  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.
