Recursive Groups

Abstract

1.1. We consider a countable “group alphabet”

$$A = \{ a_1 ,a_2 , \ldots ;a_{_1 }^{ - 1} ,a_{_2 }^{ - 1} \ldots \} .$$

The expressions in the alphabet A, including the empty expression Ø, are traditionally called words. The word a _i…a _i (m ≥ 1 times) will be written a ^m _i ; the word a ^-1 _i…a ^-1 _i (m ≥ 1 times) will be written a ^-m _i; and we agree to take a0i = Ø. We call a word a ^m1 _i1 … a ^mr _ir reduced if either it is empty or there are no subwords of the form a ^-1 _i a _i or a _i a ^-1 _i when it is written in expanded form. The operation of “joining and reducing” (by “reducing” we mean crossing out all subwords of the form a _i a ^-1 _i or a ^-1 _i a _i) defines a group structure with unit Ø (which we sometimes denote by 1) on the set of reduced words. This is a free group F with a countable set of generators {a1, …, a _n , …}. We can also consider nonreduced words as elements in F: we identify such a word with the word obtained by reducing it. We have a canonical numbering on A : N(a _i ) = 2i,N(a ^-1 _i ) = 2i ^-1. All properties related to the computability of operations and the enumerability of subsets in A and S(A) will be considered relative to any numbering of A equivalent to N and any numbering of S(A) compatible with N (see the definitions in §1 of Chapter VII). We shall continually be making use of the following facts.