Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone
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by Hyungryul Baik, Dongryul M. Kim and Chenxi Wu;
Trans. Amer. Math. Soc. 378 (2025), 4739-4764
DOI: https://doi.org/10.1090/tran/9371
Published electronically: March 19, 2025

Abstract:

In this paper, we study the asymptotic translation lengths on the sphere complexes of monodromies of a manifold fibered over the circle. Given a compact mapping torus, we define a cone in the first cohomology which we call the generalized fibered cone, and show that every primitive integral element gives a fibration over the circle. Moreover, we prove that the generalized fibered cone is a rational slice of Fried’s cone, which is defined as the dual of homological directions, an analogue of Thurston’s fibered cone.

As a consequence of our description of the generalized fibered cone, we provide each proper subcone of the generalized fibered cone with a uniform upper bound for asymptotic translation lengths of monodromies on sphere complexes of fibers in the proper subcone. Our upper bound is purely in terms of the dimension of the proper subcone. We also deduce similar estimates for asymptotic translation lengths of some mapping classes on finite graphs constructed in the works of Dowdall–Kapovich–Leininger, measured on associated free-splitting complexes and free-factor complexes.

Moreover, as an application of our result, we prove that the asymptote for the minimal asymptotic translation length of the genus $g$ handlebody group on the disk complex is $1/g^2$, the same as the one on the curve complex.

References
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Bibliographic Information
  • Hyungryul Baik
  • Affiliation: Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon 34141, South Korea
  • MR Author ID: 1033130
  • ORCID: 0000-0002-3283-7345
  • Email: hrbaik@kaist.ac.kr
  • Dongryul M. Kim
  • Affiliation: Department of Mathematics, Yale University, 219 Prospect Street, New Haven, Connecticut 06511
  • MR Author ID: 1537421
  • ORCID: 0000-0001-9898-3989
  • Email: dongryul.kim@yale.edu
  • Chenxi Wu
  • Affiliation: Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706
  • MR Author ID: 1531149
  • ORCID: 0000-0001-5856-6435
  • Email: cwu367@math.wisc.edu
  • Received by editor(s): February 1, 2024
  • Received by editor(s) in revised form: August 29, 2024, October 27, 2024, and November 6, 2024
  • Published electronically: March 19, 2025
  • Additional Notes: The first author was partially supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2020R1C1C1A01006912). The third author was partially supported by Simons Collaboration Grant 850685.
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 378 (2025), 4739-4764
  • MSC (2020): Primary 37E25, 57M60, 57K20; Secondary 37E30
  • DOI: https://doi.org/10.1090/tran/9371