Bounding geometrically of Klein’s quartic

Determine whether Klein’s quartic surface of genus 3 occurs as the unique totally geodesic boundary component of a compact hyperbolic 3-manifold, i.e., ascertain whether Klein’s quartic bounds geometrically.

Background

The paper studies when closed hyperbolic surfaces, possibly with finite group actions, embed geodesically or bound geometrically in hyperbolic 3-manifolds. A surface S (or a pair (S,G)) bounds geometrically if S is the unique totally geodesic boundary component of a compact hyperbolic 3-manifold M (with the G-action extending to M when applicable).

Klein’s quartic is the genus-3 Hurwitz surface with automorphism group of order 168. It is known that the smallest Hurwitz action on Klein’s quartic does not bound any compact 3-manifold with a unique boundary component. Nevertheless, it remains unresolved whether the surface itself bounds geometrically (i.e., as a unique totally geodesic boundary of some compact hyperbolic 3-manifold), even though every Hurwitz action embeds geodesically.