Figure-eight knot is always over there

Let ${\bf H}^{2}_{\mathbb{C}}$ be the complex hyperbolic plane, the holomorphic isometry group of ${\bf H}^{2}_{\mathbb{C}}$ is PU(2,1). The complex hyperbolic plane ${\bf H}^{2}_{\mathbb{C}}$ can be identified with the unit ball in ${\mathbb{C}}^{2}$, so the ideal boundary $\partial{\bf H}^{2}_{\mathbb{C}}$ of ${\bf H}^{2}_{\mathbb{C}}$ is the 3-sphere $\mathbb{S}^{3}$.

Thurston’s work on 3-manifolds has shown that geometry has an important role in the study of topology of 3-manifolds. We have three kinds of geometrical structures on 3-manifolds related to the pair $({\bf H}^{2}_{\mathbb{C}},\partial{\bf H}^{2}_{\mathbb{C}})$ with increasing group action constraints.

For a discrete group $\Gamma<\mbox{\rm PU(2,1)}$, the open set $\Omega$ in (2) of Definition 1.1 is a subset of the discontinuity region $\Omega_{\Gamma}$ in (3) of Definition 1.1. So for a discrete group $\Gamma$, there is at most one uniformizable spherical CR-structure associated to it, but there may be infinitely many CR-structure spherical uniformizations associated to it.

For a discrete group $\Gamma<\mbox{\rm PU(2,1)}$, the 3-manifold $M=\Omega_{\Gamma}/\Gamma$ at infinity of the 4-manifold $\mathbf{H}^{2}_{\mathbb{C}}/\Gamma$ is the analogy of the 2-manifold at infinity of a geometrically finite, infinite volume hyperbolic 3-manifold. In other words, uniformizable spherical CR-structures on 3-manifolds in $\mathbf{H}^{2}_{\mathbb{C}}$-geometry are the analogies of conformal structures on surfaces in $\mathbf{H}^{3}_{\mathbb{R}}$-geometry.

In the three kinds of geometrical structures of Definition 1.1, uniformizable spherical CR-structures on 3-manifolds seem to be the most interesting ones. But in contrast to results on other geometric structures carried on 3-manifolds, there are relatively few examples known about them. A possible way to get uniformizable spherical CR-structures is via the deformations of triangle groups in PU(2,1).

Let $T(p,q,r)$ be the abstract triangle group with presentation

where $p,q,r$ are positive integers or $\infty$ satisfying

We assume that $p\leqslant q\leqslant r$. If $p,q$ or $r$ equals $\infty$, then the corresponding relation does not appear. The ideal triangle group is the case that $p=q=r=\infty$. A $(p,q,r)$ complex hyperbolic triangle group is a representation $\rho$ of $T(p,q,r)$ into PU(2,1) where the generators fix complex lines. We denote $\rho(\sigma_{i})$ by $I_{i}$, and the image group by $\Delta(p,q,r)=\langle I_{1},I_{2},I_{3}\rangle$. It is well known [17] that the space of $(p,q,r)$-complex reflection triangle groups has real dimension one if $3\leqslant p\leqslant q\leqslant r$.

The isometry group of the real hyperbolic plane ${\bf H}^{2}_{\mathbb{R}}$ is PO(2,1), and it is well known that the ideal triangle group is rigid in PO(2,1). Goldman and Parker [7] initiated the study of the deformations of ideal triangle group into PU(2,1). They gave an interval in the moduli space of complex hyperbolic ideal triangle groups, for points in this interval the corresponding representations are discrete and faithful. They conjectured that a complex hyperbolic ideal triangle group $\Delta(\infty,\infty,\infty)=\langle I_{1},I_{2},I_{3}\rangle$ is discrete and faithful if and only if $I_{1}I_{2}I_{3}$ is not elliptic. Schwartz proved Goldman-Parker’s conjecture in [15, 18]. Furthermore, Schwartz analyzed the complex hyperbolic ideal triangle group $\Gamma$ when $I_{1}I_{2}I_{3}$ is parabolic, and showed the 3-manifold at infinity of the quotient space ${\bf H}^{2}_{\mathbb{C}}/{\Gamma}$ is commensurable with the Whitehead link complement in the 3-sphere [16]. In other words, the Whitehead link complement admits uniformizable spherical CR-structure. Seifert 3-manifolds admitting uniformizable spherical CR-structures are rather easy to construct, but the Whitehead link complement is the first example of hyperbolic 3-manifold which admits uniformizable spherical CR-structure.

