Infinite $\{3,7\}$-surface in ${\mathbb{H}}^3$

Abstract

Objects with large symmetry groups have been an interest for many mathematicians. A classical question in geometry is whether a surface with certain geometric features, such as completeness, curvature, etc..., can embed in ${\mathbb{R}}^3.$ In a recent paper, Lee constructs an infinite $\{3,7\}$-surface in ${\mathbb{R}}^3$ by gluing together prisms and antiprisms, in an attempt to find a periodic surface in ${\mathbb{R}}^3$ that is cover of Klein's quartic. While Lee's construction shows that such construction self-intersects in ${\mathbb{R}}^3$, it does not prove nor disprove the possibility of an embedding. In this paper, we explore a possible embedding of the genus three Klein's quartic, or its cover, in hyperbolic space.

6 pages, 4 figures