Journal of Lie Theory Volume 8 (1998) 51–66 ◯^{C}1998 Heldermann Verlag

Visualization of the isometry group action on the Fomenko–Matveev–Weeks manifold ^{∗}

Alexander Mednykh and Andrei Vesnin

Communicated by E. B. Vinberg

Abstract. The smallest known three-dimensional closed orientable hyperbolic manifold M_{1} , whose volume is equal to 0.94 …, was obtained independently by A. Fomenko and S. Matveev and by J. Weeks. It is known that the isometry group of the manifold M_{1} is isomorphic to the dihedral group 𝔻_{6} of order 12. The aim of the present paper is to describe the lattice of the action of the isometry group Isom(M_{1}) on the manifold M_{1} . We obtain all orbifolds which arise as quotient spaces of M_{1} by the action of the subgroups of Isom(M_{1}). In particular, we describe the manifold M_{1} as the two-fold covering of the 3-sphere branched over the knot 9_{49} and as the cyclic three-fold covering of the 3-sphere branched over the two-bridge knot 5_{2} .

1. Introduction

In the present paper we study the properties of the smallest known three-dimensional closed orientable hyperbolic manifold M_{1} that was constructed independently by A. Fomenko and S. Matveev [3] and by J. Weeks [13].

A three-dimensional hyperbolic manifold can be defined as a quotient space M = ℍ^{3}/Γ, where Γ is a discrete group of isometries of the three-dimensional Lobachevsky space ℍ^{3}, acting without fixed points. The concept of the volume in ℍ^{3} is carried over naturally to M. Further we will consider three-dimensional orientable hyperbolic manifolds of finite volume.

The structure of the set of volumes of three-dimensional hyperbolic manifolds was described by W. Thurston and T. Jørgensen. According to the Thurston– Jørgensen theorem [12], volumes of three-dimensional hyperbolic manifolds form a well-ordered non-discrete subset of the real line of the order type ω^{ω}. In particular, there exists a closed manifold with the smallest volume.

The first manifold which pretended to be the smallest closed manifold was constructed by R. Meyerhoff (see [12]). This manifold was obtained by (5, −1)

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag Dehn surgery in the figure-eight knot, and its volume is equal to 0.98 . . .. But several years later the another pretender appeared.

∗The research was supported by INTAS-94-1474 and the Russian Foundation for Basic Research.

In [13] J. Weeks calculated volumes of hyperbolic manifolds, obtained by Dehn surgeries on hyperbolic knots and links of the small order and other cusped manifolds. It was possible due to his nice computer program SnapPea [11]. As the result of the calculations, the closed hyperbolic manifold M_{1} with the volume equals to 0.94 … was found. At the same time, numerous computer calculations of volumes were given by A. Fomenko and S. Matveev [3]. This manifold also was appeared in their list (but it was obtained by another way). We remark that the manifold M_{1} can be obtained by Dehn surgeries with parameters (5, −2) and (5, −1) on both components of the Whitehead link. Further the manifold M_{1} will be referred to as the Fomenko–Matveev–Weeks manifold.

Moreover, A. Fomenko and S. Matveev [3] conjectured the structure of the initial segment of the set of volumes of three-dimensional hyperbolic manifolds. In [6] C. Hodgson and J. Weeks refined the smallest manifolds computing a lot of volumes by SnapPea program.

In [9] E. Molnar constructed the fundamental polyhedron in ℍ^{3} for the fundamental group of the manifold M_{1} and deduced that the isometry group of this manifold is the dihedral group 𝔻_{6} of order 12. Independently, using computer calculations Isom(M_{1}) = 𝔻_{6} was found in [6].

The aim of the present paper is to describe the lattice of the action of the isometry group Isom(M_{1}) on the manifold M_{1}. We obtain all orbifolds which arise as quotient spaces of M_{1} by the action of the subgroups of Isom(M_{1}). In particular, we describe the manifold M_{1} as the two-fold covering of the 3-sphere branched over the knot 9_{49} and as the cyclic three-fold covering of the 3-sphere branched over the two-bridge knot 5_{2}.

2. The lattice of the isometry group action

Let us consider the fundamental group Γ = π_{1}(M_{1}) of the Fomenko-Matveev-Weeks manifold M_{1}. According to [9], the group Γ has the following presentation:

Γ = ⟨ a, b | a b a b^{2} a^{−2} b^{2} = b a b a^{2} b^{−2} a^{2} = 1 ⟩.  (1)

It was observed by E. Molnar [9] that the isometry group Isom(M_{1}) of the manifold M_{1} is generated by the following three automorphisms of the group Γ:

\{ \begin{matrix}a → a^{−1}, \\ b → b^{−1}, \end{matrix}  \{ \begin{matrix}a → b, \\ b → a, \end{matrix}  and  \{ \begin{matrix}a → a, \\ b → a^{−1}b^{−1}. \end{matrix}  (2)

By the Mostow rigidity theorem, there exist isometries r, s and t of the Lobachevsky space ℍ^{3} such that

\{ \begin{matrix}r a r^{−1} = a^{−1}, \\ r b r^{−1} = b^{−1}, \end{matrix}  \{ \begin{matrix}s a s^{−1} = b, \\ s b s^{−1} = a, \end{matrix}  and  \{ \begin{matrix}t a t^{−1} = a, \\ t b t^{−1} = a^{−1}b^{−1}. \end{matrix} (3)

From(3) we get the properties of the products of the isometries r, s and t:

