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

TL;DR

This paper investigates the potential for embedding a highly symmetric genus three surface, related to Klein's quartic, into hyperbolic space, expanding understanding of geometric embeddings beyond Euclidean space.

Contribution

It explores the possibility of embedding Klein's quartic or its cover into hyperbolic space, addressing limitations of previous Euclidean constructions.

Findings

Constructs a theoretical framework for embedding in hyperbolic space

Identifies conditions under which the embedding may exist

Provides insights into symmetry and geometric properties of the surface

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.