Two series of polyhedral fundamental domains for Lorentz bi-quotients

TL;DR

This paper constructs two infinite series of Lorentz space forms as bi-quotients of a Lie group, providing explicit polyhedral fundamental domains linked to surface singularities and their topological links.

Contribution

It explicitly describes fundamental domains for two infinite series of Lorentz bi-quotients, connecting geometric group actions with singularity theory.

Findings

Explicit fundamental domains for the series are constructed.

The bi-quotients relate to links of surface singularities.

Results connect Lorentz geometry with algebraic singularity theory.

Abstract

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form $\Gamma_1\backslash G/\Gamma_2$, where $G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(2,{\mathbb R})}$ is a simply connected Lie group with the Lorentz metric given by the Killing form, $\Gamma_1$ and $\Gamma_2$ are discrete subgroups of $G$ and $\Gamma_2$ is cyclic. A construction of polyhedral fundamental domains for the action of $\Gamma_1\times\Gamma_2$ on $G$ via $(g,h)\cdot x=gxh^{-1}$ was given in the earlier work of the second author. In this paper we give an explicit description of the fundamental domains obtained by this construction for two infinite series of groups. These results are connected to singularity theory as the bi-quotients $\Gamma_1\backslash G/\Gamma_2$ appear as links of certain quasi-homogeneous $\mathbb Q$-Gorenstein surface singularities, i.e.\ the intersections of the singular variety with sufficiently small spheres around the isolated singular point.

Tables

level	$(p,q,r)$	type	normal form
$k$	$(k+3,3,3)$	$E_{4k+10}$	$x^3+y^{2k+6}+z^2$
$k$	$(2k+3,3,3)$	$Z_{4k+9}$	$x^{3}y+y^{2k+4}+z^2$

Table 2. Table 1. Fundamental domains for Γ ​ ( k + 3 , 3 , 3 ) k × ( C 3 ) k Γ superscript 𝑘 3 3 3 𝑘 superscript subscript 𝐶 3 𝑘 \Gamma(k+3,3,3)^{k}\times(C_{3})^{k} .

$k=4$
$k=5$
$k=7$
$k=8$

Table 3. Table 2. Fundamental domains for Γ ​ ( 2 ​ k + 3 , 3 , 3 ) k × ( C 3 ) k Γ superscript 2 𝑘 3 3 3 𝑘 superscript subscript 𝐶 3 𝑘 \Gamma(2k+3,3,3)^{k}\times(C_{3})^{k} .

$k=4$
$k=5$
$k=7$
$k=8$

Equations

$$G=\widetilde{\mathrm{SU}(1,1)}\cong \widetilde{\mathrm{SL}(2,{\mathbb{R}}^{})},$$

$$F_e^{\circ }=(E_e\cap \partial Q_u{)}^{\circ }-\bigcup _{x\in {\Gamma }_1(u)\backslash \{u\}}{Q}_x,$$

$$Q_x=\bigcap _{\genfrac{}{}{0ex}{}{(g_1,g_2)\in {\Gamma }_1\times {\Gamma }_2}{g_1(u)=x}}{H}_{g_1\cdot g_2}.$$

$$F_e^{\circ }=(E_e\cap \partial Q_u{)}^{\circ }-\bigcup _{x\in \mathcal{E}}{Q}_x,$$

$$\Gamma (k+3,3,3{)}^k\times (C_3{)}^k\text{and}\Gamma (2k+3,3,3{)}^k\times (C_3{)}^k,$$

$$F_e=E_e\cap H_D\cap H_{D^{-1}}\cap \bigcap _{m\in {\mathbb{Z}}^{}}(I_{a_m}\cup I_{b_m})$$

$$a_0$$

$$a_m$$

$$F_e=E_e\cap H_D\cap H_{D^{-1}}\cap \bigcap _{m\in {\mathbb{Z}}^{}}(I_{a_m}\cup I_{b_m}\cup I_{c_m})$$

$$a_0$$

$$a_m$$

$$\cosh L=\frac{\cos \frac{\pi }{p}\cos \frac{\pi }{q}+\cos \frac{\pi }{r}}{\sin \frac{\pi }{p}\sin \frac{\pi }{q}}.$$

$$\mathcal{E}=\{x_{0,1},x_{0,q-1},x_{1,1},x_{1,q-1},\cdots ,x_{p-1,1},x_{p-1,q-1}\}.$$

$$d=\frac{\sinh L\sin \frac{\pi }{q}}{\sqrt{{\sinh }^{2}L{\sin }^2\frac{\pi }{q}+1}}.$$

$$s=\frac{\sinh L\sin \frac{2\pi }{q}}{{\sinh }^{2}L{\sin }^2\frac{\pi }{q}+1}.$$

