A classification of fully residually free groups of rank three or less

A group G is fully residually free provided to every finite set S subset of G \ {1} of non-trivial elements of G there is a free group F-S and an epimorphism h(S): G --> F-S such that h(S)(g) not equal 1 for all g is an element of S. If n is a positive integer, then a group G is n-free provided every subgroup of G generated by n or 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. (C) 1998 Academic Press.