## Automorphism group of the turing degrees

It is conjectured that the automorphism group of the Turing degrees, $Aut(\mathcal{D})$, is trivial. However, to the best of my knowledge, the current state-of-the-art is that $Aut(\mathcal{D})$ is countable.

My question is twofold: first, is my understanding correct? Is there any countable group $G$ which we know can't be isomorphic to $Aut(\mathcal{D})$?

Second, and more interesting to me: what if anything could we conclude from some group-theoretic information about $Aut(\mathcal{D})$? I.e., what new computability theory would we know if we knew that $Aut(\mathcal{D})$ is abelian, or is simple, or is not finitely generated? (This is admittedly a slightly awkward question, given that we think $ZFC$ proves "$Aut(\mathcal{D})=\lbrace e\rbrace$.")

My reason for asking is that as a rule I am interested in interaction between computability theory and other subjects (for instance, the proof via Higman that there is a universal finitely presented group is dear to my heart), but I am especially interested in examples of other mathematics being applied to computability theory, and I'd be very interested in what the group-theoretic nature of $Aut(\mathcal{D})$ (assuming it's nontrivial) has to say about other parts of computability theory.

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Yes, your understanding is correct. This is proved (among many other related results) in Ted and Hugh's notes "Definability in degree structures", available at math.berkeley.edu/~slaman/talks – Andres Caicedo Jan 23 at 19:03
The key question here is the bi-interpretability conjecture, but as far as I know there is no concrete (interesting) information on what groups are known to be excluded as candidates. – Andres Caicedo Jan 23 at 19:05