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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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##### References:
 [1] E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459 · Zbl 1074.57001 [2] E Alibegović, Makanin-Razborov diagrams for limit groups, Geom. Topol. 11 (2007) 643 · Zbl 1185.20034 [3] M Bestvina, M Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991) 449 · Zbl 0724.20019 [4] M Bestvina, M Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995) 287 · Zbl 0837.20047 [5] M Bestvina, M Feighn, Notes on Sela’s work: limit groups and Makanin-Razborov diagrams, prepring · Zbl 1213.20039 [6] M Bestvina, Questions in geometric group theory [7] B Bowditch, Relatively hyperbolic groups, preprint · Zbl 1259.20052 [8] F Dahmani, Classifying spaces and boundaries for relatively hyperbolic groups, Proc. London Math. Soc. $$(3)$$ 86 (2003) 666 · Zbl 1031.20039 [9] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933 · Zbl 1037.20042 [10] F Dahmani, Accidental parabolics and relatively hyperbolic groups, Israel J. Math. 153 (2006) 93 · Zbl 1174.20014 [11] F Dahmani, On equations in relatively hyperbolic groups, preprint [12] C Dru\ctu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959 · Zbl 1101.20025 [13] C Dru\ctu, M Sapir, Relatively hyperbolic groups with rapid decay property, Int. Math. Res. Not. (2005) 1181 · Zbl 1077.22006 [14] M J Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449 · Zbl 0572.20025 [15] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810 · Zbl 0985.20027 [16] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 [17] D Groves, Limits of (certain) CAT(0) groups I: Compactification, Algebr. Geom. Topol. 5 (2005) 1325 · Zbl 1085.20025 [18] D Groves, Limits of certain CAT(0) groups II: The Hopf property and the shortening argument [19] D Groves, Limit groups for relatively hyperbolic groups I: The basic tools · Zbl 1231.20038 [20] V S Guba, Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Mat. Zametki 40 (1986) 321, 428 · Zbl 0621.20009 [21] G C Hruska, Nonpositively curved spaces with isolated flats, PhD thesis, Cornell University (2002) · Zbl 1063.20048 [22] O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group I: Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998) 472 · Zbl 0904.20016 [23] O Kharlampovich, A Myasnikov, Irreducible affine varieties over a free group II: Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998) 517 · Zbl 0904.20017 [24] D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) · Zbl 1093.20025 [25] E Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982) 45 · Zbl 0481.20020 [26] E Rips, Z Sela, Structure and rigidity in hyperbolic groups I, Geom. Funct. Anal. 4 (1994) 337 · Zbl 0818.20042 [27] E Rips, Z Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. $$(2)$$ 146 (1997) 53 · Zbl 0910.57002 [28] Z Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997) 527 · Zbl 0887.20017 [29] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, Geom. Funct. Anal. 7 (1997) 561 · Zbl 0884.20025 [30] Z Sela, Endomorphisms of hyperbolic groups I: The Hopf property, Topology 38 (1999) 301 · Zbl 0929.20033 [31] Z Sela, Diophantine geometry over groups I: Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31 · Zbl 1018.20034 [32] Z Sela, Diophantine geometry over groups VII: The elementary theory of a hyperbolic group, preprint · Zbl 1241.20049 [33] Z Sela, Diophantine geometry over groups: a list of research problems · Zbl 1285.20042 [34] J P Serre, Trees, Springer (1980) · Zbl 0548.20018 [35] A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41 · Zbl 1043.20020
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