  Consider $G=SL_2(\mathbb{Z})/\{\pm I\}$ acting on the complex upper half
plane $H$ by $h_M(z)=\frac{az+b}{cz+d},$ for $M \in G$. Let $D=\{z \in H:
|z|\geq 1, |\Re(z)|\leq 1/2\}$. We consider the set $\mathcal{E} \subset G$
with the $9$ elements $M$, different from the identity, such that $(MM^T)\leq
3$. We equip the tiling of $H$ defined by $\mathbb{D}=\{h_M(D), M \in G\}$ with
a graph structure where the neighbours are defined by $h_M(D) \cap h_{M'}(D)
\neq \emptyset$, equivalently $M^{-1}M' \in \mathcal{E}$.
  The present paper studies several Markov chains related to the above
structure. We show that the simple random walk on the above graph converges
a.s. to a point $X$ of the real line with the same distribution of $S_2
W^{S_1}$, where $S_1,S_2,W$ are independent with $\Pr (S_i=\pm 1)=1/2$ and
where $W$ is valued in $(0,1)$ with distribution $\Pr(W<w)=?(w)$. Here $?$ is
the Minkowski function. If $K_1, K_2, \ldots$ are i.i.d with distribution $\Pr
(K_i=n)= 1/2^n$ for $n=1,2,\ldots$, then $W= \frac{1}{K_1+\frac
{1}{K_2+\ldots}}$: this known result (Isola (2014)) is derived again here.
