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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 $M^{-1}M' \\in \\mathcal{E}$.\n  The present paper studies several Markov chains related to the above structure. 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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 $M^{-1}M' \\in \\mathcal{E}$.\n  The present paper studies several Markov chains related to the above structure. 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