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Let Gamma be a torsion-free group which is hyperbolic relative to acollection of free abelian subgroups. We construct Makanin-Razborov diagramsfor Gamma. We also prove that every system of equations over Gamma isequivalent to a finite subsystem, and a number of structural results aboutGamma-limit groups.

Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams