A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS⊂G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G→F_Ssuch thath_S(g)≠1 for allg∈S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.