This has been posted on TCS stack exchange for a while here and hasn't gotten any answers, so I'm trying again here.
It has been known for a long time that, given any r.e. Turing degree, there is a finitely presented group whose word problem is in that degree. My question is whether the same thing is true for arbitrary polynomial time Turing degrees. Specifically, given a decidable set, $A$, does there exist a finitely presented group, with word problem, $W$, such that $W\leq_T^P A$ and $A\leq_T^P W$? I would also be willing to relax finitely presented to recursively presented.
I suspect that the answer is yes, and I have heard others say they read this somewhere, but I haven't been able to chase down a reference.
EDIT: As per the comments, here, $\leq_T^P$ means polynomial-time Turing reducibility. See here for more info.