A small cusped hyperbolic 4-manifold

Abstract

By gluing some copies of a polytope of Kerckhoff and Storm's, we build the smallest known orientable hyperbolic 4-manifold that is not commensurable with the ideal 24-cell or the ideal rectified simplex. It is cusped and arithemtic, and has twice the minimal volume.

Figures (5)

• Figure 1: The Coxeter diagram of the reflection group $\Gamma$ of $Q$. If two nodes are joined by a thin, thick, or dashed edge, then the two corresponding bounding hyperplanes meet with angle $\pi/3$, are tangent at infinity, are ultraparallel, respectively. There is no edge joining two nodes if the corresponding hyperplanes are orthogonal.
• Figure 2: The top, upper, and central facets $E_4$, $H_4$, and $C_{12}$ of $P$. The ideal vertices are drawn as white dots, while the top vertex $V$ and the type-2 and type-3 vertices are in blue, green and black, respectively. The black, red, and yellow edges have dihedral angle $\pi/2$, $2\pi/3$, and $\mathrm{arcos}(-1/3)$, respectively. The red pentagons are $2\pi/3$-angled ridges of $P$, while the other 2-faces in the picture are right angled.
• Figure 3: The vertex links of $P$ up to symmetry: three spherical tetrahedra for the finite vertices (from left to right, of type 1, 2 and 3), and a Euclidean parallelepiped for the ideal vertices. The black edges are right angled, while the red ones are $2\pi/3$ angled.
• Figure 4: The complete graph $K_5$, with its nodes and half-edges labelled to remind the "more abstract" construction of $X$ and $\tilde{P}$, which follows here. To build $X$ (resp. $\tilde{P}$), we glue the extremal (resp. top) facets of 5 abstract copies $P_0, \ldots, P_4$ of $P$ as follows. Consider the edge of $K_5$ joining the nodes $i$ and $j$, where $i < j$. If $i = 0$ (blue edge), glue the facets $E_j$ and $E'_j$ (resp. the facet $E_j$) of $P_0$ to the corresponding facets of $P_j$ via the map induced by the identity of $P$. If instead $i \neq 0$ (black edge), glue the facets $E_j$ and $E'_j$ (resp. the facet $E_j$) of $P_i$ to the facets $E_i$ and $E'_i$ (resp. the facet $E_i$) of $P_j$ via the map induced by the reflection $(ij) \in \mathfrak{S}_4 = G_I$ of $P$.
• Figure 5: A schematic picture of the "more concrete" construction of $X$ from $\tilde{P} = P \cup P_1 \cup \ldots \cup P_4 \subset {\mathbb{H}}^4$.

Theorems & Definitions (12)

• Theorem
• Proposition 1.1
• proof
• Lemma 1.2
• proof
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3