On hyperbolic cobweb manifolds

TL;DR

This paper constructs a specific compact hyperbolic cobweb manifold of type Cw(6,6,6), explores its geometric properties, and relates it to an infinite series of similar manifolds, expanding understanding of hyperbolic space forms.

Contribution

It introduces a new explicit construction of a hyperbolic cobweb manifold Cw(6,6,6) as part of an infinite series, based on Coxeter orthoschemes and reflection groups.

Findings

Constructed a compact hyperbolic cobweb manifold Cw(6,6,6).

Determined the maximal inscribed ball and minimal covering ball in the manifold.

Linked the specific case to an infinite series of similar hyperbolic manifolds.

Abstract

A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product of our investigations \cite{MSz16}. In that work dense ball packings and coverings of hyperbolic space $\HYP$ have been constructed on the base of complete hyperbolic Coxeter orthoschemes $\mathcal{O}=W_{uvw}$ and its extended reflection groups $\bG$ (see diagram in Fig.~3. and picture of fundamental domain in Fig.~2). Now $u=v=w=6 (=2p)$. Thus the maximal ball contained in $Cw(6,6,6)$, moreover its minimal covering bal l (so diameter) can also be determined. The algorithmic procedure provides us with the proof of our statements.