M. Feighn: Limit groups

Abstract: A natural question (and one that comes up in the Tarski problem) is:

Given a finitely generated group G and a free group F, what can we say about Hom(G,F)?

It turns out that Hom(G,F) has a beautiful structure (discovered independently by Kharlampovich-Myasnikov and Sela) that surprisingly can be explained geometrically. The question reduces to the case where G is a "limit group", i.e. a group with the property that, for every finite subset S of G, there is an element of Hom(G,F) that is injective on S. In these talks, we will give an account of this structure and explore properties of these limit groups.