Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn't be upset if we just took $G=\operatorname{O}(1,n)$ and $K = \operatorname{O}(n)$ so that $M$ is a finite volume hyperbolic orbifold.

Edit: as pointed out by @MoisheKohan in the comments, I really shouldn't be asking about the real hyperbolic setting. Let's instead take $G=\operatorname{U}(1,n)$ and $K = \operatorname{U}(n)$ so that $M$ is a finite volume *complex* hyperbolic orbifold.

Now suppose that $M' \subset M$ is a connected component of an orbifold locus. I believe it follows that $M'$ is also a lattice quotient (of smaller dimension), and feel free to correct me if I'm wrong about that. Now what can be said about the relationship between the arithmeticity of $M$ and $M'$? E.g. does the (non)arithmeticity of one imply the (non)arithmeticity of the other?

5more comments