$$⟨(z_1,w_1),(z_2,w_2)⟩=\mathrm{Re}(z_1{\bar{z}}_2-w_1{\bar{w}}_2).$$

$${\bar{G}}=\left\{a\in E^{2,2}\,\,\big{|}\,\,\langle a,a\rangle=-1\right\}=\left\{(z,w)\in E^{2,2}\,\,\big{|}\,\,|z|^{2}-|w|^{2}=-1\right\}.$$

$$\bar{L}$$

$$\displaystyle=\left\{a\in E^{2,2}\,\,\big{|}\,\,\langle a,a\rangle<0\right\}=\left\{(z,w)\in E^{2,2}\,\,\big{|}\,\,|z|<|w|\right\}.$$

$$\left(\begin{array}{cc}\bar{w}& z\\ \bar{z}& w\end{array}\right)\mapsto (z,w),$$

$${\bar{E}}_{\pi(g)}=\{a\in{\bar{L}}\,\,\big{|}\,\,\langle\pi(g),a\rangle=-1\}$$

$${\bar{I}}_{\pi(g)}=\{a\in{\bar{L}}\,\,\big{|}\,\,\langle\pi(g),a\rangle\leqslant-1\}.$$

$$\pi((E_{e}\cap\partial Q_{u})^{\circ})=\left\{(z,w)\,\,\big{|}\,\,\operatorname{Re}(w)=1,~{}|\operatorname{Im}(w)|<\tan{\frac{\pi k}{2p}}\right\}.$$

$$(a,b)\cdot x=\frac{\bar{b}x+a}{\bar{a}x+b}.$$

$$-1=|a{|}^2-|b{|}^2=|-\bar{b}x{|}^2-|b{|}^2=-|b{|}^2\cdot (1-|x{|}^2)$$

$$|w^{\prime }|=|\bar{a}z+bw|=|-b\bar{x}z+bw|=\frac{|w-\bar{x}z|}{\sqrt{1-|x{|}^2}}⩽1,$$

$${\pi }^{-1}(z,w)={\pi }^{-1}(\pi (g{)}^{-1}\cdot (z^{\prime },w^{\prime }))=g^{-1}\cdot {\pi }^{-1}(z^{\prime },w^{\prime })\subset g^{-1}\cdot Q_u=Q_x.\text{qed}$$

$$F_e^{\circ }=(E_e\cap \partial Q_u{)}^{\circ }-\bigcup _{x\in {\Gamma }_1(u)\backslash \{u\}}{Q}_x.$$

$$F_{\mathcal{E}}=(E_e\cap \partial Q_u{)}^{\circ }-\bigcup _{x\in \mathcal{E}}{Q}_x.$$

$${\ell }^{\pm }(t)=\frac{1}{\tanh L}\pm \frac{1}{\sinh L\cdot \sin \frac{\pi }{q}}\cdot \sqrt{\frac{1}{t^2}-{\cos }^2\frac{\pi }{q}},$$

$${\ell }^{-}\left(\sec \frac{\pi k}{2p}\right)⩽1.$$

$$(z,w)\in \pi (F_{\mathcal{E}})⟹|z|<|w|\cdot {\ell }^{-}(|w|).$$

Full text

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through the research group No (RG-1440-142). To produce the images of polyhedra we used Geomview, a programme developed at the Geometry Centre, University of Minnesota, Minneapolis.

1. Introduction

In this paper we study the Lorentz space forms obtained as bi-quotients ${\Gamma}_{1}\backslash G/{\Gamma}_{2}$, where $G$ is the universal cover

[TABLE]

${\Gamma}_{1}$ and ${\Gamma}_{2}$ are discrete subgroups of $G$ of the same finite level, and ${\Gamma}_{2}$ is cyclic. The group ${\Gamma}_{1}\times{\Gamma}_{2}$ acts on $G$ via $(g,h)\cdot x=gxh^{-1}$. The level of a subgroup ${\Gamma}$ of $G$ is the index of ${\Gamma}\cap Z(G)$ in the centre $Z(G)\cong{{\mathbb{Z}}^{\vphantom{x}}}$. Subgroups of $G$ of finite level $k$ are those subgroups that can be obtained as pre-images of subgroups of the $k$-fold covering of $\operatorname{PSU}(1,1)$. We use the construction introduced in [Pra11] of polyhedral fundamental domains for this action which generalizes the results of [Pra07a], [BPR], [BKNRS]. This construction leads to a description of the bi-quotients ${\Gamma}_{1}\backslash G/{\Gamma}_{2}$ as polyhedra with faces identified according to some gluing rules on the boundary.

