An introductory text on lambda-trees. It should be suitable for research students in algebra and topology, and covers: the construction of lambda-trees; the isometries of lambda-trees; aspects of group actions on lambda-trees; free actions; and Rips' Theorem.
The theory of Λ-trees has its origin in the work of Lyndon on length functions in groups. The first definition of an R-tree was given by Tits in 1977. The importance of Λ-trees was established by Morgan and Shalen, who showed how to compactify a generalisation of Teichmüller space for a finitely generated group using R-trees. In that work they were led to define the idea of a Λ-tree, where Λ is an arbitrary ordered abelian group. Since then there has been much progress in understanding the structure of groups acting on R-trees, notably Rips'' theorem on free actions. There has also been some progress for certain other ordered abelian groups Λ, including some interesting connections with model theory.Introduction to Λ-Trees will prove to be useful for mathematicians and research students in algebra and topology.
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