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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Number of orbits of branch points of $ {\bf R}$-trees


Author: Renfang Jiang
Journal: Trans. Amer. Math. Soc. 335 (1993), 341-368
MSC: Primary 20E08; Secondary 20E06, 20F32, 57M07
DOI: https://doi.org/10.1090/S0002-9947-1993-1107026-4
MathSciNet review: 1107026
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Abstract: An $ R$-tree is a metric space in which any two points are joined by a unique arc. Every arc is isometric to a closed interval of $ R$ . When a group $ G$ acts on a tree ($ Z$-tree) $ X$ without inversion, then $ X/G$ is a graph. One gets a presentation of $ G$ from the quotient graph $ X/G$ with vertex and edge stabilizers attached. For a general $ R$-tree $ X$, there is no such combinatorial structure on $ X/G$. But we can still ask what the maximum number of orbits of branch points of free actions on $ R$-trees is. We prove the finiteness of the maximum number for a family of groups, which contains free products of free abelian groups and surface groups, and this family is closed under taking free products with amalgamation.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1107026-4
Article copyright: © Copyright 1993 American Mathematical Society