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The precise question is the following:

Is there a closed aspherical manifold $M$ of dimension $n\geq 3$ such that Out($\pi_1(M)$) is infinite and $\pi_1(M)$ does not contain $\mathbb Z \times \mathbb Z$?

Every surface of genus $g\geq 2$ is of course an example of such manifold, but I cannot find any other example in higher dimension. Note that $\pi_1(M)$ cannot be hyperbolic, otherwise Out($\pi_1(M)$) would be finite. There are no examples in dimension $n=3$ because of Geometrization. I cannot find any reference to this natural question in the literature, but I am probably missing something.

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    $\begingroup$ I do not think it is known, but my guess would be that such manifolds exist and you might be the one who will construct such examples. :) $\endgroup$
    – Moishe Kohan
    Commented Oct 4 at 11:21
  • $\begingroup$ Have you tried asking Tom Farrell? I would imagine this is the kind of question he's thought about. $\endgroup$
    – Ryan Budney
    Commented Oct 4 at 18:36
  • $\begingroup$ I don't have potential examples for the moment :-) I am probably going to cite this question as open in a paper I am writing. $\endgroup$
    – Bruno Martelli
    Commented Oct 8 at 11:53
  • $\begingroup$ I just want to point out the relation with a question I asked here. If the manifold $M$ exists and the center $Z\pi_1(M)$ is not trivial (thus $Z\pi_1(M)\cong\mathbb{Z}$), the inner automorphism group $Inn(\pi_1(M))$ is infinite periodic, since the subgroup generated by $Z\pi_1(M)$ and any element $\gamma\in\pi_1(M)$ is abelian and hence cyclic by hypotheis. The image of $\gamma$ in $Inn(\pi_1(M))$ needs to be a torsion element. So probably, $Z\pi_1(M)$ will be trivial. $\endgroup$
    – Jordi Daura
    Commented Oct 14 at 10:58

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