\{ \begin{matrix}r^{2} a r^{−2} = a, \\ r^{2} b r^{−2} = b, \end{matrix} \{ \begin{matrix}s^{2} a s^{−2} = a, \\ s^{2} b s^{−2} = b, \end{matrix} \{ \begin{matrix}t^{2} a t^{−2} = a = a^{−1}aa, \\ t^{2} b t^{−2} = a^{−1}ba, \end{matrix}

\{ \begin{matrix}(rs) a (rs)^{−1} = b^{−1}, \\ (rs) b (rs)^{−1} = a^{−1}, \end{matrix} \{ \begin{matrix}(rs)^{2} a (rs)^{−2} = a, \\ (rs)^{2} b (rs)^{−2} = b, \end{matrix}

\{ \begin{matrix}(rt) a (rt)^{−1} = a^{−1}, \\ (rt) b (rt)^{−1} = ab, \end{matrix} \{ \begin{matrix}(rt)^{2} a (rt)^{−2} = a, \\ (rt)^{2} b (rt)^{−2} = b, \end{matrix}

\{ \begin{matrix}(st) a (st)^{−1} = b, \\ (st) b (st)^{−1} = b^{−1}a^{−1}, \end{matrix} \{ \begin{matrix}(st)^{2} a (st)^{−2} = b^{−1}a^{−1}, \\ (st)^{2} b (st)^{−2} = a, \end{matrix} \{ \begin{matrix}(st)^{3} a (st)^{−3} = a, \\ (st)^{3} b (st)^{−3} = b. \end{matrix}

Because the centralizer of any discrete cocompact subgroup of the group Isom(ℍ^{3}) is trivial, we have Inn(Γ) ≅ Γ. Whence, using the above formulae, we get:

r^{2} = s^{2} = (rs)^{2} = (rt)^{2} = (st)^{3} = 1, t^{2} = a^{−1}.  (4)

We recall that the group of isometries of a hyperbolic 3-manifold is canonically isomorphic to the group of outer automorphisms of its fundamental group. Therefore,

Isom(M_{1}) ≅ N(Γ) / Γ,

where N(Γ) is the normalizer of the group Γ in the isometry group Isom(ℍ^{3}).

Let us consider the cosets R = rΓ, S = sΓ and T = tΓ in the group N(Γ)/Γ. Then the group Isom(M_{1}) is generated by R, S and T. According to (4), the group Isom(M_{1}) is a Coxeter group with the following Coxeter diagram:

[figure: R; S; T]

Therefore, the group Isom(M_{1}) is isomorphic to the dihedral group 𝔻_{6} of order 12. Another proof was given independently by C. Hodgson and J. Weeks [6] using computer program SnapPea [11]. Thus,

Isom(M_{1}) ≅ G = ⟨ R, S, T ⟩ ≅ 𝔻_{6},  (5)

and

N(Γ) ≅ ⟨ a, b, r, s, t ⟩,  (6)

where relations between generators a, b, r, s, and t are given by (1), (3) and (4).

The following theorem describes the action of the group Isom(M_{1}) on the manifold M_{1}.

Theorem. The lattice of the action of the isometry group Isom(M_{1}) on the manifold M_{1} is presented in Fig. 1, where 9_{49}(2), 5_{2}(3), 7_{1}^{2}(2, 3), Θ_{1}(2, 2, 3), Θ_{2}(2, 2, 3), Θ_{3}(2, 2, 2), and T (2, 2, 2, 2, 2, 3) are three-dimensional hyperbolic orbifolds whose underlying space is the 3-sphere and whose singular sets are pictured in Figures 3 – 9; L_{5,−1}(2) and L_{5,−2}(2) are orbifolds whose underlying spaces are the lens spaces L_{5,−1} and L_{5,−2}. In this diagram notations A ⟶ B, A ⟶^{3} B

and A ⟶^{(3)} B correspond to a two-fold covering, to a regular three-fold covering and to an irregular three-fold covering, respectively.

[figure: M_{1}; L_{5,−1}(2); L_{5,−2}(2); 9_{49}(2); 5_{2}(3); Θ_{3}(2, 2, 2); Θ_{1}(2, 2, 3); Θ_{2}(2, 2, 3); 7_{1}^{2}(2, 3); T (2, 2, 2, 2, 2, 3); 3; 3; (3); (3); (3); Fig. 1. The lattice of the action of G.]

Proof. The proof of the theorem is the immediate consequence of Propositions 1 and 2, and Lemmata 1 – 9 given below.

3. The lattice of subgroups of the group G

In this section we consider the subgroup structure of the group G ≅ Isom(M_{1}). We observe that the group G = ⟨ R, S, T ⟩ ≅ 𝔻_{6} has exactly 10 subgroups up to conjugations in G. These subgroups are the following: the trivial group ⟨ 1 ⟩; three subgroups ⟨ R ⟩, ⟨ S ⟩ and ⟨ T ⟩ of order 2; the subgroup ⟨ ST ⟩ of order 3; the subgroup ⟨ R, S ⟩ of order 4; three subgroup ⟨ S, T ⟩, ⟨ RS, RT ⟩ and ⟨ R, RT ⟩ of order 6; and the group G = ⟨ R, S, T ⟩ of order 12. By direct considerations we see that these subgroups of the group G form the lattice with respect to subgroup inclusion which is pictured in Fig. 2. In this diagram notations A ⟶ B, A ⟶^{3} B and A ⟶^{(3)} B correspond respectively to the case when A is a subgroup of index 2 in B; to the case when A is a normal subgroup of index 3 in B; and to the case when A is a subgroup of index 3 in B, but not normal.

[figure: ⟨ 1 ⟩; ⟨ S ⟩; ⟨ RS ⟩; ⟨ R ⟩; ⟨ ST ⟩; ