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Consider the third homology group of a real special linear group $H_3 (SL(2,\mathbb{R}),\mathbb{F}_p)$. It is known$[1]$ that for $p=2$ the third homology group of $SL(2,\mathbb{R})$ vanishes.

Question: is it true that there exists a prime number $p>2$ such that $H_3 (SL(2,\mathbb{R}),\mathbb{F}_p) \neq 0$?

$[1]$ Walter Parry, Chih-Han Sah, Third homology of $SL(2,\mathbb{R})$ made discrete.

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