# references for the quotient of a product of hyperbolic 2- and 3- spaces by $SL_2$ over a number ring

Let $F$ be a number field of degree $r+2s$ with ring of integers $\mathcal{O}_F$, $G=SL_2(F\otimes_{\mathbb{Q}}\mathbb{R})\cong SL_2(\mathbb{R})^r\times SL_2(\mathbb{C})^s$, $K=SO_2(\mathbb{R})^r\times SU_2(\mathbb{C})^s\subseteq G$, and $\Gamma=SL_2(\mathcal{O}_F)$.

Is there a name for the locally symmetric space $X=\Gamma\backslash G/K\cong\Gamma\backslash(\mathbb{H}^2)^r\times(\mathbb{H}^3)^s$? When $F$ is real quadratic (or more generally totally real) these go by the name of Hilbert or Hilbert-Blumenthal modular surfaces (varieties). When $F$ has complex places, these seem to be less well-studied (probably because they lack complex structure).

In any case, I'm looking for references on the geometry of $X$ (especially any discussion of compact totally geodesic subspaces).

[Edit: For instance, when $F$ has exactly one place, i.e. $F=\mathbb{Q}$ or $F=\mathbb{Q}(\sqrt{-d})$, $d>0$, we get the modular surface and the Bianchi orbifolds respectively. The closed geodesics on the modular surface and in the Bianchi orbifolds are associated to anisotropic indefinite binary quadratic forms, and the closed geodesic surfaces in the Bianchi orbifolds are associated to anisotropic indefinite binary Hermitian forms.]

## 1 Answer

When $r=0$ and $s=1$ the groups are known as Bianchi groups and the quotient space $X$ is a Bianchi manifold.

Perhaps as a class the spaces you are asking about are loosely called "arithmetic manifolds", but I'm not sure.

You might look up the references on that wikipedia page linked above, also the book by Dave Witte-Morris entitled "Introduction to arithmetic groups", although perhaps those references concentrate more on the groups than on the quotient spaces.