Let ${\Gamma}(p_{1},3,3)^{k}$ be a subgroup of $G$ of level $k$ such that its image under the projection to $\operatorname{PSU}(1,1)$ is a triangle group ${\Gamma}(p_{1},3,3)$, see section 2 for more details. Subgroups of $G$ and $\operatorname{PSU}(1,1)$ act on the disc model ${{\mathbb{D}}^{\vphantom{x}}}$ of the hyperbolic plane since $\operatorname{Isom}^{+}({{\mathbb{D}}^{\vphantom{x}}})\cong\operatorname{PSU}(1,1)$. Let $u\in{{\mathbb{D}}^{\vphantom{x}}}$ be the fixed point of a generator of ${\Gamma}(p_{1},3,3)$ of order $p_{1}$. Let $(C_{p_{2}})^{k}$ be a subgroup of $G$ of level $k$ such that its image under the projection to $\operatorname{PSU}(1,1)$ is a cyclic group of order $p_{2}$ generated by an elliptic element with fixed point $u$. The aim of this paper is to give explicit descriptions of fundamental domains for two infinite series of groups ${\Gamma}(k+3,3,3)^{k}\times(C_{3})^{k}$ and ${\Gamma}(2k+3,3,3)^{k}\times(C_{3})^{k}$, where $k$ is a positive integer not divisible by $3$.

The bi-quotients ${\Gamma}_{1}\backslash G/{\Gamma}_{2}$ are the links of certain quasi-homogeneous ${{\mathbb{Q}}^{\vphantom{x}}}$-Gorenstein surface singularities as explained in [Pra07b], see also [Dol83]. The motivation for the choice of the series ${\Gamma}(k+3,3,3)^{k}\times(C_{3})^{k}$ and ${\Gamma}(2k+3,3,3)^{k}\times(C_{3})^{k}$ is that they correspond to ${{\mathbb{Q}}^{\vphantom{x}}}$-Gorenstein surface singularities that are obtained as quotients of certain singularities in the series $E$ and $Z$ according to the classification by V.I. Arnold. We listed in the table below the normal form of the singularity as well as the level $k$ of ${\Gamma}_{i}$ and the signature $(p,q,r)$ of the image of ${\Gamma}_{1}$ in $\operatorname{PSU}(1,1)$, for more details see [AGZV], [Dol74], [Dol75] and [Pra01].

We will now outline the fundamental domain construction described in [Pra11]. Suppose that ${\Gamma}_{1}$ and ${\Gamma}_{2}$ are discrete subgroups of level $k$ in $G$ and ${\Gamma}_{2}$ is cyclic. We assume the existence of a joint fixed point $u\in{{\mathbb{D}}^{\vphantom{x}}}$ of ${\Gamma}_{1}$ and ${\Gamma}_{2}$. Let $p_{i}=|(\overline{\Gamma}_{i})_{u}|$ be the size of the isotropy group of $u$ in the image $\overline{\Gamma}_{i}$ of ${\Gamma}_{i}$ in $\operatorname{PSU}(1,1)$ for $i=1,2$. This construction works if $p=\operatorname{lcm}(p_{1},p_{2})>k$.

We think of the group $G$ as a hypersurface embedded in the bundle $L={{\mathbb{R}}^{\vphantom{x}}}_{+}\times G$. The Killing form on $G$ induces a pseudo-Riemannian metric of signature $(2,1)$ on $G$ and of signature $(2,2)$ on $L$. The hypersurface $G$ is replaced with its piece-wise totally geodesic polyhedral model, specially adapted to the action of ${\Gamma}_{1}\times{\Gamma}_{2}$. The fundamental domains $F_{g_{1}g_{2}}$, $g_{1}\in{\Gamma}_{1}$, $g_{2}\in{\Gamma}_{2}$ for the action of ${\Gamma}_{1}\times{\Gamma}_{2}$ on the model polyhedron are its faces, which are then projected onto $G$ to obtain fundamental domains ${\mathcal{F}}_{g_{1}g_{2}}$ for the action of ${\Gamma}_{1}\times{\Gamma}_{2}$ on $G$. Similar ideas of projecting an affine construction with half-planes onto a quadric were used in the algorithmic construction of Voronoi diagrams for point sets in the Euclidean and hyperbolic plane, compare [BY98].

For $g\in G$ let $E_{g}$ be the embedded tangent space of $G$ at the point $g$, i.e. the $3$-dimensional totally geodesic submanifold of $L$ which is tangent to $G$ at the point $g$. The hypersurface $E_{g}$ divides $L$ into two connected components. We will refer to the closures of these connected components as half-spaces and denote them by $H_{g}$ and $I_{g}$, see section 3 for more details.

