  In this paper, we first generalize a result of Bishop and Steger
[Representation theoretic rigidity in PSL(2, R). Acta Math., 170, (1993),
121-149] by proving that for a Fuchsian group $G$ of divergence type and
non-lattice, if $h$ is a quasi-symmetric homeomorphism of the real axis
$\mathbb{R}$ corresponding to a quasi-conformal compact deformation of $G$.
Then for any $E\subset \mathbb{R}$, we have max(dim$E$,
dim$h(\mathbb{R}\setminus E))=1$. Furthermore, we showed that Bishop and
steger's result does not hold for the covering groups of all '$d$-dimensional
jungle gym' (d is any positive integer) which generalizes G\"onye's results [
Differentiability of quasi-conformal maps on the jungle gym. Trans. Amer. Math.
Soc. Vol 359 (2007), 9-32] where the author discussed the case of
'$1$-dimensional jungle gym'.
