A Slick Proof of the Unsolvability of the Word Problem for Finitely Presented Groups

A famous theorem of P. Novikov 1955 and W. W. Boone 1959 asserts the existence of a finitely presented group with unsolvable word problem. In my Spring 2005 topics course (MATH 574, Topics in Mathematical Logic), I presented Boone’s proof, as simplified by J. L. Britton, 1963. In this seminar I shall present a truly slick, streamlined proof, due to S. Aanderaa and D. E. Cohen, 1980. Instead of Turing machines or register machines, the Aanderaa-Cohen proof uses another kind of machines, called modular machines, which I shall discuss in detail. In addition, the Aanderaa-Cohen proof uses Britton’s Lemma. I shall omit the proof of Britton’s Lemma, which can be found in my course notes [3] at http://www.math.psu.edu/simpson/notes/. We present the Aanderaa-Cohen [1] simplified proof of the unsolvability of the word problem for finitely presented groups. Like the original Boone-Britton proof, the Aanderaa-Cohen proof is based on HNN extensions and Britton’s Lemma. The statement and proof of Britton’s Lemma are in [3]. Here we mention some consequences of Britton’s Lemma which we shall need. Definition 1. Let G be any group, and let φi : Hi ∼= Ki, i ∈ I, be a family of isomorphisms between subgroups of G. Then the group G = 〈G, pi, i ∈ I | p −1 i hpi = φi(h), h ∈ Hi, i ∈ I〉 is called an HNN extension of G with stable letters pi, i ∈ I. By Britton’s Lemma, G ⊆ G. Definition 2. A good subgroup of G is a subgroup A ⊆ G such that φi(A∩Hi) = A∩Ki for all i ∈ I. Let A ′ be the subgroup of G generated by A, pi, i ∈ I, i.e., A plus the stable letters. By Britton’s Lemma, A is an HNN extension of A with the same stable letters, and A ∩G = A.