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Fangting Zheng

Publications and source records attributed to Fangting Zheng.

8 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 $2π/p$. 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 $Π_p$. Dimension five is optimal, since $Π_p$ does not embed in $\mathbb S^3$. For $p=2,3$, the constructions are analytic and yield cyclically symmetric infinite families.

math.GT↗
arXiv2026

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↗
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↗
arXiv2020

Algebraic fibrations of certain hyperbolic 4-manifolds

Algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let $M(\mathcal{P})$ and $M(\mathcal{E})$ be the cusped and compact hyperbolic real moment-angled manifolds associated to the hyperbolic right-angled 24-cell $\mathcal{P}$ and the hyperbolic right-angled 120-cell $\mathcal{E}$, respectively. Jankiewicz-Norin-Wise showed in [13] that $π_1(M(\mathcal{P}))$ and $π_1(M(\mathcal{E}))$ are algebraic fibered. Namely, there are two exact sequences $$1\rightarrow H_{\mathcal{P}}\rightarrow π_1(M(\mathcal{P}))\xrightarrow{ϕ_{\mathcal{P}}} \mathbb{Z}\rightarrow 1,$$ $$1\rightarrow H_{\mathcal{E}}\rightarrow π_1(M(\mathcal{E}))\xrightarrow{ϕ_{\mathcal{E}}} \mathbb{Z}\rightarrow 1,$$ where $H_{\mathcal{P}}$ and $H_{\mathcal{E}}$ are finitely generated. In this paper, we furtherly show that the groups $H_{\mathcal{P}}$ and $H_{\mathcal{E}}$ are not $FP_2$. In particular, those fiber-kernel groups are finitely generated, but not finitely presented.

math.GT↗
arXiv2018

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 $β^1(M)=k$ 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].

math.GT↗