Richard Schwartz [17] has also conjectured the necessary and sufficient condition for a general complex hyperbolic triangle group

to be a discrete and faithful representation of $T(p,q,r)$. Schwartz’s conjecture has been proved in a few cases [8, 13, 14].

The critical point of the 1-dimensional deformation space of complex hyperbolic triangle groups is a point such that some preferred word $W_{A}$ or $W_{B}$ is accidental parabolic. For more details, see [17]. People found several more examples of cusped hyperbolic 3-manifolds which admit uniformizable spherical CR-structures at these critical points [1, 3, 4, 8, 10, 11]. Almost all of the examples of uniformizable spherical CR-structures gotten now are via difficult and sophisticated analysis. But we do not know the topological/geometrical reason the 3-manifolds at infinity of the groups associated to critical points should be the ones we got. Falbel-Guilloux-Will [5] proposed a method to predict the 3-manifold when there is an accidental parabolic element.

We now just consider the representations of $T(3,3,4)$ into PU(2,1) with complex reflection generators $I_{1},I_{2},I_{3}$. We can parametrize the representations by $t\in[1/3,\sqrt{2}-1]$, and the even subgroup of the image group is denoted by $\Gamma_{t}$, see Section 3 for more details. Moreover,

• •

  When $t=\sqrt{2}-1$, the image group lies in PO(2,1). So we have the classical $\mathbb{R}$-Fuchsian group;

• •

  For any $t\in(3/8,\sqrt{2}-1]$, $I_{3}I_{2}I_{1}I_{2}$ is loxodromic;

• •

  When $t=3/8$, $I_{3}I_{2}I_{1}I_{2}$ is parabolic. This is an accidental parabolicity, so $t=3/8$ corresponds to the critical point in the moduli space of $\Delta(3,3,4)$ in our parameterization;

• •

  When $t\in[1/3,3/8)$, $I_{3}I_{2}I_{1}I_{2}$ is elliptic. We will not consider representations in this interval.

It is showed by Parker-Wang-Xie [13] for each $t\in[3/8,\sqrt{2}-1]$, the corresponding representation is discrete and faithful. Since when $t=\sqrt{2}-1$, we have a $\mathbb{R}$-Fuchsian group, so 3-manifold at infinity of the corresponding group is just the unit tangent bundle over the real hyperbolic $(3,3,4)$-orbisurface. But when $t=3/8$, there is a new parabolic element $I_{3}I_{2}I_{1}I_{2}$, so the 3-manifold at infinity of the corresponding group must change. It is showed by Deraux-Falbel [4] the 3-manifold at infinity of the even subgroup $\Gamma_{3/8}$ is the figure-eight knot complement. But we do not know the reason that the 3-manifold at infinity of $\Delta(3,3,4)$ when $I_{3}I_{2}I_{1}I_{2}$ is parabolic should be this one. Falbel-Guilloux-Will [5] proposed an explanation of this phenomenon.

