Typos: (1) Factor in front of dS metric should be (R_dS)^2
(2) Red equation for hyperboloid should have minus sign in front of R_AdS^2.
Also note that R_ab = g_ab Λ is missing the proper normalization factor.
The comment below about the centering of the group being Z is slightly more subtle than I presented in my lecture
due to the option to include in the universal cover of SO(2,d) either time reflections (T)
or spatial reflections (P) or both.
The group element which translates you to the next antipodal point consists of a π rotation in the t-u plane
times a reflection of the x,y,z... coordinates. This commutes with all elements of the AdS group only if
we exclude discrete time reversal symmetry (T). Furthermore, if AdS has an odd number of spatial
dimensions, then the map to the next antipodal point is only an element of the group if we include discrete
spatial reflection symmetry (P). However, if we don't include P or T in our group, then the center is still Z,
but now the smallest element of the center is a 2π rotation in the t-u plane (to the next cover of the AdS hyperboloid).
