17 - A cutpoint tree for a continuum

Summary

Abstract. Given a Hausdorff continuum X and a set of cut points C of X, we construct a “tree” T ⊃ C and a relation between X and T which preserves the separation properties of elements of C. We then give an application of this result which simplifies the proof of the cut point conjecture for negatively curved groups.

INTRODUCTION

The cut point conjecture for negatively curved groups has been proven. The pieces of the proof appear in, culminating in. The steps in the proof are as follows:

1. Start with the convergence action of the negatively curved group G on the continuum X = ∂G, and assume that X has a cutpoint.

2. Construct an ℝ-tree R on which R acts in a non-nesting stable and virtually cyclic fashion.

3. Construct from R, an ℝ-tree T on which G acts by isometries.

4. Apply the Rips machine to this action to obtain a contradiction.

This paper provides a short easy version of step 2. The first written proof of step 2 appeared in. The approach in this paper was first developed while the author was at Michigan Tech. in the winter of 94. The author was however unable to complete step 3, and so gave it up without ever writing up step 2. The present treatment was developed as part of a presentation of the cut point theorem in a graduate class at the university of Wisconsin, Milwaukee. The author was attempting to present the contents of when it became apparent that it was simply too long to present in the allotted time.