Let $e$ be the identity in $G$. The interior of the fundamental domain $F_{e}$ can be described as

[TABLE]

where ${{\Gamma}_{1}(u)}\subset{{\mathbb{D}}^{\vphantom{x}}}$ is the orbit of the point $u$ under the action of ${\Gamma}_{1}$ and $Q_{x}$ for every $x\in{{\Gamma}_{1}(u)}$ is the prism-like polyhedron

[TABLE]

According to Theorem B in [Pra11], if ${\Gamma}_{1}$ is co-compact we can replace the union of $Q_{x}$ over all $x\in{{{\Gamma}_{1}(u)}\backslash\{u\}}$ by the union over some finite subset $N\subset{{{\Gamma}_{1}(u)}\backslash\{u\}}$, which is a crucial step on the way to obtain an explicit description of the fundamental domain $F_{e}$. However, the proof of this fact uses a compactness argument which in general does not provide an explicit description of such a finite subset. The first main result of this paper is to show that for a certain family of groups such a finite subset of ${{{\Gamma}_{1}(u)}\backslash\{u\}}$ is given by the edge corona of ${\Gamma}_{1}$ (see section 2 for the definition).

Theorem 1**.**

Let ${\Gamma}_{1}={\Gamma}(p_{1},3,3)^{k}$ with $p_{1}\geqslant k+3$ and ${\Gamma}_{2}=(C_{3})^{k}$. Assume that a generator of ${\Gamma}_{1}$ of order $p_{1}$ and the cyclic group ${\Gamma}_{2}$ have a shared fixed point $u\in{{\mathbb{D}}^{\vphantom{x}}}$. Then

[TABLE]

where the finite set ${\mathcal{E}}$ is the edge corona of ${\Gamma}_{1}(p_{1},3,3)$ with respect to $u$.

Theorem 1 reduces the description of the fundamental domain to an intersection of finitely many unions of finitely many half-spaces. For two infinite series of groups,

[TABLE]

we can reduce the description further after some long but elementary computations with systems of linear inequalities, for details see [BT]. The results are summarized in Theorem 2. The conjectural description of these fundamental domains for $k=2$ was given in [Pra11]. In this paper, following  [BT], we confirm these conjectures and extend them to two infinite series. We also describe some methods that can be used to determine the fundamental domains for other series of groups.

To state the results we will first introduce some notation. We make use of the following construction to describe certain elements of $G$: Given a base-point $x\in{{\mathbb{D}}^{\vphantom{x}}}$ and a real number $t$, let $\rho_{x}(t)\in\operatorname{PSU}(1,1)$ denote the rotation through the angle $t$ about the point $x$. Thus we obtain a homomorphism $\rho_{x}:{{\mathbb{R}}^{\vphantom{x}}}\to\operatorname{PSU}(1,1)$, which lifts to the unique homomorphism $R_{x}:{{\mathbb{R}}^{\vphantom{x}}}\to G$ into the universal covering group. The element $C=R_{u}(2\pi)$ is one of the two generators of the centre of $G$. The stabilisers $({\Gamma}_{1})_{u}$ and $({\Gamma}_{2})_{u}={\Gamma}_{2}$ generate a cyclic group with generator $D=R_{u}(2\pi k/p)$, where $p=\operatorname{lcm}(p_{1},p_{2})$.

Theorem 2**.**

Consider the group ${\Gamma}_{1}\times{\Gamma}_{2}={\Gamma}(p,3,3)^{k}\times(C_{3})^{k}$. Consider a triangle generating ${\Gamma}(p,3,3)$. Let $u$ and $v$ be vertices of this triangle with angles $\pi/p$ and $\pi/3$ respectively. Let $\lambda=1$ if $k=1\mod 3$ and $\lambda=2$ if $k=2\mod 3$.

1. $\bullet$

If $p=k+3$ then

[TABLE]

is a fundamental domain for ${\Gamma}_{1}\times{\Gamma}_{2}$, where

[TABLE] 2. $\bullet$

If $p=2k+3$ then

[TABLE]

is a fundamental domain for ${\Gamma}_{1}\times{\Gamma}_{2}$, where

[TABLE]

The general results in Theorem 2 are illustrated by the images of fundamental domains for $\Gamma(k+3,3,3)^{k}\times(C_{3})^{k}$ and $\Gamma(2k+3,3,3)^{k}\times(C_{3})^{k}$ for $k=2$ in Figures 1 and 2 respectively. Some explanations of the figures are required. We project the fundamental domain $F_{e}$ to the Lie algebra of $\operatorname{SU}(1,1)$, which is a $3$-dimensional flat Lorentz space of signature $(2,1)$. The result is a compact polyhedron with flat faces and a distinguished rotational axis of symmetry. The direction of this axis is negative definite, and the orthogonal complement is positive definite. Changing the sign of the pseudo-metric in the direction of the rotational axis transforms Lorentz space into a well-defined Euclidean space. The image of the fundamental domain is then transformed into a polyhedron in Euclidean space with dihedral symmetry. The direction of the rotational axis is vertical. The top and bottom faces are removed to make the structure of the polyhedra clearer.

