Characterization of separable metric -trees

Authors:
J. C. Mayer, L. K. Mohler, L. G. Oversteegen and E. D. Tymchatyn

Journal:
Proc. Amer. Math. Soc. **115** (1992), 257-264

MSC:
Primary 54F50; Secondary 54E35, 54F65

MathSciNet review:
1124147

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Abstract | References | Similar Articles | Additional Information

Abstract: An -tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. -trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized -trees topologically among metric spaces. The purpose of this paper is to provide a simpler proof of this characterization for separable metric spaces. The main theorem is the following: *Let* *be a separable metric space. Then the following are equivalent:*

(1) *admits an equivalent metric* *such that* *is an* *-tree*.

(2) *is locally arcwise connected and uniquely arcwise connected*. The method of proving that (2) implies (1) is to "improve" the metric through a sequence of equivalent metrics of which the first is monotone on arcs, the second is strictly monotone on arcs, and the last is convex, as desired.

**[AB]**Roger Alperin and Hyman Bass,*Length functions of group actions on Λ-trees*, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265–378. MR**895622****[Be]**Mladen Bestvina,*Degenerations of the hyperbolic space*, Duke Math. J.**56**(1988), no. 1, 143–161. MR**932860**, 10.1215/S0012-7094-88-05607-4**[MO]**John C. Mayer and Lex G. Oversteegen,*A topological characterization of 𝑅-trees*, Trans. Amer. Math. Soc.**320**(1990), no. 1, 395–415. MR**961626**, 10.1090/S0002-9947-1990-0961626-8**[Mr]**J. W. Morgan,*Deformations of algebraic and geometric structures*, CBMS Lectures, UCLA (Summer, 1986), preprint.**[MrS]**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. MR**769158**, 10.2307/1971082

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1124147-5

Keywords:
-tree,
convex metric,
uniquely arcwise connected,
locally arcwise connected

Article copyright:
© Copyright 1992
American Mathematical Society