Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T20:42:03.259Z Has data issue: false hasContentIssue false

17 - A cutpoint tree for a continuum

Published online by Cambridge University Press:  06 July 2010

Michael Atkinson
Affiliation:
University of St Andrews, Scotland
Nick Gilbert
Affiliation:
Heriot-Watt University, Edinburgh
James Howie
Affiliation:
Heriot-Watt University, Edinburgh
Steve Linton
Affiliation:
University of St Andrews, Scotland
Edmund Robertson
Affiliation:
University of St Andrews, Scotland
Get access

Summary

Abstract. Given a Hausdorff continuum X and a set of cut points C of X, we construct a “tree” TC 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.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×