## Length functions of group actions on $$\Lambda$$-trees.(English)Zbl 0978.20500

Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 265-378 (1987).
[For the entire collection see Zbl 0611.00010.]
The following is a brief outline, by sections, of results from this long and technical paper. Throughout, $$\Lambda$$ is an ordered Abelian group and $$\Gamma$$ is any group.
(0) Introduction. The connection with the work of J. W. Morgan and P. B. Shalen [Ann. Math. (2) 120, No. 3, 401-476 (1984; Zbl 0583.57005)] and others is discussed. Chapter I. $$\Lambda$$-trees. (1) Functions $$l\colon\Gamma\to\Lambda$$ such that $$l(s)=|h(s)|$$ for some homomorphism $$h\colon\Gamma\to\Lambda$$ are characterized. (2) $$\Lambda$$-trees are defined and shown to have many of the properties of $$\mathbb{Z}$$-trees and $$\mathbb{R}$$-trees. (3) Given a set $$X$$ and a function $$\wedge\colon X\times X\to\Lambda$$ satisfying three (common segment) axioms, I. M. Chiswell’s construction [Math. Proc. Camb. Philos. Soc. 80, 451-463 (1976; Zbl 0351.20024)] gives a $$\Lambda$$-tree $$T$$ and a function $$\varphi\colon X\to T$$ with universal properties. Appendix A. This is applied to $$F^2$$, where $$F$$ has a $$\Lambda$$-valuation. (4) Associated with any order-preserving homomorphism $$h\colon\Lambda\to\Lambda'$$ there is a functor from $$\Lambda$$-trees with $$\Lambda$$-metric morphisms to $$\Lambda'$$-trees and morphisms. Chapter II. Tree actions and length functions. (5) Every $$\Lambda$$-valued Lyndon length function of a group $$\Gamma$$ arises essentially uniquely from an action of $$\Gamma$$ on a $$\Lambda$$-tree $$T$$. (6) The hyperbolic length of elements of $$\Gamma$$ is defined from the action of $$\Gamma$$ on a $$\Lambda$$-tree. Appendix B. Hyperbolic lengths are calculated for elements of $$\text{GL}_2(F)$$ acting on the $$\Lambda$$-tree of Appendix A. (7) Properties of the hyperbolic length give information about the action of $$\Gamma$$ on the $$\Lambda$$-tree. (8) The hyperbolic length of a product, needed in Section 7, is calculated. (9) Necessary conditions for a function $$l\colon\Gamma\to\Lambda$$ to be the hyperbolic length of some $$\Lambda$$-tree action are discussed but their sufficiency is left open.

### MSC:

 20E08 Groups acting on trees 20F65 Geometric group theory 20F05 Generators, relations, and presentations of groups 57M07 Topological methods in group theory

### Citations:

Zbl 0611.00010; Zbl 0583.57005; Zbl 0351.20024