In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.

Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff dimension $\delta(\Lambda) > \frac{1}{2}$ ...

*Are these necessarily Fuchsian groups?*

I would like to know what it means for such a group to have a Hausdorff dimension and whether any examples which have known $\delta$ (larger or smaller than $ \frac{1}{2}$). Guessing here we identify these $2 \times 2$ matrices as fractional linear transformations in $PSL(2, \mathbb{Z})$ acting on $\mathbb{H}$.

With some searching I found that $SL(2, \mathbb{Z})$ is a Fuchsian group itself with a limit set of $\mathbb{R}$. This seems to be related to the fact that all real numbers have continued fraction expansions.

Apparently there are subgroups $\Lambda \subset SL(2, \mathbb{Z}) $ where the limit set is smaller than $\mathbb{R}$ - in fact a Cantor set. What are some known examples where this is the case?

Here are notes handling the case where the limit set has $0,1,2$ points. Referring is to Svetlana Katok's book for the others. The notes claim:

$$ z \mapsto 4z, z \mapsto \frac{7z-6}{3z-2} $$

is a Fuchsian group with a cantor set as a limit set. Now I am worried these are not subgroups of $SL(2,\mathbb{Z})$, but $PSL(2,\mathbb{Z})$.

Are there any limit sets with Hausdorff dimension known to be on either side of $\frac{1}{2}$?