Figures 3 and 4 illustrate the identification schemes on the boundary of the polyhedron for the infinite series $\Gamma(k+3,3,3)^{k}\times(C_{3})^{k}$ and $\Gamma(2k+3,3,3)^{k}\times(C_{3})^{k}$ respectively. The face identifications are equivariant with respect to the dihedral symmetry of the polyhedron. The faces of the same shading/colour are identified. Numbers on the edges of shaded/coloured faces indicate the identified flags (face, edge, vertex).

Tables 1 and 2 at the end of the paper show several fundamental domains from each of the infinite series $\Gamma(k+3,3,3)^{k}\times(C_{3})^{k}$ and $\Gamma(2k+3,3,3)^{k}\times(C_{3})^{k}$ respectively. We can see that the combinatorial structure of the fundamental domains and their identification schemes are similar in each of the two series, only the order of the dihedral symmetry increases with $k$. For $\Gamma(k+3,3,3)^{k}\times(C_{3})^{k}$ the fundamental polyhedron has the combinatorial structure of an anti-prism with two triangular faces sharing an edge which are then rotated around the vertical axis. For $\Gamma(2k+3,3,3)^{k}\times(C_{3})^{k}$ we see a quadrangular face to which two triangular faces are attached and all three faces are then rotated around the vertical axis. Note that while the combinatorics of the fundamental domains of all triangle groups in ${{\mathbb{D}}^{\vphantom{x}}}$ is the same, the combinatorics of the fundamental domains which we constructed for $\Gamma(k+3,3,3)^{k}\times(C_{3})^{k}$ and $\Gamma(2k+3,3,3)^{k}\times(C_{3})^{k}$ in $G$ is different. Hence the combinatorics of our fundamental domains shows the structure of the groups which is not apparent in their fundamental domains in the hyperbolic plane.

2. Triangle Groups

A triangle group of signature $(p,q,r)$ is a group of orientation-preserving isometries of the hyperbolic plane, generated by the rotations through $\frac{2\pi}{p}$, $\frac{2\pi}{q}$, $\frac{2\pi}{r}$ about the vertices of a hyperbolic triangle with angles $\frac{\pi}{p}$, $\frac{\pi}{q}$ and $\frac{\pi}{r}$. All such groups are conjugate to each other and we will denote such a group ${\Gamma}(p,q,r)$.

The following existence result for discrete subgroups of finite level in $G$ can be found in [Pra01] (section 2.8, Satz 38).

Proposition 3**.**

If $\gcd(k,p)=\gcd(k,q)=\gcd(k,r)=1$ and $pqr-pq-qr-rp$ is divisible by $k$ then there exists a unique subgroup of $G$ of level $k$, denoted ${\Gamma}(p,q,r)^{k}$, such that its image under the projection to $\text{PSU}(1,1)$ is the triangle group ${\Gamma}(p,q,r)$. In particular the conditions for the existence of ${\Gamma}(p,3,3)^{k}$ are that $k$ is not divisible by $3$ and $p-3$ is divisible by $k$.

Definition**.**

For a Fuchsian group $\overline{\Gamma}\in\operatorname{PSU}(1,1)$, the edge corona ${\mathcal{E}}$ with respect to a point $u\in{{\mathbb{D}}^{\vphantom{x}}}$ consists of all those points in $\overline{\Gamma}(u)\backslash\{u\}$ whose Dirichlet region shares at least an edge with the Dirichlet region of $u$, compare [GSh].

Figures 5 and 6 show as an example the edge coronas for the triangle groups ${\Gamma}(5,3,3)$ and ${\Gamma}(7,3,3)$ respectively. The Dirichlet region with respect to the origin is red/dark grey, while the union of Dirichlet regions of the points of the edge corona is green/lighter grey.

We have the following description of the edge corona, for details of the proof see Propositions 41 and 42 in [Pra01]:

Proposition 4**.**

Let ${\Gamma}(p,q,r)$ be a triangle group generated by rotations $\rho_{u}$, $\rho_{v}$, $\rho_{w}$ through $\frac{2\pi}{p}$, $\frac{2\pi}{q}$, $\frac{2\pi}{r}$ around the vertices $u$, $v$, $w$ of a hyperbolic triangle with angles $\frac{\pi}{p}$, $\frac{\pi}{q}$, $\frac{\pi}{r}$ respectively. Let $L$ be the hyperbolic distance between the vertices $u$ and $v$. Recall that $L$ is determined by the angles of the hyperbolic triangle

[TABLE]

Then the edge corona consists of the following points of the form $x_{m,l}=(\rho^{m}_{u}\rho^{l}_{v})(u)$:

[TABLE]

All points of the edge corona are on the circle with centre at the origin of (Euclidean) radius

[TABLE]

Moreover, the largest (Euclidean) distance between subsequent points of the edge corona on this circle is

[TABLE]

