|dc.description.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