  Let $\mathcal{T}_K(D)$ be the class of $K$-quasiconformal automorphisms of a
domain $D\subsetneq \mathbb{R}^n$ with identity boundary values.
Teichm\"uller's problem is to determine how far a given point $x\in D$ can be
mapped under a mapping $f\in \mathcal{T}_K(D)$. We estimate this distance
between $x$ and $f(x)$ from the above by using two different metrics, the
distance ratio metric and the quasihyperbolic metric. We study
Teichm\"{u}ller's problem for Gromov hyperbolic domains in $\mathbb{R}^n$ with
identity values at the boundary of infinity. As applications, we obtain results
on Teichm\"{u}ller's problem for $\psi$-uniform domains and inner uniform
domains in $\mathbb{R}^n$.
