Complete Trees for Groups with a Real-Valued Length Function

Abstract

Following Lyndon's axiomatic treatment of Nielsen's cancellation arguments for free products [5], I. M. Chiswell showed the equivalence of the Bass-Serre theory of group actions (without inversions) on a tree and integer-valued length functions on a group [2]. In the process Chiswell defined, for a group with a real-valued length function, a contractible metric space X on which there is an action of the group. For integer-valued length functions, X\s a tree in the ordinary simplicial sense. The results of our first section show that X, the metric completion of X, has the following treelike properties:(i) any two points of X are the endpoints of the image of an isometric embedding of a closed interval s u,

Covering R-trees, R-free groups, and dendrites

“…The term "R-tree" was coined by Morgan and Shalen [33] in 1984 to describe a type of space that was first defined by Tits [36] in 1977. In the last three decades R-trees have played a prominent role in topology, geometry, and geometric group theory (see, for example, [2,9,13,33,22]). They are the most simple of geodesic spaces, and yet Theorem 1 shows that every length space, no matter how complex, is an orbit space of an R-tree.…”

“…If X is also separable, in particular compact, then ind(X) = Ind(X) = dim(X) (see [23]). The above theorem and the fact that a (non-trivial) R-tree X is simply connected with ind(X) = 1 [2,31,5] give us:…”

“…In fact, let us take first an increasing integer sequence {i(n), n ∈ N} so that s i(n) < 1 2 n . One can define a path that wraps one of two ways around C i (1) , then one of two ways around C i (2) , and so on. Then reverse the parameterization, so that C i(1) is wrapped around last.…”

Connected components of the compactification of representation spaces of surface groups

“…C1, so that lim n!C1 d.h n / D C1; hence there exists an accumulation point .T; h 1 / 2 @m u g .3/ of this sequence of representations. We claim that no orientation on T is preserved by 1 Note that when A; B are hyperbolic and do not have any common fixed points in @H 2 , then the repulsive fixed point of AB n , as n ! C1, converges to the one of B , whereas the attractive fixed point of AB n converges to the image, by A, of the one of B .…”

A small unstable action on a tree

“…The theory of groups acting on R-trees was initiated by Tits [15] and by Alperin and Moss [1]. It was developed by Morgan and Shalen [10,11] and others.…”