Do there exist groups with word problems in arbitrary P-degrees? 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.
 A: The answer is "yes" (for finitely presented groups). It follows from the main result of Sapir, Mark V.; Birget, Jean-Camille; Rips, Eliyahu Isoperimetric and isodiametric functions of groups. Ann. of Math. (2) 156 (2002), no. 2, 345–466. 
 Edit.  Here is the combination of me explanations in comments below.
You should look at the construction of the group in the paper. First there is a modification of a Turing machine so that the input configurations contribute to the time function the most. Then an S-machine is constructed which by prop. 4.1  is working in polynomial time comparing to the Turing machine and has the same property re input vs arbitrary configurations. Then a group is constructed using the S-machine. Then the Dehn function of the group is estimated (you need the estimate from above). The last step is done by the snowman decomposition. The snowman decomposition decomposes every van Kampen diagram into a linear number of discs and a diagram without hubs (whose area is at most cubic). The perimeters of the discs are linear in terms of the length of the boundary of the diagram. Given a boundary label of a disc, deciding that the disc with this boundary label exists is essentially the same as to decide that certain word belongs to L. Thus the snowman decomposition is a certificate that the word problem is in $L^P$.
A: I think the question is also answered positively by the main result in a paper of
mine -- Efficient computation in groups and simplicial complexes. 
Trans. Amer. Math. Soc. 276 (1983), no. 2, 715–727 -- where it is shown that any Turing
machine may be simulated by a finitely presented group in linear space and cubic time.
