Inheritance of arithmeticity properties in orbifold strata

Question

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?

Answer 1

Here is what I think is the correct setup:

Let $X$ be a symmetric space of noncompact type, $\Gamma$ is a lattice in the isometry group of $X$. Then $\Gamma$ has finitely many $\Gamma$-conjugacy classes of finite subgroups. Let $\Phi$ be one of these finite subgroups (I will pick one from each conjugacy class). Then $\Gamma$ fixes a symmetric subspace $Y\subset X$. The (set-wise) $\Gamma$-stabilizer $\Gamma_Y$ of $Y$ is a sublattice in $\Gamma$: It acts on $Y$ as a lattice. The quotient-orbifold $Y/\Gamma_Y$ has a natural projection $Q_Y$ to the orbifold $X/\Gamma$, the projection map $Y/\Gamma_Y\to X/\Gamma$ is an isometric immersion in the sense of orbifolds. This defines a (finite) stratification of the orbifold locus of $X/\Gamma$: The strata are $Q_Y$'s.

This picture serves a correction to the erroneous claim in the post "I believe it follows that..."

Now, one can ask meaningful questions relating arithmeticity of $\Gamma$ and the sublattices $\Gamma_Y$. If memory serves me well, arithmeticity of $\Gamma$ implies arithmeticity of each $\Gamma_Y$. However, the converse is false: It is surely false for $X={\mathbb H}^n$: One can build examples in all dimensions using the Gromov--Piatetsky-Shapiro construction. I did not check Deligne-Mostov nonarithmetic examples, but, likely, you will find some arithmetic complex one-dimensional immersed totally geodesic suborbifolds. Since you like DM-examples, you should try finding such.