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Suppose the AdS$_3$ metric is given by $$ ds^2 =d\rho^2+cosh^2\rho d\psi^2 +sinh^2 \rho d\phi^2 $$

what is the n-sheeted space of it? Can the n-sheeted BTZ be constructed from it by identifications as $n=1$ case?

Thanks in advance.

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