Elementary Properties of Free Groups

In this paper we show that several classes of elementary properties (properties definable by sentences of a first order logic) of groups hold for all nonabelian free groups. These results are obtained by examining special embeddings of these groups into one another which preserve the properties in question. I. It has long been conjectured that the finitely generated nonabelian free groups are elementarily equivalent. (That is, they satisfy the same sentences of first order group theory.) The truth of this conjecture would yield the elementary equivalence of all of the nonabelian free groups. There have been several partial results in this area. The most notable are (') (1) (Oral tradition) All nonabelian free groups satisfy the same IL sentences; (2) (Merzlyakov's Theorem) If m > m' > 2 ate integers, then a positive sentence <D in the language of F / (the free group of rank m') holds in F (under the standard embedding) if and only if it holds in Fm" In this paper we shall apply the graph-theoretic techniques of small cancellation theory as developed by Lyndon, Schupp, and Weinbaum to prove a general lemma (hereafter called the Principal Lemma) from which both of these results and others can easily be obtained. More precisely, we shall prove A. Any n, sentence in the language of F , which holds in F (under the standard embedding) holds in F , . B. Let O be a positive sentence in the language of a free group F , ; then 0 holds in F , if and only if an instance of 0, obtained by replacing the existentially bound variables of <D by terms involving the generators and the preceding (in the prefix of <D) universally bound variables and deleting the quantifiers, is valid in Fm-' C. Merzlyakov's Theorem. D. Given a positive sentence 0 in the language of F , there is an embedding of F into F / such that F satisfies $ if and only if F / satisfies $. ° m m m J m In addition we will prove Received by the editors December 7, 1971. AMS (MOS) subject classifications (1970). Primary 02H05, 02H15, 20A05, 20E05. (l)Mal cev has attributed the following result to A. D. Taimanov. All nonabelian free groups satisfy the same n4 sentences. Unfortunately this proof is unavailable to the author. Copyright © 1973, Amtúcan Mathematical Society