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There is a classic result of Baumslag which states,

Thm: If $G$ is residually finite then so is $\operatorname{Aut}(G)$.

While Grossman proved the (essentially) analogous result for $\operatorname{Out}(G)$,

Thm: If $G$ is conjugacy separable and every conjugating automorphism is inner then $\operatorname{Out}(G)$ is residually finite.

(A conjugating automorphism is an automorphism $\delta: g\mapsto g^{w_g}$ where $w_g$ is dependent on the $g\in G$ - so every element is sent to a conjugate of itself).

I was wondering if it was possible to "go the other way", so to speak,

What conditions, if any, can we put on $\operatorname{Aut}(G)$ or $\operatorname{Out}(G)$ to ensure that $G$ is residually finite?

One obvious condition is that if $\operatorname{Aut}(G)$ is residually finite and $G$ is centerless then $G$ is residually finite. However, you have the added stipulation that $G$ is centerless, which is a condition on $G$ not on $\operatorname{Aut}(G)$. That said, this is a relatively harmless condition on $G$ (checking $Z(G)=1$ is often easier than checking $G$ is residually finite). So I suppose my question can be taken up to "relatively harmless conditions on $G$".

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    $\begingroup$ It seems relevant to note that Bumagina and Wise proved that every finitely presentable group arises as the outer automorphism group of a finitely generated residually finite group. $\endgroup$
    – HJRW
    Sep 29, 2011 at 13:39

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First, Baumslag's result is for finitely generated groups only. HW already says that essentially the Out of a residually finite group can be ``arbitrary". Now if you take a Tarski monster with trivial Out, then the direct product of it with the residually finite group above gives a non-residualy finite group with an arbitrary Out. Another way is to use Minasyan's result from Minasyan, Ashot, Groups with finitely many conjugacy classes and their automorphisms. Comment. Math. Helv. 84 (2009), no. 2, 259–296. He constructed a 2-generated group with two conjugacy classes (hence simple and not residually finite) with arbitrary countable Out. So there is no connection between properties of Out and residual finiteness of the group.

With Aut the situation is more complicated since if the group is not res. finite and does not have center then Aut is not residually finite. On the other hand, there are non-residually finite f.g.groups with residualy finite Aut. For example, Anna Erschler, I think, constructed such a group as a central extension of a Grigorchuk group. See Erschler, Anna Not residually finite groups of intermediate growth, commensurability and non-geometricity. J. Algebra 272 (2004), no. 1, 154–172.

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  • $\begingroup$ Mark, how do you show that Aut of Erschler's example is residually finite? I can see that it is finite-by-(res. finite), but this is not enough in general... $\endgroup$ Oct 6, 2011 at 18:19
  • $\begingroup$ @Ashot: It follows from Anna's construction, I think. The Out of Grigorchuk's group is a direct product of ${\mathbb Z}_2$ (Grigorchuk-Sidki), and the Out of Erschler's extension can be described explicitly. There are many (continuum) groups, and at least for some of them Out is residually finite. But I did not check carefully all the details. $\endgroup$
    – user6976
    Oct 6, 2011 at 18:26

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