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 \(S\).

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 \(S\).

Reviewer: B.Zimmermann (Trieste)

##### MSC:

57M60 | Group actions on manifolds and cell complexes in low dimensions |

20F65 | Geometric group theory |