3. The Elements of the Construction

In this section we will add some notation to the setting that we sketched in the introduction. Let us describe the embedding of $G$ in $L$ and the totally geodesic subspaces $E_{g}$ in $L$. We consider the $4$-dimensional pseudo-Euclidean space $E^{2,2}$ of signature $(2,2)$. We think of $E^{2,2}$ as the real vector space ${{\mathbb{C}}^{\vphantom{x}}}^{2}\cong{{\mathbb{R}}^{\vphantom{x}}}^{4}$ with the symmetric bilinear form

[TABLE]

In $E^{2,2}$ we consider the quadric

[TABLE]

We consider the cone over ${\bar{G}}$

[TABLE]

The bilinear form on $E^{2,2}$ induces a pseudo-Riemannian metric of signature $(2,1)$ on ${\bar{G}}$ and of signature $(2,2)$ on ${\bar{L}}$.

Let $\pi:G\to{\bar{G}}$ be the universal covering. Henceforth we identify the Lie group $\operatorname{SU}(1,1)$ with ${\bar{G}}$ via

[TABLE]

and $\widetilde{\operatorname{SU}(1,1)}$ with $G$. The bi-invariant metrics on ${\bar{G}}$ and $G$ are proportional to the Killing forms. The covering $G\to{\bar{G}}$ extends to the universal covering $L\to{\bar{L}}$. The covering space $L$ inherits canonically a pseudo-Riemannian metric from ${\bar{L}}$. There exist canonical trivializations ${\bar{L}}\cong{\bar{G}}\times{{\mathbb{R}}^{\vphantom{x}}}_{+}$ and $L\cong G\times{{\mathbb{R}}^{\vphantom{x}}}_{+}$.

For $g\in G$, consider the intersection with ${\bar{L}}$ of the affine tangent space of ${\bar{G}}$ at the point $\pi(g)$

[TABLE]

and the intersection with ${\bar{L}}$ of the half-space of ${{\mathbb{C}}^{\vphantom{x}}}^{2}$ bounded by ${\bar{E}}_{\pi(g)}$ and not containing the origin

[TABLE]

The sets ${\bar{E}}_{\pi(g)}$ and ${\bar{I}}_{\pi(g)}$ are simply connected and even contractible, hence their pre-images under the covering map $\pi$ consist of infinitely many connected components, one of them containing $g$. Let $E_{g}$ and $I_{g}$ be those connected components of $\pi^{-1}({\bar{E}}_{\pi(g)})$ and $\pi^{-1}({\bar{I}}_{\pi(g)})$ respectively that contain $g$. The three-dimensional submanifold $E_{g}$ divides $L$ into two connected components, the closure of one of which is $I_{g}$. Let $H_{g}$ be the closure of the other component of the complement of $E_{g}$ in $L$. The boundaries of $I_{g}$ and of $H_{g}$ are equal to $E_{g}$.

The set $\pi((E_{e}\cap\partial Q_{u})^{\circ})$ can be described as

[TABLE]

4. Approximating prisms $Q_{x}$ by their inscribed and subscribed cylinders

The following estimates are obtained by approximating the prisms $Q_{x}$ via their inscribed and subscribed cylinders:

Proposition 5**.**

Let $x\in{{{\Gamma}_{1}(u)}\backslash\{u\}}$ and $(z,w)\in{\bar{L}}$. Then

1. (i)

If $(z,w)\in\pi(Q_{x})$ then $|w|-|z|<|w|-|x|\cdot|z|\leqslant|w-\bar{x}z|\leqslant\sec{\frac{\pi k}{2p}}\cdot\sqrt{1-|x|^{2}}$. 2. (ii)

If $|w-\bar{x}z|\leqslant\sqrt{1-|x|^{2}}$ then $\pi^{-1}(z,w)\subset Q_{x}$.

Proof.

For the proof of (i) see Lemma 1(i) in [Pra11]. The same idea works for (ii).

Let us first consider the case $x=u=0$. In this case the inequality in (ii) reduces to $|w|\leqslant 1$. Let $g=(z,w)\in{\bar{L}}$ with $|w|\leqslant 1$. Then we have $g\in\bar{H}_{a}$ for all $a\in(\overline{\Gamma}_{1})_{u}$ and therefore $\pi^{-1}(g)\subset H_{a}$ for all $a\in({\Gamma}_{1})_{u}$. Thus $\pi^{-1}(g)\subset Q_{u}$.

