| Let L be a limit group. By constructing a hyperbolic space on which L acts freely, properly discontinuously by isometries we show that L is hyperbolic relative to the collection of its maximal noncyclic abelian subgroups.; For any finitely generated group G we describe the set Hom(G, L) using the Makanin-Razborov diagram. This is a finite diagram consisting of limit groups, except for the initial group G. Each group in this diagram has a finite number of directed edges issuing from it. This number will be zero if the group in question is a free group. Each edge represents a quotient map, and all quotients are proper, except for possibly one that represents an embedding into L. Every homomorphism G → L, possibly redefined on subgroups whose images contain parabolic elements, can be written as a composition of these quotient maps, elements of modular automorphism groups of groups within the diagram and either a map from a free group into L or an embedding of some limit group from the diagram into L. |
