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$$\Lambda$$-trees and their applications. (English) Zbl 0767.05054
Automorphism groups of trees (called simplicial trees in the paper) are studied by methods from combinatorial group theory. The notion of a tree is generalized to a tree over $$\mathbb{Z}$$ ($$\mathbb{Z}$$-tree) or $$\mathbb{R}$$ ($$\mathbb{R}$$- tree) and the corresponding generalized theory is discussed.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20F65 Geometric group theory 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54H12 Topological lattices, etc. (topological aspects) 20G99 Linear algebraic groups and related topics
##### Keywords:
automorphism groups; trees; combinatorial group theory
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##### References:
 [1] Roger Alperin and Hyman Bass, Length functions of group actions on \Lambda -trees, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265 – 378. · Zbl 0978.20500 [2] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5 – 251 (French). · Zbl 0254.14017 [3] Marc Culler and John W. Morgan, Group actions on \?-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571 – 604. · Zbl 0658.20021 [4] Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109 – 146. · Zbl 0529.57005 [5] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91 – 119. · Zbl 0589.20022 [6] Henri Gillet and Peter B. Shalen, Dendrology of groups in low \?-ranks, J. Differential Geom. 32 (1990), no. 3, 605 – 712. · Zbl 0732.20011 [7] H. Gillet, P. Shalen, and R. Skora, preprint. [8] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75 – 263. · Zbl 0634.20015 [9] Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 48 – 106. · Zbl 0664.53023 [10] John W. Morgan, Ergodic theory and free actions of groups on \?-trees, Invent. Math. 94 (1988), no. 3, 605 – 622. · Zbl 0676.57001 [11] John W. Morgan, Group actions on trees and the compactification of the space of classes of \?\?(\?,1)-representations, Topology 25 (1986), no. 1, 1 – 33. · Zbl 0595.57030 [12] -, Trees and degenerations of hyperbolic structures, CBMS Lecture Note Series (to appear). [13] John W. Morgan and Peter B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2) 120 (1984), no. 3, 401 – 476. · Zbl 0583.57005 [14] John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. II. Measured laminations in 3-manifolds, Ann. of Math. (2) 127 (1988), no. 2, 403 – 456. , https://doi.org/10.2307/2007061 John W. Morgan and Peter B. Shalen, Degenerations of hyperbolic structures. III. Actions of 3-manifold groups on trees and Thurston’s compactness theorem, Ann. of Math. (2) 127 (1988), no. 3, 457 – 519. · Zbl 0661.57004 [15] -, Valuations, trees, and degenerations of hyperbolic structures, III: action of 3-manifold groups on trees and Thurson’s compactness theorem, Ann. of Math. (2) 127 (1988), 467-519. [16] John W. Morgan and Peter B. Shalen, Free actions of surface groups on \?-trees, Topology 30 (1991), no. 2, 143 – 154. · Zbl 0726.57001 [17] John W. Morgan and Richard K. Skora, Groups acting freely on \?-trees, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 737 – 756. · Zbl 0766.57021 [18] G. D. Mostow, Quasi-conformal mappings in \?-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53 – 104. · Zbl 0189.09402 [19] Walter Parry, Axioms for translation length functions, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 295 – 330. · Zbl 0829.20039 [20] Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. · Zbl 0548.20018 [21] Peter B. Shalen, Dendrology of groups: an introduction, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 265 – 319. · Zbl 0649.20033 [22] Richard K. Skora, Splittings of surfaces, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 85 – 90. · Zbl 0708.30044 [23] W. Thurston, Geometry and topology of 3-manifolds, Princeton Univ., preprint, 1980. [24] J. Tits, A ”theorem of Lie-Kolchin” for trees, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 377 – 388.
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