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.

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.

Reviewer: A.H.M.Hoare (MR 89c:20005)

##### MSC:

20E08 | Groups acting on trees |

20F65 | Geometric group theory |

20F05 | Generators, relations, and presentations of groups |

57M07 | Topological methods in group theory |