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What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$
Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...
