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Jiming Ma

Publications and source records attributed to Jiming Ma.

At least 19 recordsLinked to original sources
arXiv2026

Discrete embeddings of hyperbolic groups with Pontryagin-surface boundaries

Let $X_p$ be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2\pi/p$. For every $2\leq p\leq8$, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to $X_p$ that admits a discrete, faithful, convex cocompact reflection representation into Isom$(\mathbf H^5)$, whose limit set is homeomorphic to the index-$p$ Pontryagin surface $\Pi_p$. Dimension five is optimal, since $\Pi_p$ does not embed in $\mathbb S^3$. For $p=2,3$, the constructions are analytic and yield cyclically symmetric infinite families.

math.GT↗
arXiv2026

Spherical CR uniformizations of a sequence of hyperbolic 3-manifolds

Let $s782$ be the 2-cusped hyperbolic 3-manifold in the SnapPy census. Its spherical CR uniformization was established in \cite{JWX2023} using the Ford domain of the complex hyperbolic triangle group $\Delta_{4,4,\infty;\infty}$. By comparing the combinatorial structures of the Ford domain of $\Delta_{4,4,\infty;\infty}$ and the Dirichlet domain of $\Delta_{4,4,n;\infty}$, we prove that for each $n \geq 5$, the Dehn filling of $s782$ along the slope $(n-1)\mathcal{m}_1 + \mathcal{l}_1$ on its second cusp admits a spherical CR uniformization, where $(\mathcal{m}_1, \mathcal{l}_1)$ denotes the meridian-longitude system of a cusp in SnapPy notation.

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arXiv2026

Several families of incommensurable noncompact hyperbolic Coxeter polytopes

We classify all 141 finite-volume hyperbolic Coxeter five-dimensional polytopes with eight facets, of which 125 are noncompact. Using maximal-cusp density and a noncompact analog of Bogachev-Douba-Raimbault's argument, we construct infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4, 5, 6, 7, and 9, with the number of commensurability classes growing at least exponentially in volume.

math.GT↗
arXiv2024

Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold

Let $$G_{6,3}=\langle a_0, \cdots, a_5| a_{i}^{3}=id, a_{i} a_{i+1}= a_{i+1} a_{i}, i \in \mathbb{Z}/6\mathbb{Z}\rangle$$ be a hyperbolic group with boundary the Menger curve. J. Granier \cite{Granier} constructed a discrete, convex cocompact and faithful representation $ρ$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $ρ(G_{6,3})$ is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the $\mathbb{Z}_3$-coned chain-link $C(6,-2)$. This answers the second part of Misha Kapovich's Conjecture 10.6\cite{Kapovich}.

math.GT↗
arXiv2024

A note on the finitely generated fixed subgroup property

We study when a group of form $G\times\mathbb{Z}^m (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\rm{FGFP}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples.

math.GR↗
arXiv2024

Complex Hyperbolic Geometry of Chain Links

The complex hyperbolic triangle group $Γ=Δ_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane ${\bf H}^2_{\mathbb C}$ is generated by complex reflections $I_1$, $I_2$, $I_3$ such that the product $I_2I_3$ has order four, the products $I_3I_1$, $I_1I_2$ are parabolic and the product $I_1I_3I_2I_3$ is an accidental parabolic element. Unexpectedly, the product $I_1I_2I_3I_2$ is a hidden accidental parabolic element. We show that the 3-manifold at infinity of $Δ_{4,\infty,\infty;\infty}$ is the complement of the chain link $8^4_1$ in the 3-sphere. In particular, the quartic cusped hyperbolic 3-manifold $S^3-8^4_1$ admits a spherical CR-uniformization. The proof relies on a new technique to show that the ideal boundary of the Ford domain is an infinite-genus handlebody. Motivated by this result and supported by the previous studies of various authors, we conjecture that the chain link $C_p$ is an ancestor of the 3-manifold at infinity of the critical complex hyperbolic triangle group $Δ_{p,q,r;\infty}$, for $3 \leq p \leq 9$.

math.GT↗
arXiv2023

Figure-eight knot is always over there

It is well-known that complex hyperbolic triangle groups $Δ(3,3,4)$ generated by three complex reflections $I_1,I_2,I_3$ in $\mbox{PU(2,1)}$ has 1-dimensional moduli space. Deforming the representations from the classical $\mathbb{R}$-Fuchsian one to $Δ(3,3,4; \infty)$, that is, when $I_3I_2I_1I_2$ is accidental parabolic, the 3-manifolds at infinity change from a Seifert 3-manifold to the figure-eight knot complement. When $I_3I_2I_1I_2$ is loxodromic, there is an open set $Ω\subset \partial\mathbf H^{2}_{\mathbb C}=\mathbb S^3$ associated to $I_3I_2I_1I_2$, which is a subset of the discontinuous region. We show the quotient space $Ω/ Δ(3,3,4)$ is always the figure-eight knot complement in the deformation process. This gives the topological/geometrical explain that the 3-manifold at infinity of $Δ(3,3,4; \infty)$ is the figure-eight knot complement. In particular, this confirms a conjecture of Falbel-Guilloux-Will.

