Geometrically bounding 3-manifold, volume and Betti number

It is well known that an arbitrary closed orientable $3$-manifold can be realized as the unique boundary of a compact orientable $4$-manifold, that is, any closed orientable $3$-manifold is cobordant to zero. In this paper, we consider the geometric cobordism problem: a hyperbolic $3$-manifold is geometrically bounding if it is the only boundary of a totally geodesic hyperbolic 4-manifold. However, there are very rare geometrically bounding closed hyperbolic 3-manifolds according to the previous research [11,13]. Let $v \approx 4.3062\ldots$ be the volume of the regular right-angled hyperbolic dodecahedron in $\mathbb{H}^{3}$, for each $n \in \mathbb{Z}_{+}$ and each odd integer $k$ in $[1,5n+3]$, we construct a closed hyperbolic 3-manifold $M$ with $\beta^1(M)=k$ and $vol(M)=16nv$ that bounds a totally geodesic hyperbolic 4-manifold. The proof uses small cover theory over a sequence of linearly-glued dodecahedra and some results of Kolpakov-Martelli-Tschantz [9].Comment: the latest version that adjust some figures and add more detail description

The moduli space of the modular group in three-dimensional complex hyperbolic geometry

We study the moduli space of discrete, faithful, type-preserving representations of the modular group $\mathbf{PSL}(2,\mathbb{Z})$ into $\mathbf{PU}(3,1)$. The entire moduli space $\mathcal{M}$ is a union of $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$, $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ and some isolated points. This is the first Fuchsian group such that its $\mathbf{PU}(3,1)$-representations space has been entirely constructed. Both $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3})$ and $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ are parameterized by a square, where two opposite sides of the square correspond to representations of $\mathbf{PSL}(2,\mathbb{Z})$ into the smaller group $\mathbf{PU}(2,1)$. In particular, both sub moduli spaces $\mathcal{M}(0,\frac{2\pi}{3},\frac{4\pi}{3} )$ and $\mathcal{M}(\frac{2\pi}{3},\frac{4\pi}{3},\frac{4\pi}{3})$ interpolate the geometries studied in \cite{FalbelKoseleff:2002} and \cite{Falbelparker:2003}

Three-dimensional complex reflection groups via Ford domains

We initiate the study of deformations of groups in three-dimensional complex hyperbolic geometry. Let $$G=\left\langle \iota_1, \iota_2, \iota_3, \iota_4 \Bigg| \begin{array}{c} \iota_1^2= \iota_2^2 = \iota_3^2=\iota_4^2=id,\\ (\iota_1 \iota_3)^{2}=(\iota_1 \iota_4)^{3}=(\iota_2 \iota_4)^{2}=id \end{array}\right\rangle$$ be an abstract group. We study representations $\rho: G \rightarrow \mathbf{PU}(3,1)$, where $\rho( \iota_{i})=I_{i}$ is a complex reflection fixing a complex hyperbolic plane in ${\bf H}^{3}_{\mathbb C}$ for $1 \leq i \leq 4$, with the additional condition that $I_1I_2$ is parabolic. When we assume two pairs of hyper-parallel complex hyperbolic planes have the same distance, then the moduli space $\mathcal{M}$ is parameterized by $(h,t) \in [1, \infty) \times [0, \pi]$ but $t \leq \operatorname{arccos}(-\frac{3h^2+1}{4h^2})$. In particular, $t=0$ and $t=\operatorname{arccos}(-\frac{3h^2+1}{4h^2})$ degenerate to ${\bf H}^{3}_{\mathbb R}$-geometry and ${\bf H}^{2}_{\mathbb C}$-geometry respectively. Using the Ford domain of $\rho_{(\sqrt{2},\operatorname{arccos}(-\frac{7}{8}))}(G)$ as a guide, we show $\rho_{(h,t)}$ is a discrete and faithful representation of $G \rightarrow \mathbf{PU}(3,1)$ when $(h,t) \in \mathcal{M}$ is near to $(\sqrt{2}, \operatorname{arccos}(-\frac{7}{8}))$. This is the first nontrivial example of the Ford domain of a subgroup in $\mathbf{PU}(3,1)$ that has been studied

Complexification of an infinite volume Coxeter tetrahedron

Let $T$ be an infinite volume Coxeter tetrahedron in three dimensional real hyperbolic space ${\bf H}^{3}_{\mathbb R}$ with two opposite right-angles and the other angles are all zeros. Let $G$ be the Coxeter group of $T$, so $$G=\left\langle \iota_1, \iota_2, \iota_3, \iota_4 \Bigg| \begin{array} {c} \iota_1^2= \iota_2^2 = \iota_3^2=\iota_4^2=id, \\ (\iota_1 \iota_3)^{2}=(\iota_2 \iota_4)^{2}=id \end{array}\right\rangle$$ as an abstract group. We study type-preserving representations $\rho: G \rightarrow \mathbf{PU}(3,1)$, where $\rho( \iota_{i})=I_{i}$ is a complex reflection fixing a complex hyperbolic plane in three dimensional complex hyperbolic space ${\bf H}^{3}_{\mathbb C}$ for $1 \leq i \leq 4$. The moduli space $\mathcal{M}$ of these representations is parameterized by $\theta \in [\frac{5 \pi}{6}, \pi]$. In particular, $\theta=\frac{5 \pi}{6}$ and $\theta=\pi$ degenerate to ${\bf H}^{2}_{\mathbb C}$-geometry and ${\bf H}^{3}_{\mathbb R}$-geometry respectively. Via Dirichlet domains, we show $\rho=\rho_{\theta}$ is a discrete and faithful representation of the group $G$ for all $\theta \in [\frac{5 \pi}{6}, \pi]$. This is the first nontrivial moduli space in three dimensional complex hyperbolic space that has been studied completely.Comment: arXiv admin note: text overlap with arXiv:2306.1524

High distance Heegaard splittings from involutions

AbstractFixed an oriented handlebody H=H+ with boundary F, let η(H+)=H− be the mirror image of H+ along F, so η(F) is the boundary of H−, for a map f:F→F, we have a 3-manifold by gluing H+ and H− along F with attaching map f, and denote it by Mf=H+∪f:F→FH−. In this note, we show that there are involutions f:F→F which are also reducible, such that Mf have arbitrarily high Heegaard distances

Compact hyperbolic Coxeter four-polytopes with eight facets

In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.Comment: We corrected some textual typos about the cited authors/papers. Many thanks to Nikolay Bogache