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Axioms for translation length functions. (English) Zbl 0829.20039
Arboreal group theory, Proc. Workshop, Berkeley/CA (USA) 1988, Publ., Math. Sci. Res. Inst. 19, 295-330 (1991).
[For the entire collection see Zbl 0744.00026.]
Following Culler and Morgan, define a pseudo-length function to be a function \(|\;|:G\to\Lambda\) from a group \(G\) to the nonnegative elements of an ordered Abelian group \(\Lambda\) which satisfies certain axioms. The object of this paper is to prove that every pseudo-length function is a translation length function. The main theorem can be formally stated as follows. Main theorem. Let \(G\) be a group, and let \(|\;|\) be a pseudo-length function from \(G\) to an ordered Abelian group \(\Lambda\). Then there exists a \(\Lambda\)-tree \(T\) and an action of \(G\) on \(T\) by isometries such that \(|\;|\) is the translation length function for this action.

20E08 Groups acting on trees
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups