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

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
Fibrations of $3$–manifolds and asymptotic translation length in the arc complex

Balázs Strenner

Algebraic & Geometric Topology 23 (2023) 4087–4142

Given a 3–manifold M fibering over the circle, we investigate how the asymptotic translation lengths of pseudo-Anosov monodromies in the arc complex vary as we vary the fibration. We formalize this problem by defining normalized asymptotic translation length functions μd for every integer d 1 on the rational points of a fibered face of the unit ball of the Thurston norm on H1(M; ). We show that, even though the functions μd themselves are typically nowhere continuous, the sets of accumulation points of their graphs on d–dimensional slices of the fibered face are rather nice and in a way reminiscent of Fried’s convex and continuous normalized entropy function. We also show that these sets of accumulation points depend only on the shape of the corresponding slice. We obtain a particularly concrete description of these sets when the slice is a simplex. We also compute μ1 at infinitely many points for the mapping torus of the simplest pseudo-Anosov braid to show that the values of μ1 are rather arbitrary. This suggests that giving a formula for the functions μd seems very difficult even in the simplest cases.

3–manifold, fibrations, veering triangulation
Mathematical Subject Classification
Primary: 57M10, 57M50, 57M60
Secondary: 11H06, 11P21
Received: 14 March 2021
Revised: 30 January 2022
Accepted: 22 February 2022
Published: 23 November 2023
Balázs Strenner
Department of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States

Open Access made possible by participating institutions via Subscribe to Open.