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arXiv · 1306.1424

Geometry of the Gromov product : Geometry at infinity of Teichm\"uller space

Hideki Miyachi
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Abstract

This paper is devoted to study of 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 Teichm\"uller 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 Teichm\"uller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichm\"uller space does not admit (quasi)-invertible rough-homothety.

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Hideki Miyachi. 2013-06-06. Geometry of the Gromov product : Geometry at infinity of Teichm\"uller space. https://arxiv.org/abs/1306.1424

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