For all $t\in(3/8,\sqrt{2}-1]$, $I_{3}I_{2}I_{1}I_{2}$ is loxodromic. Let $p_{1}$ and $p_{2}$ be the attractive and repulsive fixed points of it, they determine a $\mathbb{C}$-circle. We denote by $\alpha_{1}$ a preferred one of the two arcs with end points $p_{1}$ and $p_{2}$ in the $\mathbb{C}$-circle (see Section 5 for this arc). Let $\Lambda_{t}$ be the limit set of $\Gamma_{t}$. Then it is a topological circle. The crown associated to $I_{3}I_{2}I_{1}I_{2}$ is the subset of $\mathbb{S}^{3}$ defined as

We denote $\Omega_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}\subset\Omega_{\Gamma_{t}}$ as the complement of ${\rm Crown}_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}$ in $\mathbb{S}^{3}$. Recall that $\Omega_{\Gamma_{t}}=\mathbb{S}^{3}-\Lambda_{t}$ is the discontinuous region of $\Gamma_{t}$’s action on $\mathbb{S}^{3}$.

It was shown in [2] that $\Omega_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}/\Gamma_{t}$ is homeomorphism to the figure-eight knot complement when $t=\sqrt{2}-1$. In fact Falbel-Guilloux-Will [5] identified this manifold as drilling out the unit tangent bundle of $(3,3,4)$-orbisurface a certain closed orbit associated to $I_{3}I_{2}I_{1}I_{2}$. Moreover, Falbel-Guilloux-Will [5] conjectured that the quotient space of $\Omega_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}$ by $\Gamma_{t}$ is always the figure-eight knot complement for any $t\in(3/8,\sqrt{2}-1)$. So each of them gives a CR-structure spherical uniformization of figure-eight knot complement. The last one, that is when $t=3/8$, gives the uniformizable spherical CR-structure on the figure-eight knot complement. Which corresponds to pinching on the limit set of $\Gamma_{t}$ to the limit set of $\Gamma_{3/8}$. So this conjecture explains how to get the 3-manifold at infinity of $\Gamma_{3/8}$ from the 3-manifold at infinity of a $\mathbb{R}$-Fuchsian group. Falbel-Guilloux-Will [5] confirmed the conjecture when $t$ is near to $\sqrt{2}-1$.

We certificate Falbel-Guilloux-Will’s conjecture totally in this paper:

So in the deformation process, the figure-eight knot is always over there! This explains the 3-manifold at infinity of the even subgroup $\Gamma_{3/8}$ (with accidental parabolic element) is the figure-eight knot [4].

We prove Theorem 1.2 in the following steps:

• •

  For $\Delta(3,3,4)=\langle I_{1},I_{2},I_{3}\rangle$ depends on $t\in(3/8,\sqrt{2}-1]$, $I_{1}I_{2}$ has order 4, and $I_{1}I_{2}$ has fixed point $p_{0}\in\mathbf{H}^{2}_{\mathbb{C}}$;

• •

  Consider the Dirichlet domain $D_{t}$ of $\Gamma_{t}<\Delta(3,3,4)$ with center $p_{0}$, $D_{t}$ has eight facets [13];

• •

  The ideal boundary $\partial_{\infty}D_{t}=D_{t}\cap\partial\mathbf{H}^{2}_{\mathbb{C}}$ is a solid torus. Moreover, the boundary of $\partial_{\infty}D_{t}$ consists of eight annuli, the side-pairing pattern on them is independent of $t\in(3/8,\sqrt{2}-1]$. So the 3-manifold at infinity of the group $\Gamma_{t}$ is independent of $t$. This proves (1) of Theorem 1.2;

• •

  We then consider the complement of the crown in $\partial_{\infty}D_{t}$, that is,

  $$\partial_{\infty}D_{t}-{\rm Crown}_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}.$$

  Which is a fundamental domain of $\Gamma_{t}$’s action on $\Omega_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}$. In fact $\partial_{\infty}D_{t}\cap{\rm Crown}_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}$ are exactly eight arcs. We will show the topology and the side-pairing pattern on $\partial_{\infty}D_{t}-{\rm Crown}_{\Gamma_{t},I_{3}I_{2}I_{1}I_{2}}$ are independent of $t$. This in turn proves (2) of Theorem 1.2.

