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

TL;DR

This paper explores the asymptotic geometry of Teichmüller space through the Gromov product, characterizing transformations at infinity and revealing rigidity properties of its quasi-isometries.

Contribution

It introduces a qualitative approach to quasi-isometries in hyperbolic spaces, linking them to simplicial automorphisms and establishing rigidity results for Teichmüller space.

Findings

Transformations at infinity are characterized by simplicial automorphisms.

Teichmüller space admits no (quasi)-invertible rough-homothety.

A quotient semigroup of transformations is identified with automorphisms of the curve complex.

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.