Volume 21, issue 6 (2017)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 533–1073
Issue 1, 1–532

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
Thurston norm via Fox calculus

Stefan Friedl, Kevin Schreve and Stephan Tillmann

Geometry & Topology 21 (2017) 3759–3784

In 1976 Thurston associated to a 3–manifold N a marked polytope in H1(N; ), which measures the minimal complexity of surfaces representing homology classes and determines all fibered classes in H1(N; ). Recently the first and third authors associated to a presentation π with two generators and one relator a marked polytope in H1(π; ) and showed that it determines the Bieri–Neumann–Strebel invariant of π. We show that if the fundamental group of a 3–manifold N admits such a presentation π, then the corresponding marked polytopes in H1(N; ) = H1(π; ) agree.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Thurston norm, $3$–manifold, Novikov ring, Fox calculus
Mathematical Subject Classification 2010
Primary: 20J05, 57M05, 57M27, 57R19
Received: 5 June 2016
Revised: 25 October 2016
Accepted: 29 November 2016
Published: 31 August 2017
Proposed: Ian Agol
Seconded: Peter Teichner, András I. Stipsicz
Stefan Friedl
Fakultät für Mathematik
Universität Regensburg
Kevin Schreve
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Stephan Tillmann
School of Mathematics and Statistics
The University of Sydney
Sydney, NSW