Finitely Generated Groups and First-order Logic

We prove that the following classes of finitely generated (f.g.) groups have Π1–complete first–order theories: all f.g. groups, the n–generated groups, and the strictly n–generated groups (n > 2). Moreover, all those theories are distinct. Similar techniques show that quasi-finitely axiomatizable (QFA) groups have a hyperarithmetical word problem, where a f.g. group is QFA if it is the only f.g. group satisfying an appropriate first–order sentence [8]. The Turing degrees of word problems of QFA groups form a cofinal set in the Turing degrees of hyperarithmetical sets. Given a first order theory, two fundamental tasks are to determine its computational complexity and its expressivity. The theory Th(E) of a class E of groups is the set of first–order sentences which are true in all the members of E . We determine the complexity for various theories Th(E). Our main results in this direction are: – The theory T of the class of finitely generated (f.g.) groups is Π1-complete. – For n ≥ 2, the theories Tn of all n-generated groups, and the theories T ! n of all strictly n-generated groups are Π1-complete. Here a group is strictly n-generated if it is n-generated but not (n− 1)-generated. An example of a Π1-complete set from arithmetic is the set of all sentences of the form ∀Xφ(X) which hold in N, for any Σ1-formula φ involving the arithmetical operations and expressions “t ∈ X” for some term t. For another related example, the set of (indices for) recursive subtrees of ω which have no infinite path is Π1complete. We first verify that all those theories are in Π1. Then we prove that it is as hard as it could possibly be to determine whether a first–order sentence φ holds in all f.g. groups, and similarly for the other classes: the theories are Π1-complete. Theories of classes of groups and their complexity have been studied for a long time. A. Tarski [16] was the first to prove undecidability of the theory of groups in 1949. Many other results on decidability and undecidability of theories of classes of groups were obtained by A. I. Mal’cev and Yu. L. Ershov in the 1960’s (see [17] for details and bibliography) and later. Szmielev [15] proved that the theory of all abelian groups is decidable, and O. Kharlampovich and A. Myasnikov announced that the theory of a free groups is decidable [3]. Other theories, while being undecidable, have a comparatively low complexity, like the theory of all groups, which is in Σ1, or the theory of finite groups, which is in Π 0 1. Nies [8, Cor 5.5] showed that the theory of many classes of groups have the same complexity as true arithmetic Th(N,+,×), for example the class of finitely presented groups, or the class of f.g. groups of nilpotency class c (c ≥ 2 fixed). In one case the theory of a class of 2000 Mathematics Subject Classification 03D35 (primary), 03D40, 20A15, 20F10 (secondary). The first author was partially supported by binational NSF grant DMS-0075899 and by RFBR (Russian Fund for Basic Research) grant No. 02-01-00593. The second author was partially supported by binational NSF grant DMS-0075899. 2 a. morozov, a. nies groups was shown to be Π1–complete. A.S.Morozov proved this for the class of all subgroups of the group of computable permutations [6]. Fragments of theories have been considered as well. The universal theory of the f.p. groups is undecidable by the unsolvability of the word problem. Slobodskoi [14] showed that the universal theory of finite groups is undecidable. Besides the complexity, one wants to determine the expressive power of first order theories. Kharlampovich and Myasnikov [3] also announced that all non-abelian free groups have the same first–order theory, thereby answering a long–open question of Tarski. The result was later confirmed by Sela [12]. This exposes a weakness in the expressiveness of first–order logic for free groups. On the other hand, Nies [8] shows that many natural classes of groups have distinct theories, for instance the classes of finite, finitely presented (f.p.), f.g., and of all groups (or, equivalently, all countable groups). In this paper we also show that all the theories T, Tn and T ! n introduced above are distinct. Thus all the inclusions in the diagram below are proper: