Embedding closed totally geodesic surfaces in Bianchi orbifolds

Abstract

We study embedding of closed totally geodesic hyperbolic 2-orbifolds in the Bianchi orbifolds ${\mathbb{H}}^3/PSL(2,{\mathcal{O}}_d)$. Our main result shows that there is a constant $c$ such that for $d$ large enough there are at least $cd$ closed embedded totally geodesic hyperbolic 2-orbifolds. Moreover we provide a list which conjecturally consists of those $d$ for which ${\mathbb{H}}^3/PSL(2,{\mathcal{O}}_d)$ does not contain a closed embedded totally geodesic hyperbolic 2-orbifold.

22 pages