In the general case $x\in{\Gamma}_{1}(u)\backslash\{u\}$, let $g\in{\Gamma}_{1}$ be an element such that $g(x)=u$ and let $\pi(g)=(a,b)$. The element $(a,b)\in{\bar{G}}$ acts on ${{\mathbb{D}}^{\vphantom{x}}}$ by

[TABLE]

The property $(a,b)\cdot x=u=0$ implies $a=-\bar{b}x$. From $(a,b)\in{\bar{G}}$ we conclude

[TABLE]

and hence $|b|=(1-|x|^{2})^{-1/2}$. Let us consider $(z,w)$ with $|w-\bar{x}z|\leqslant\sqrt{1-|x|^{2}}$. Let $(z^{\prime},w^{\prime})=\pi(g)\cdot(z,w)$. Then

[TABLE]

hence $\pi^{-1}(z^{\prime},w^{\prime})\subset Q_{u}$ and

[TABLE]

5. Reduction of the description of $F_{e}$

The interior of the fundamental domain $F_{e}$ is

[TABLE]

The aim of this section is to show that in the case ${\Gamma}_{1}={\Gamma}(p_{1},q,r)^{k}$ and ${\Gamma}_{2}=(C_{p_{2}})^{k}$ with $p=\operatorname{lcm}(p_{1},p_{2})\geqslant kq/2$ it is sufficient to consider the prisms $Q_{x}$ with $x$ in the edge corona ${\mathcal{E}}$, i.e. $F_{e}^{\circ}=F_{{\mathcal{E}}}$, where

[TABLE]

We will separate the set $\pi(F_{{\mathcal{E}}})$ from the sets $\pi(Q_{x})$ for sufficiently large $|x|$ by enclosing them within cylinders. We will first describe a cylinder around $\pi(F_{{\mathcal{E}}})$. To this end we are going to derive an estimate for the distance from the vertical axis to the points in $\pi(F_{{\mathcal{E}}})$ by approximating the sets $\pi(Q_{x})$ through their inscribed cylinders.

Proposition 6**.**

Let the functions ${\ell}^{\pm}$ be defined as

[TABLE]

where $L$ is the distance defined in Proposition 4. Suppose that

[TABLE]

Then for any $(z,w)\in\pi((E_{e}\cap\partial Q_{u})^{\circ})$

[TABLE]

Proof.

Proposition 4 says that the points of the edge corona ${\mathcal{E}}$ are all on a circle. Let $x_{1},\cdots,x_{m}$ be the points of ${\mathcal{E}}$ in the anti-clockwise direction. Recall that

[TABLE]

Let $z_{i}=w/\bar{x}_{i}$ and $r=\sqrt{1-d^{2}}/d$. Proposition 5(ii) implies that $(z,w)\in\pi(Q_{x})$ for any $z\in B(z_{i},r)$ with $|z|<|w|$. Let $\hat{d}=|z_{i}|=|\frac{w}{x_{i}}|=\frac{|w|}{d}$ and

[TABLE]

We want to show that $\hat{s}\leqslant r$, i.e. that the neighbouring disks $B(z_{i},r)$ and $B(z_{i+1},r)$ intersect. Straightforward computation shows that

[TABLE]

We know that $(z,w)\in\pi((E_{e}\cap\partial Q_{u})^{\circ})$, hence $\operatorname{Re}(w)=1$ and $|\operatorname{Im}(w)|\leqslant\tan{\frac{\pi k}{2p}}$. Then,

[TABLE]

since ${\frac{\pi k}{2p}}\leqslant\frac{\pi}{q}$. Therefore

[TABLE]

This means that any point $z\in{{\mathbb{C}}^{\vphantom{x}}}$ with $\frac{|z|}{|w|}$ between ${\ell}^{-}(|w|)$ and ${\ell}^{+}(|w|)$ is contained in the union of the disks $B(z_{i},r)$. Functions ${\ell}^{-}$ and ${\ell}^{+}$ are monotone increasing and decreasing respectively, therefore any point $z$ with $\frac{|z|}{|w|}$ between ${\ell}^{-}\left(\sec{\frac{\pi k}{2p}}\right)$ and ${\ell}^{+}\left(\sec{\frac{\pi k}{2p}}\right)$ is contained in the union of the disks $B(z_{i},r)$ and hence in $\bigcup_{x\in{\mathcal{E}}}\pi(Q_{x})$. It follows that $|z|<|w|\cdot{\ell}^{-}(|w|)$ for any point $(z,w)$ in $\pi(F_{{\mathcal{E}}})$. ∎

We have now enclosed $F_{{\mathcal{E}}}$ within a cylinder, it remains to estimate the position of $Q_{x}$ for sufficiently large $|x|$.

Proposition 7**.**

Let the function $f$ be defined as

[TABLE]

Let $x$ be a point in ${\Gamma}_{1}(u)\backslash\{u\}$ such that $|x|\geqslant R>0$. Then for any $(z,w)\in\pi((E_{e}\cap\partial Q_{u})^{\circ})$

[TABLE]

Proof.

