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Virtual homological spectral radius and mapping torus of pseudo-Anosov maps


Author: Hongbin Sun
Journal: Proc. Amer. Math. Soc. 145 (2017), 4551-4560
MSC (2010): Primary 57M10, 57M27
DOI: https://doi.org/10.1090/proc/13564
Published electronically: May 4, 2017
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Abstract: In this note, we show that if a pseudo-Anosov map $ \phi :S\to S$ admits a finite cover whose action on the first homology has spectral radius greater than $ 1$, then the monodromy of any fibered structure of any finite cover of the mapping torus $ M_{\phi }$ has the same property.


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  • [Ag1] Ian Agol, The virtual Haken conjecture, with an appendix by Agol, Daniel Groves, and Jason Manning, Doc. Math. 18 (2013), 1045-1087. MR 3104553
  • [Ag2] Ian Agol, Virtual properties of $ 3$-manifolds, Proceedings of the International Congress of Mathematicians 1 (2014), 141-170.
  • [Bo] David W. Boyd, Kronecker's theorem and Lehmer's problem for polynomials in several variables, J. Number Theory 13 (1981), no. 1, 116-121. MR 602452, https://doi.org/10.1016/0022-314X(81)90033-0
  • [DY] Jérôme Dubois and Yoshikazu Yamaguchi, The twisted Alexander polynomial for finite abelian covers over three manifolds with boundary, Algebr. Geom. Topol. 12 (2012), no. 2, 791-804. MR 2914618, https://doi.org/10.2140/agt.2012.12.791
  • [FV] Stefan Friedl and Stefano Vidussi, A survey of twisted Alexander polynomials, The mathematics of knots, Contrib. Math. Comput. Sci., vol. 1, Springer, Heidelberg, 2011, pp. 45-94. MR 2777847, https://doi.org/10.1007/978-3-642-15637-3_3
  • [Ha] A. Hadari, Every infinite order mapping class has an infinite order action on the homology of some finite cover, available at arXiv:math.GT/1508.01555.
  • [Ko1] Thomas Koberda, Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification, Geom. Dedicata 156 (2012), 13-30. MR 2863543, https://doi.org/10.1007/s10711-011-9587-y
  • [Ko2] Thomas Koberda, Alexander varieties and largeness of finitely presented groups, J. Homotopy Relat. Struct. 9 (2014), no. 2, 513-531. MR 3258692, https://doi.org/10.1007/s40062-013-0037-4
  • [Le] Thang Le, Homology torsion growth and Mahler measure, Comment. Math. Helv. 89 (2014), no. 3, 719-757. MR 3260847, https://doi.org/10.4171/CMH/332
  • [McM1] Curtis T. McMullen, Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 4, 519-560 (English, with English and French summaries). MR 1832823, https://doi.org/10.1016/S0012-9593(00)00121-X
  • [McM2] Curtis T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 2, 153-171 (English, with English and French summaries). MR 1914929, https://doi.org/10.1016/S0012-9593(02)01086-8
  • [McM3] Curtis T. McMullen, Entropy on Riemann surfaces and the Jacobians of finite covers, Comment. Math. Helv. 88 (2013), no. 4, 953-964. MR 3134416, https://doi.org/10.4171/CMH/308
  • [SW] Daniel S. Silver and Susan G. Williams, Mahler measure, links and homology growth, Topology 41 (2002), no. 5, 979-991. MR 1923995, https://doi.org/10.1016/S0040-9383(01)00014-3
  • [Th] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. MR 956596, https://doi.org/10.1090/S0273-0979-1988-15685-6

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Additional Information

Hongbin Sun
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
Email: hongbins@math.berkeley.edu, hongbin.sun2331@gmail.com

DOI: https://doi.org/10.1090/proc/13564
Keywords: Pseudo-Anosov maps, fibered $3$-manifolds, Alexander polynomial, Mahler measure.
Received by editor(s): August 28, 2016
Received by editor(s) in revised form: September 27, 2016, October 18, 2016, and October 21, 2016
Published electronically: May 4, 2017
Additional Notes: The author was partially supported by NSF grant No. DMS-1510383.
Communicated by: David Futer
Article copyright: © Copyright 2017 American Mathematical Society

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