Define the $s$-dimensional Hausdorff measure $\mathcal{H}^{s}(F)$ of a set $F \subset \mathbb{R}^{N}$. Explain briefly how properties of this measure may be used to define the Hausdorff dimension $\operatorname{dim}_{H}(F)$ of such a set.

Prove that the limit sets of conjugate Kleinian groups have equal Hausdorff dimension. Hence, or otherwise, prove that there is no subgroup of $\mathbb{P} S L(2, \mathbb{R})$ which is conjugate in $\mathbb{P} S L(2, \mathbb{C})$ to $\mathbb{P} S L(2, \mathbb{Z} \oplus \mathbb{Z} i)$.