Teichmüller's problem for Gromov hyperbolic domains

Let $\mathcal{T}_K(D)$ be the class of $K$-quasiconformal automorphisms of a domain $D\subsetneq \mathbb{R}^n$ with identity boundary values. Teichmüller'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 Teichmüller's problem for Gromov hyperbolic domains in $\mathbb{R}^n$ with identity values at the boundary of infinity. As applications, we obtain results on Teichmüller's problem for $ψ$-uniform domains and inner uniform domains in $\mathbb{R}^n$.