Limit groups and Makanin-Razborov diagrams for hyperbolic groups

Abstract

This thesis gives a detailed description of Zlil Sela’s construction of Makanin-Razborov diagrams which describe Hom(G, T), the set of all homomorphisms from G to T, where G is a finitely generated group and T is a hyperbolic group. Moreover, while Sela’s construction requires T to be torsion-free, this thesis removes this condition and addresses the case of arbitrary hyperbolic groups. Sela’s shortening argument, which is the main tool in the construction of the Makanin- Razborov diagrams, relies on the Rips machine, a structure theorem for finitely generated groups acting stably on real trees. As homomorphisms from a f.g. group G to a hyperbolic group T give rise to stable actions of G on real trees, which appear topologically as limits of the G-actions on the Cayley graph of T, the Rips machine and the shortening argument allow us to explore the structure of Hom(G, T) and construct Makanin-Razborov diagrams which encode all homomorphisms from G to T. While Sela’s version of the Rips machine allows the formulation of the shortening argument only in the case where T is torsion-free, Guirardel has presented a generalized version of the Rips machine, which we exploit to generalize the shortening argument and the construction of Makanin-Razborov diagrams to the case of arbitrary hyperbolic groups.