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The Gromov topology on \({\mathbb{R}}\)-trees. (English) Zbl 0675.20033
Author’s summary: “We are interested in isometric actions of a fixed finitely generated group on R-trees. Using metric methods inspired by Gromov’s work, we define a more geometric topology on sets of such objects. We prove it to be the same as the Morgan-Shalen topology, defined by the translation lengths of the group elements, in the case of minimal irreducible actions.
This paper has been written simultaneously with the papers of R. Alperin and H. Bass [in Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 265-378 (1987; Zbl 0647.20016)] and M. Culler and J. Morgan [Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)], and independently.
Reviewer: T.M.Rassias

MSC:
20F65 Geometric group theory
57M05 Fundamental group, presentations, free differential calculus
57N10 Topology of general \(3\)-manifolds (MSC2010)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C05 Trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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References:
[1] Alperin, R.; Bass, H., Length functions of group actions on ʌ-trees, (), 265-378 · Zbl 0978.20500
[2] Bestvina, M., Degenerations of the hyperbolic space, Duke math. J., 56, 143-161, (1988) · Zbl 0652.57009
[3] Chiswell, I.M., Length functions and free products with amalgamation of groups, Proc. London math. soc., 42, 42-58, (1981) · Zbl 0454.20030
[4] Chiswell, I.M., Abstract length functions in groups, Math. proc. camb. phil. soc., 80, 451-463, (1976) · Zbl 0351.20024
[5] Culler, M.; Morgan, J., Group actions on \(R\)-trees, Proc. London math. soc., 55, 571-604, (1987) · Zbl 0658.20021
[6] Culler, M.; Vogtmann, K., Moduli of graphs and outer automorphisms of free groups, Invent. math., 84, 91-119, (1986) · Zbl 0589.20022
[7] Gromov, M., Hyperbolic manifolds, groups and actions, (), 183-213 · Zbl 0467.53035
[8] Gromov, M.; Lafontaine, J.; Pansu, P., Structures Métriques pour LES variétés riemanniennes, (1981), Cedic/Fernand Nathan Paris
[9] Gromov, M., Groups of polynomial growth and expanding maps, Publ. math. I.H.E.S., 53, 53-78, (1981) · Zbl 0474.20018
[10] Lyndon, R.C., Length functions in groups, Math. scand., 12, 209-234, (1963) · Zbl 0119.26402
[11] J. Morgan and J.P. Otal, Non-archimedian measured laminations and degenerations of surfaces, to appear.
[12] Morgan, J., Groups actions on trees and the compactification of the space of conjugary classes of SO(n, 1)-representations, Topology, 25, 1-33, (1986)
[13] Morgan, J.; Shalen, P.B., Valuations, trees and degeneration of hyperbolic structures I, Ann. math., 122, 398-476, (1985)
[14] Morgan, J.; Shalen, P.B., Valuations, trees and degeneration of hyperbolic structures II, III, Ann. math., Ann. math., 127, 457-519, (1988) · Zbl 0661.57004
[15] Paulin, F., Topologies de Gromov équivariantes, structures hyperboliques et arbres réels, Invent. math., 94, 53-80, (1988) · Zbl 0673.57034
[16] Paulin, F., Topologies de Gromov équivariantes, structures hyperboliques et arbres réels, ()
[17] Serre, J.P., Trees, (1980), Springer Berlin
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