# zbMATH — the first resource for mathematics

Makanin-Razborov diagrams for limit groups. (English) Zbl 1185.20034
Summary: We give a description of $$\operatorname{Hom}(G,L)$$, where $$L$$ is a limit group (fully residually free group). We construct a finite diagram of groups, the Makanin-Razborov diagram, that gives a convenient representation of all such homomorphisms.

##### MSC:
 20E36 Automorphisms of infinite groups 20E26 Residual properties and generalizations; residually finite groups 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 57M07 Topological methods in group theory
Full Text:
##### References:
 [1] E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459 · Zbl 1074.57001 [2] J M Alonso, e al., Notes on word hyperbolic groups, World Sci. Publ., River Edge, NJ (1991) 3 · Zbl 0849.20023 [3] M Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988) 143 · Zbl 0652.57009 [4] M Bestvina, $$\mathbbR$$-trees in topology, geometry, and group theory, North-Holland (2002) 55 · Zbl 0998.57003 [5] M Bestvina, M Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995) 287 · Zbl 0837.20047 [6] M Bestvina, M Feighn, Notes on Sela’s work: Limit groups and Makanin-Razborov diagrams (2003) · Zbl 1213.20039 [7] M R Bridson, G A Swarup, On Hausdorff-Gromov convergence and a theorem of Paulin, Enseign. Math. $$(2)$$ 40 (1994) 267 · Zbl 0846.20038 [8] I Chiswell, Introduction to $$\Lambda$$-trees, World Scientific Publishing Co. (2001) · Zbl 1004.20014 [9] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933 · Zbl 1037.20042 [10] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 [11] D Groves, Limit groups for relatively hyperbolic groups. II. Makanin-Razborov diagrams, Geom. Topol. 9 (2005) 2319 · Zbl 1100.20032 [12] 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 [13] 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 [14] J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. $$(2)$$ 120 (1984) 401 · Zbl 0583.57005 [15] F Paulin, Outer automorphisms of hyperbolic groups and small actions on $$\mathbbR$$-trees, Math. Sci. Res. Inst. Publ. 19, Springer (1991) 331 · Zbl 0804.57002 [16] Z Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. (2001) 31 · Zbl 1018.20034 [17] H Zieschang, Alternierende Produkte in freien Gruppen, Abh. Math. Sem. Univ. Hamburg 27 (1964) 13 · Zbl 0135.41805
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.