Olga Kharlampovich, McGill University

Abstract:
Finitely generated fully residually free groups (limit groups) play a crucial role in the theory of equations and first-order formulas over a free group. It is remarkable that these groups, which have been widely studied before, turn out to be the basic objects in newly developing areas of algebraic geometry and model theory of free groups. Recall that a group $G$ is called fully residually free if for any finitely many non-trivial elements in G there exists a homomorphism of $G$ into a free group such that the images are non-trivial. We will give a survey of the results on these groups and their subgroups. We begin with the works by Baumslag, Kharlampovich, Myasnikov, Remeslennikov. We explain K-M characterization of f.g. fully resifually free groups as subgroups of the Lyndon group $F^{Z[t]}$, talk about algorithmic problems for these groups. Then we survey recent results by Bridson, Howie, Miller, Short, Wilton, Serbin, Nikolaev and others about subgroups of fully residually free and residually free groups.