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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., Comment: 9 pages

The complex hyperbolic triangle group $\Gamma=\Delta_{4,\infty,\infty;\infty}$ 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 $\Delta_{4,\infty,\infty;\infty}$ 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 $\Delta_{p,q,r;\infty}$, for $3 \leq p \leq 9$., Comment: 30 pages, 13 figures

In this paper, we obtain a complete classification of 331 finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets.

It is well-known that complex hyperbolic triangle groups $\Delta(3,3,4)$ 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 $\Delta(3,3,4; \infty)$, 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 $\Omega \subset \partial\mathbf H^{2}_{\mathbb C}=\mathbb S^3$ associated to $I_3I_2I_1I_2$, which is a subset of the discontinuous region. We show the quotient space $\Omega/ \Delta(3,3,4)$ is always the figure-eight knot complement in the deformation process. This gives the topological/geometrical explain that the 3-manifold at infinity of $\Delta(3,3,4; \infty)$ is the figure-eight knot complement. In particular, this confirms a conjecture of Falbel-Guilloux-Will.

The Eisenstein-Picard modular surface $M$ is the quotient space of the complex hyperbolic plane by the modular group $\rm PU(2,1; \mathbb{Z}[\omega])$. We determine the global topology of $M$ as a 4-orbifold.

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:2306.15240

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.

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}.

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., Comment: arXiv admin note: text overlap with arXiv:2011.03784

We show the manifolds at infinity of the complex hyperbolic triangle groups $\Delta_{3,4,4;\infty}$ and $\Delta_{3,4,6;\infty}$,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 $\Delta_{3,4,n;\infty}$ is the one-cusped hyperbolic 3-manifold obtained by Dehn surgery on the first cusp of $m295$ with slope $n-2$.

In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets., Comment: arXiv admin note: substantial text overlap with arXiv:2201.00154

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 $\rho$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $\rho(G_{6,3})$ 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}., Comment: 28 pages. arXiv admin note: text overlap with arXiv:1401.0308 by other authors

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 Bogachev

We show the 3-manifold at infinity of the complex hyperbolic triangle group $\Delta_{3,\infty,\infty;\infty}$ is the three-cusped "magic" 3-manifold $6_1^3$. We also show the 3-manifold at infinity of the complex hyperbolic triangle group $\Delta_{3,4,\infty;\infty}$ 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 $\Delta_{3,n,m;\infty}$ 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., Comment: 64 pages, 34 figures. Comments are welcome!

It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is $ \mathbb{H}^{4}$, $\mathbb{H}^{2}_{\mathbb{C}}$ or $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold $M$, we show that whether $M$ is geometric could be detected by the profinite completion of its fundamental group.

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 $\pi_1(M(\mathcal{P}))$ and $\pi_1(M(\mathcal{E}))$ are algebraic fibered. Namely, there are two exact sequences $$1\rightarrow H_{\mathcal{P}}\rightarrow \pi_1(M(\mathcal{P}))\xrightarrow{\phi_{\mathcal{P}}} \mathbb{Z}\rightarrow 1,$$ $$1\rightarrow H_{\mathcal{E}}\rightarrow \pi_1(M(\mathcal{E}))\xrightarrow{\phi_{\mathcal{E}}} \mathbb{Z}\rightarrow 1,$$ 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.

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.

In this paper, we construct two families of satellite constructions for Brunnian links, called the satellite sum and the satellite tie. An interesting fact is that by applying the satellite sum and the satellite tie constructions, we can build infinitely many new Brunnian links from any given Brunnian links. With the helps of the satellite sum and the satellite tie, we give a new decomposition theorem for Brunnian links. We prove that every Brunnian link determines a unique labelled "tree-arrow structure" such that each vertex of the tree represents a generalized Hopf link, a hyperbolic Brunnian link, or a hyperbolic Brunnian link in an unlink-complement.

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 $\frac{34\pi^2}{3}\cdot 16$ 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., Comment: In this version, we corret some drawing typos in figures of adjacent matrices of $X_{45}$, $X_{55}$, $X_{56}$ and $X_{57}$. The author are grateful to Leonardo Ferrari who pointed out the mistakes

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 descriptions

We show that a random link defined by random bridge splitting is hyperbolic with asymptotic probability 1., Comment: 9 pages

We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances., Comment: 6 pages, one picture

Let $M=H_{+}\cup_{S} H_{-}$ be a genus $g$ Heegaard splitting with Heegaard distance $n\geq \kappa+2$: (1) Let $c_{1}$, $c_{2}$ be two slopes in the same component of $\partial_{-}H_{-}$, such that the natural Heegaard splitting $M^{i}=H_{+}\cup_{S} (H_{-}\cup_{c_{i}} 2-handle)$ has distance less than $n$, then the distance of $c_{1}$ and $c_{2}$ in the curve complex of $\partial_{-}H_{-}$ is at most $3\mathfrak{M}+2$, where $\kappa$ and $\mathfrak{M}$ are constants due to Masur-Minsky. (2) Let $M^{*}$ be the manifold obtained by attaching a collection of handlebodies $\mathscr{H}$ to $\partial_{-} H_{-}$ along a map $f$ from $\partial \mathscr{H}$ to $\partial_{-} H_{-}$. If $f$ is a sufficiently large power of a generic pseudo-Anosov map, then the distance of the Heegaard splitting $M^{*}=H_{+}\cup (H_{-}\cup_{f} \mathscr{H})$ is still $n$. The proofs rely essentially on Masur-Minsky's theory of curve complex., Comment: V2; Added references and corrected typos

Using alternating Heegaard diagrams, we construct some 3-manifolds which admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms are composed of Smale-Williams solenoid attractors and repellers, an interesting example is the truncated-cube space. In addition, we prove that if the nonwandering set of the diffeomorphism consists of genus two Smale-Williams solenoids, then the Heegaard genus of the closed manifold is at most two., Comment: 18 pages, corrected typos

In this paper, we shall prove that any Heegaard splitting of a $\partial$-reducible 3-manifold $M$, say $M=W\cup V$, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of $n$ manifolds $M_{1},..., M_{n}$ where $M_{i}$ is either a solid torus or a $\partial$-irreducible manifold. Furthermore, $W\cup V$ is stabilized if and only if one of the factors is stabilized., Comment: 12 pages, 4 figures