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I am trying to compute the group $H_1(Sl_2(\mathbb{Z}_2),M)$, where $\mathbb{Z}_2$ are $2$-adic integers and M is a module $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. I suppose that the group acts on $M$ by matrix multiplication.

I found a similar-looking computation in the paper of Dupont and Sah "Homology of Euclidean groups of motion made discrete and Euclidean scissors congruences". It was shown there that $H_1(SO_3(\mathbb{R}),\mathbb{R^3}) = \Omega^1_\mathbb{R}.$

I would be very grateful for any help with computing the group or for any interpretation of its elements.

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