42

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