Journal of the Mathematical Society of Japan

Geometry of the Gromov product: Geometry at infinity of Teichmüller space


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This paper is devoted to studying transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a “coarsification” of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller space does not admit (quasi)-invertible rough-homothety.

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J. Math. Soc. Japan, Volume 69, Number 3 (2017), 995-1049.

First available in Project Euclid: 12 July 2017

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Primary: 30F60: Teichmüller theory [See also 32G15] 54E40: Special maps on metric spaces
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems 51M10: Hyperbolic and elliptic geometries (general) and generalizations 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds

Teichmüller space Teichmüller distance Gromov hyperbolic space Gromov product complex of curves mapping class group


MIYACHI, Hideki. Geometry of the Gromov product: Geometry at infinity of Teichmüller space. J. Math. Soc. Japan 69 (2017), no. 3, 995--1049. doi:10.2969/jmsj/06930995.

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