The Gromov topology on \({\mathbb{R}}\)-trees. (English) Zbl 0675.20033

Paulin, Frédéric

Topology Appl. 32, No. 3, 197-221 (1989).

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

Keywords:

isometric actions; finitely generated group; R-trees; Morgan-Shalen topology; translation lengths

Citations:

Zbl 0647.20016; Zbl 0658.20021

References:

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