Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be finite?

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Brieskorn homology sphere, which in particular has trivial $H_1$. Taking $\Gamma$ to be the preimage of $\Gamma_0$ in $SL_2(\mathbb{R})$ obviously gives the same quotient. $\endgroup$1more comment