Decision Problems in Group Theory

At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an exposition of modular machines, will be found in a paper by Aanderaa and Cohen [2]. The aim of this paper is to use modular machines to give short and fairly simple proofs of some of the well-known results on decision problems in group theory. Most of these results will be proved in the setting of recursively enumerable (r.e.) Turing degrees (for the rest of the paper 'degree' will mean 'Turing degree' unless otherwise stated). The other main tool used here is the concept of 'group with a standard basis' introduced by Bokut' in [4] (a more detailed account is given by him in [5]). This is a method for obtaining and computing (usually with respect to an oracle) unique normal forms in certain types of Britton towers. The method gives a unified approach to the paper. We give a brief, but for our purposes adequate, account of both modular machines and Bokut' normal forms in §§ 2 and 3 respectively. Background results on HNN extensions and free products may be found in the books of Lyndon and Schupp [19] and Rotman [23].