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Limit groups for relatively hyperbolic groups. II: Makanin-Razborov diagrams. (English) Zbl 1100.20032
Z. Sela [Diophantine geometry over groups: a list of research problems, http://www.ma.huji.ac.il/sim zlil/problems.dvi] asked whether a group acting properly cocompactly on a CAT(0) space with isolated flats is Hopfian and whether one can construct Makanin-Razborov diagrams for it. In the prequel to this paper, the author develops tools to approach this problem that allow Sela’s program to be applied, despite additional technical difficulties that arise in the case under consideration.
Let \(\Gamma\) be a torsion-free group that is hyperbolic relative to a collection of free Abelian subgroups. One wants to understand the group \(\operatorname{Hom}(G,\Gamma)\) where \(G\) is an arbitrary finitely generated group. A Makanin-Razborov diagram is a finite directed tree associated to such a \(G\) that encodes the set \(\operatorname{Hom}(G,\Gamma)\). The main result of this paper is to construct such diagrams for groups \(\Gamma\) in the given class.
The general approach is to reduce the description of \(\operatorname{Hom}(G,\Gamma)\) to a description of a finite collection of \(\operatorname{Hom}(L_i,\Gamma)\) where the \(L_i\) are proper quotients of \(G\). The \(L_i\) are \(\Gamma\)-limit groups. This procedure is again applied to each of the \(\operatorname{Hom}(L_i,\Gamma)\). Thus, the author obtains a descending sequence of \(\Gamma\)-limit groups. Then, the author shows that any descending sequence of \(\Gamma\)-limit groups with \(\Gamma\) as above must terminate after finitely many steps and uses this to construct Makanin-Razborov diagrams over \(\Gamma\).

MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
20E08 Groups acting on trees
57M07 Topological methods in group theory
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