Virtual homological spectral radii for automorphisms of surfaces

Yi Liu International Center for Mathematical Research (BICMR), Beijing (Peking) University, Beijing, PEOPLES REPUBLIC OF CHINA

Dynamical Systems Geometric Analysis and Geometric Topology mathscidoc:2203.11004

Journal of the Americal Mathematical Society, 33, (4), 1167-1227, 2020.10
Let S be a compact, orientable surface of hyperbolic type and let ψ be a mapping class of S. The present article shows that there exists a finite cover X of S and a lift ψ_X of ψ to X such that the spectral radius of ψ_X, viewed as an automorphism of H_1(X,C), is greater than one if and only if ψ has infinite order and is not a Dehn multitwist. Among the corollaries of this result is the fact that a compact, irreducible 3-manifold with toroidal or empty boundary and positive simplicial volume admits a finite cover for which the multivariable Alexander polynomial is not identically zero and has Mahler measure strictly greater than one; such a manifold also admits a finite cover whose integral first homology has nontrivial torsion. An independent proof of the main result of this paper was given by A. Hadari [Geom. Topol. 24 (2020), no. 4, 1717–1750; MR4173920]. In that paper, the surface was assumed to have at least one boundary component, but stronger control over the required finite covers was obtained.
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  title={Virtual homological spectral radii for automorphisms of surfaces},
  author={Yi Liu},
  booktitle={Journal of the Americal Mathematical Society},
Yi Liu. Virtual homological spectral radii for automorphisms of surfaces. 2020. Vol. 33. In Journal of the Americal Mathematical Society. pp.1167-1227.
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