Turing Degrees and the Word and Conjugacy Problems for Finitely Presented Groups

The unsolvability of the conjugacy problem for finitely presented groups was first shown by Novikov [26]. Shortly thereafter the corresponding result for the word problem was proved by Novikov [27] and Boone [6] (see also Boone [7], Higman [20] and Britton [9]). Friedberg [19] and Mucnik [25] revitalised the theory of recursively enumerable (r.e.) Turing degrees by showing that there are degrees which are incomparable. A trivial corollary of this is that there are degrees which lie strictly between 0 and 1. It was then natural to ask whether or not each such degree contains a word problem and a conjugacy problem (of finitely presented groups). Fridman [17, 18] gave an affirmative answer for the word problem. The same result was proved also by Clapham [10], Bokut’ [3] and Boone [8]. The question for the conjugacy problem was settled, also in the affirmative, by Bokut’ [4], Collins [13] and Miller [22] (for finitely generated recursively presented groups—see also Miller [24]). Let G be a recursively presented group whose word problem has degree a and conjugacy problem has degree b. Naturally a, b are recursively enumerable. Moreover a ≤T b since a word of G is equal to the identity if and only if it is conjugate to it. The question now arises of whether or not the converse is true. The fact that we can have a 6= b was shown by Fridman [16] who proved that there is a finitely presented group with solvable word problem but unsolvable conjugacy problem. Miller [23] showed that the full converse holds for finitely generated recursively presented groups (see also Miller [24]). Later on Collins [15] was able to extend his analysis from [13] to prove the following.