Introduction to Lambda TreesThe theory of o-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 o-trees was established by Morgan and Shalen, who showed how to compactify a generalisation of Teichmller space for a finitely generated group using R -trees. In that work they were led to define the idea of a o-tree, where o 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 o, including some interesting connections with model theory. Introduction to o-Trees will prove to be useful for mathematicians and research students in algebra and topology. Contents: o-Trees and Their Construction; Isometries of o-Trees; Aspects of Group Actions on o-Trees; Free Actions; Rips'' Theorem. Readership: Mathematicians and research students in algebra and topology." |
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5-orbit A-metric space action of G assume Borel measure Borel set Cayley graph choose closed subtree compact component contains contradiction Corollary countable defined definition denote domain edges equivalence finite finitely generated group fixed point flow box follows easily free abelian groups free group free product fully residually free fundamental group G acts g G G G-invariant geodesic geodesic curve group acting group G hence homeomorphism homotopy hyperbolic isometries hyperbolic length function hyperbolic line hyperbolic surface induced integer intersection interval inversions isomorphic lamination leaf Lemma Let G lexicographic ordering linear subtree Lyndon length function metric NEC group non-degenerate non-empty non-trivial normal subgroup Note obtained orbits ordered abelian group parametrisation R-tree real(r remark preceding restriction segment sequence Similarly simplicial tree subset system of isometries Theorem topology transverse transverse measure triangle inequality ultrapower unique vertex vertices x,gx
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Page 253 - L(a, ß, у, п) is said to be isomorphic vnth another if and only if one can be obtained from the other by a sequence of transformations of the type described in Lemma 1.
Page 21 - ... if two segments intersect in a single point, which is an endpoint of both, then their union is a segment.
Page 3 - A of a topological space X is said to be dense in X if A = X, Exercise 2.12.
Page 74 - The action of G on X induces an action of G on the group of measurable functions on X of absolute value 1.
Page 3 - Nowadays this completeness property is usually expressed by saying that a non-empty subset of R which is bounded above has a least upper bound, equivalently a non-empty subset of 1 which is bounded below has a greatest lower bound.
Page 42 - X be a continuous map. If x = a(a) and y = a(b), then [x,y] C Im(a). Proof. Let A = Im(a). Since A is a closed subset of X (being compact), it is enough to show that every point of [x, y] is within distance e of A, for all real e >0.
Page vi - Morgan [42], which confines attention to R-trees. However, the most interesting results in this paper concern spaces of actions on R-trees, motivated by the fact that the compactification by Morgan and Shalen mentioned previously involves points in this space. They show that, if G is a finitely generated group, then the space PLF(G) of projectivised nonzero hyperbolic length functions for actions of G on R-trees is itself compact.
Bibliographic information
| Title | Introduction to Lambda Trees |
| Author | Ian Chiswell |
| Publisher | World Scientific, 2001 |
| ISBN | 9812810536, 9789812810533 |
| Length | 327 pages |
| Export Citation | BiBTeX EndNote RefMan |