Acknowledgement: Part of the work was carried out when Jiming Ma was visiting Hunan University in the summer of 2022, the hospitality is gratefully appreciated. The second author thanks John Parker for a useful discussion about the parametrization of the deformation space of the triangle group $\Delta(3,3,4)$ several years ago.

We will briefly introduce some background of complex hyperbolic geometry in this section. One can refer to Goldman’s book [6] for more details.

Let $I_{i}$ be a reflection along the complex line $C_{i}$ for $i=1,2,3$. We assume that $C_{i-1}$ and $C_{i}$ either meet at the angle $\pi/p_{i}$ for some integer $p_{i}\geqslant 3$ or else $C_{i-1}$ and $C_{i}$ are asymptotic, in which case they make an angle 0 and we write $p_{i}=\infty$, where the indices are taken mod 3. The subgroup $\Delta(p_{1},p_{2},p_{3})$ of PU(2,1) generated by $I_{1},I_{2}$ and $I_{3}$ is called a complex hyperbolic triangle group. For fixed $p_{1},p_{2},p_{3}$, modulo conjugacy in PU(2,1), there exists in general a $1$-parameter family of complex hyperbolic triangle group $\Delta(p_{1},p_{2},p_{3})$.

We consider the deformation space of complex hyperbolic triangle group $\Delta(3,3,4)$, generated by three complex reflections $I_{1},I_{2}$ and $I_{3}$. As an abstract group, it is given by

We will describe a parametrization of the deformation space of $\Delta(3,3,4)$, which is a little different from that in [13].

Suppose that the polar vectors $n_{1}$, $n_{2}$ of the complex lines $C_{1}$, $C_{2}$ are given by

Then the corresponding complex reflections $I_{1}$ and $I_{2}$ are given by

We may also suppose that the polar vector $n_{3}$ of $C_{3}$ is

Furthermore, we can assume that $a$, $b$, $d$ are nonnegative real numbers by conjugating a diagonal map ${\rm Diag}(e^{i\beta},e^{i\beta},e^{-2i\beta})$ if necessary. After a normalization of $n_{3}$, we have

The matrix for the complex reflection $I_{3}$ is given by

One may always assume $\theta\in[0,\pi]$ by complex conjugating if necessary.

The condition that $I_{1}I_{3}$ and $I_{2}I_{3}$ have order 3 is equivalent to $tr(I_{1}I_{3})=tr(I_{2}I_{3})=0$. That is,

and

Since we know that $a^{2}+b^{2}=d^{2}+1$, we have

We also have that $d^{2}$ is nonnegative and $\left|1/2-2a^{2}\right|\leqslant 2a$ if and only if $\sqrt{3}/2\leqslant a\leqslant(\sqrt{2}+1)/2$. In other words, our parametrization of the deformation space of $\Delta(3,3,4)$ is given by

In particular, the entries of $n_{3}$ are all real when $a=(\sqrt{2}+1)/2$. Thus the complex hyperbolic triangle group $\Delta(3,3,4)$ lies in $SO(2,1)$ when $a=(\sqrt{2}+1)/2$.

So our parameter space for $\left\langle I_{1},I_{2},I_{3}\right\rangle$ with $I_{1}I_{3}I_{2}I_{3}$ nonelliptic is given by

To make the computation simpler, we write $a=\frac{1}{2\sqrt{1-2t}}$. Thus the parameter space for the triangle group $\Delta(3,3,4)$ becomes

with the new parameter $t$.

Most calculations are carried out in the Siegel model. From now on, we will work on this model.

It is convenient to introduce some notations that are used throughout the paper. We define

In the Siegel model, the polar vectors $n_{1},n_{2}$ and $n_{3}$ are given by

The corresponding complex reflections $I_{1},I_{2}$ and $I_{3}$ are given by the matrices

and

respectively.