Some cusp-transitive hyperbolic 4-manifolds

Abstract

We realize 4 of the 6 closed orientable flat 3-manifolds as a cusp section of an orientable finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps.

Figures (21)

• Figure 1: The link $L_0$ (left) and the link $L_1$ (right).
• Figure 2: The facets of $L_0$ (top) and the facets of $L_1$ (bottom).
• Figure 5: The link $L_7$ (top-left), the $3$-torus (top-right), the $\frac{1}{2}$-twist manifold (bottom-left), the $\frac{1}{4}$-twist manifold (bottom-right). In the last three pictures, if two opposite facets do not have a letter inside, we glue them with a translation, otherwise we glue them as indicated with the letters.
• Figure 6: The link $L_7$ and the fixed planes of the reflections used to define $R_T,R_{\frac{1}{2}},R_{\frac{1}{4}}$.
• Figure 7: The link $L_8$ (left) and the Hantzsche-Wendt manifold (right), with the same notation of Figure \ref{['f14']}. Moreover, we see how the facet $6_{4,5}$ is divided in the two parts $6_{4,5}^U$ and $6_{4,5}^D$. Similarly the facet $6_{6_5,4,5}$ is divided in the two parts $6_{6_5,4,5}^U$ and $6_{6_5,4,5}^D$

Theorems & Definitions (51)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• proof
• Proposition 3.1
• proof
• Definition 3.2