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.
Do there exist groups with word problems in arbitrary P-degrees?
cc.complexity theory;computability;gr.group theory