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A note on a theorem of Chiswell. (English) Zbl 0841.20029

Summary: We give an alternative proof of a theorem of I. M. Chiswell which states that every finitely generated group which acts non-trivially on a \(\Lambda\)-tree admits a non-trivial action on an \(\mathbb{R}\)-tree.

MSC:

20E08 Groups acting on trees
20F65 Geometric group theory
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
20F05 Generators, relations, and presentations of groups
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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] I. M. Chiswell, Nontrivial group actions on \Lambda -trees, Bull. London Math. Soc. 24 (1992), no. 3, 277 – 280. · Zbl 0791.20022 · doi:10.1112/blms/24.3.277
[3] Herbert B. Enderton, A mathematical introduction to logic, Academic Press, New York-London, 1972. · Zbl 0298.02002
[4] 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 · doi:10.1007/978-1-4613-9586-7_4
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