We are looking for examples of groups $G$ such that $G$ is "big", but $Out(G)$ is trivial. By "big" we mean things like virtually free, or large, or Golod-Shafarevich. However, we would like our groups to be residually-finite.

**Edit**: Henry, Igor and Lee thank you for all your help. Eventually, we only needed finite $Out(G)$ with $G$ having a trivial center. Thus, $A*B$ was the example we used. It is included in "Large normal subgroup growth and large characteristic subgroup growth" (sorry for the self-promotion, but I thought people might be interested in our motivation and the results of their help).

innerin $GL(2, \mathbb{Z}),$ but the conjugation is by a matrix of determinant $-1.$ $\endgroup$7more comments