Geometry of the Gromov product: geometry at infinity of Teichmüller space. (English) Zbl 1378.30018

Miyachi, Hideki

J. Math. Soc. Japan 69, No. 3, 995-1049 (2017).

Summary: 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.

MSC:

30F60	Teichmüller theory for Riemann surfaces
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Keywords:

Teichmüller space; Gromov product