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Branch points and free actions on $$\mathbb{R}$$-trees. (English) Zbl 0792.57003
Arboreal group theory, Proc. Workshop, Berkeley/CA (USA) 1988, Publ., Math. Sci. Res. Inst. 19, 251-293 (1991).
[For the entire collection see Zbl 0744.00026.]
An $$\mathbb{R}$$-tree is a metric space in which every 2 points are connected by a unique arc, i.e. a subspace homeomorphic to a closed interval, and every arc is isometric to a closed interval. For various reasons, group actions on $$\mathbb{R}$$-trees have attracted much attention recently (see for example the survey of P. B. Shalen in: “Group theory from a geometrical viewpoint”, ICTP Trieste, 1990 (Editors: E. Ghys, A. Haefliger and A. Verjovsky), World Scientific, Singapore 1991). One of the main conjectures stating that every finitely generated group acting freely on an $$\mathbb{R}$$-tree is a free product of surface groups and free abelian groups has been settled recently by Rips. The present paper is devoted to the study of branch points of free actions on $$\mathbb{R}$$-trees: these are points of index (or valence) at least 3, i.e. the complement of the point has at least 3 components. It is proved that if $$G$$ is the free product of $$n$$ free abelian groups and acts freely and minimally on an $$\mathbb{R}$$-tree $$X$$ then $$\sum(\text{index } - 2)$$, the sum taken over all $$G$$-orbits of branch points of $$X$$, is bounded by $$2n-2$$; in particular, there are only finitely many such orbits and finitely many branches at each branch point. In the second part of the paper, a special class of $$G$$-actions is studied in detail called freely branched actions. It is shown that such actions are free and that $$G$$ splits in a natural way as a free product of free abelian groups. Moreover these actions are determined up to isometry, by a finite set $$S$$ of numerical data, and the translation length function takes values in the subgroup of $$\mathbb{R}$$ generated by $$S$$.

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 20F65 Geometric group theory