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. 1. 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) (1) (Oral tradition) All nonabelian free groups satisfy the same fI2 sentences; (2) (Merzlyakov's Theorem) If m > m' > 2 are integers, then a positive sentence (D in the language of Fm, (the free group of rank m') holds in Fm (under the standard embedding) if and only if it holds in