math.GT↗
arXiv2023

Spherical CR uniformization of the magic 3-manifold

We show the 3-manifold at infinity of the complex hyperbolic triangle group $Δ_{3,\infty,\infty;\infty}$ is the three-cusped "magic" 3-manifold $6_1^3$. We also show the 3-manifold at infinity of the complex hyperbolic triangle group $Δ_{3,4,\infty;\infty}$ is the two-cusped 3-manifold $m295$ in the Snappy Census, which is a 3-manifold obtained by Dehn filling on one cusp of $6_1^3$. In particular, hyperbolic 3-manifolds $6_1^3$ and $m295$ admit spherical CR uniformizations. These results support our conjecture that the 3-manifold at infinity of the complex hyperbolic triangle group $Δ_{3,n,m;\infty}$ is the one-cusped hyperbolic 3-manifold from the "magic" $6_1^3$ via Dehn fillings with filling slopes $(n-2)$ and $(m-2)$ on the first two cusps of it.

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arXiv2023

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 $\mathcal{M}(0,\frac{2π}{3},\frac{4π}{3})$, $\mathcal{M}(\frac{2π}{3},\frac{4π}{3},\frac{4π}{3})$ and some isolated points. This is the first Fuchsian group such that its $\mathbf{PU}(3,1)$-representations space has been entirely constructed. Both $\mathcal{M}(0,\frac{2π}{3},\frac{4π}{3})$ and $\mathcal{M}(\frac{2π}{3},\frac{4π}{3},\frac{4π}{3})$ 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 $\mathcal{M}(0,\frac{2π}{3},\frac{4π}{3} )$ and $\mathcal{M}(\frac{2π}{3},\frac{4π}{3},\frac{4π}{3})$ interpolate the geometries studied in \cite{FalbelKoseleff:2002} and \cite{Falbelparker:2003}.

math.GT↗
arXiv2023

Three-dimensional complex reflection groups via Ford domains

We initiate the study of deformations of groups in three-dimensional complex hyperbolic geometry. Let $$G=\left\langle ι_1, ι_2, ι_3, ι_4 \Bigg| \begin{array}{c} ι_1^2= ι_2^2 = ι_3^2=ι_4^2=id,\\ (ι_1 ι_3)^{2}=(ι_1 ι_4)^{3}=(ι_2 ι_4)^{2}=id \end{array}\right\rangle$$ be an abstract group. We study representations $ρ: G \rightarrow \mathbf{PU}(3,1)$, where $ρ( ι_{i})=I_{i}$ 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 $(h,t) \in [1, \infty) \times [0, π]$ 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 $ρ_{(\sqrt{2},\operatorname{arccos}(-\frac{7}{8}))}(G)$ as a guide, we show $ρ_{(h,t)}$ 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.

math.GT↗
arXiv2023

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 $$G=\left\langle ι_1, ι_2, ι_3, ι_4 \Bigg| \begin{array} {c} ι_1^2= ι_2^2 = ι_3^2=ι_4^2=id, \\ (ι_1 ι_3)^{2}=(ι_2 ι_4)^{2}=id \end{array}\right\rangle$$ as an abstract group. We study type-preserving representations $ρ: G \rightarrow \mathbf{PU}(3,1)$, where $ρ( ι_{i})=I_{i}$ 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 $θ\in [\frac{5 π}{6}, π]$. In particular, $θ=\frac{5 π}{6}$ and $θ=π$ degenerate to ${\bf H}^{2}_{\mathbb C}$-geometry and ${\bf H}^{3}_{\mathbb R}$-geometry respectively. Via Dirichlet domains, we show $ρ=ρ_θ$ is a discrete and faithful representation of the group $G$ for all $θ\in [\frac{5 π}{6}, π]$. This is the first nontrivial moduli space in three dimensional complex hyperbolic space that has been studied completely.

math.GT↗
arXiv2023

On profinite rigidity of 4-dimensional Seifert manifolds

There are many results showing the connection and phenomenon between some low-dimensional manifolds with the profinite completions of their fundamental groups. We focus on some Seifert 4-manifolds about the extent of their profinite completion to detect one, giving classification of monodromies and conditions for them to be profinitely rigid.

math.GT↗
arXiv2022

Orientable hyperbolic 4-manifolds over the 120-cell

Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34π^2}{3}\cdot 16$ by using the small cover theory. In particular, we classify all of the orientable four-dimensional small covers over the right-angled 120-cell up to homeomorphism; these are all with even intersection forms.

math.GT↗
arXiv2022

Three-manifolds at infinity of complex hyperbolic orbifolds

We show the manifolds at infinity of the complex hyperbolic triangle groups $Δ_{3,4,4;\infty}$ and $Δ_{3,4,6;\infty}$,are one-cusped hyperbolic 3-manifolds $m038$ and $s090$ in the Snappy Census respectively.That is,these two manifolds admit spherical CR uniformizations. Moreover, these two hyperbolic 3-manifolds above can be obtained by Dehn surgeries on the first cusp of the two-cusped hyperbolic 3-manifold $m295$ in the Snappy Census with slopes $2$ and $4$ respectively. In general,the main result in this paper allow us to conjecture that the manifold at infinity of the complex hyperbolic triangle group $Δ_{3,4,n;\infty}$ is the one-cusped hyperbolic 3-manifold obtained by Dehn surgery on the first cusp of $m295$ with slope $n-2$.

math.GT↗