Let $(z,w)$ be a point in $\pi(Q_{x})$. Proposition 5(i) implies that

[TABLE]

hence

[TABLE]

Note that for $s\in(0,1)$ and $t\in\left[1,\sec{\frac{\pi k}{2p}}\right]$ we have

[TABLE]

hence $f(\cdot,t)$ is monotone increasing for $t\in\left[1,\sec{\frac{\pi k}{2p}}\right]$.

For $(z,w)\in\pi((E_{e}\cap\partial Q_{u})^{\circ})\cap\pi(Q_{x})$ we have $|w|\in\left[1,\sec{\frac{\pi k}{2p}}\right]$ and therefore $f(|x|,|w|)>f(R,|w|)$ and

[TABLE]

In Proposition 6 we found an upper bound on the distance from the vertical axis to the points of the set $\pi(F_{{\mathcal{E}}})$. On the other hand, in proposition 7 we provide a lower bound for the distance from the vertical axis to the points of the set $\pi(Q_{x})$. Combining these two estimates we can show under certain conditions that the sets $\pi(Q_{x})$ with $x\not\in{\mathcal{E}}\cup\{u\}$ share no points with $\pi(F_{{\mathcal{E}}})$ and therefore with $\pi(F_{{e}})\subset\pi(F_{{\mathcal{E}}})$. That would mean $\pi(F_{e})=\pi(F_{{\mathcal{E}}})$ and hence $F_{e}=F_{{\mathcal{E}}}$.

Theorem 8**.**

Let $R\geqslant\tanh L$ be such that

[TABLE]

and

[TABLE]

Then

[TABLE]

Proof.

Let $x$ be a point in ${\Gamma}_{1}(u)\backslash\{u\}$ such that $|x|\geqslant R$. Let $(z,w)\in\pi((E_{e}\cap\partial Q_{u})^{\circ})$. We have ${\ell}^{-}\left(\sec{\frac{\pi k}{2p}}\right)\leqslant\frac{1-\sqrt{1-R^{2}}}{R}\leqslant 1$. According to Proposition 6,

[TABLE]

On the other hand Proposition 7 implies that

[TABLE]

So to prove that $\pi(Q_{x})\cap\pi(F_{{\mathcal{E}}})=\varnothing$ for all $x\in{\Gamma}_{1}(u)$ with $|x|\geqslant R$, it is sufficient to show that

[TABLE]

Expressing the function $f(R,t)-{\ell}^{-}(t)$ as

[TABLE]

we see that it is monotone decreasing on $\left[1,\sec\frac{\pi}{q}\right]$ and hence on $\left[1,\sec{\frac{\pi k}{2p}}\right]$ as ${\frac{\pi k}{2p}}\leqslant\frac{\pi}{q}$. Therefore for all $t\in[1,\sec{\frac{\pi k}{2p}}]$ we have

[TABLE]

Thus,we proved that

[TABLE]

for all $x\in{\Gamma}_{1}(u)$ such that $|x|\geqslant R$, i.e. for all $x\in{\Gamma}_{1}(u)\backslash\left({\mathcal{E}}(u)\cup\{u\}\right)$. Therefore,

[TABLE]

6. Proof of Theorem 1

In this section we will apply Theorem 8 to the special case ${\Gamma}_{1}={\Gamma}(p_{1},3,3)^{k}$ with $p_{1}\geqslant k+3$ and ${\Gamma}_{2}=(C_{3})^{k}$ to prove Theorem 1. We have $p=\operatorname{lcm}(p_{1},3)=3p_{1}$. Let ${\alpha}=\frac{\pi}{2p_{1}}\in\left(0,\frac{\pi}{10}\right)$. In the case $q=r=3$ the formulas in Proposition 4 become

[TABLE]

To use Theorem 8 in this case, we need to find $R\geqslant\tanh L$ such that

[TABLE]

A careful study of the structure of the orbit ${\Gamma}_{1}(u)$ (for details see Proposition 72 in [Pra01]) shows that $|x|\geqslant R$ for every point $x\in{\Gamma}_{1}(u)\backslash({\mathcal{E}}\cup\{u\})$, where

[TABLE]

It is easy to check that $R>\tanh L$. We then compute

[TABLE]

hence the inequality ${\ell}^{-}\left(\sec{\frac{\pi k}{2p}}\right)\leqslant\frac{1-\sqrt{1-R^{2}}}{R}$ is equivalent to

[TABLE]

which is equivalent to

[TABLE]

We have

[TABLE]

since $k\leqslant p_{1}-3$ and ${\alpha}<\pi/6$. Thus

[TABLE]

It remains to show that

[TABLE]

Note that

[TABLE]

since $\sin{\alpha}\in(0,\frac{1}{2})$. Thus we have checked that the conditions of Theorem 8 are satisfied in this case, hence $F_{e}^{\circ}=F_{{\mathcal{E}}}$.

Acknowledgements: We would like to thank the referees for their helpful comments.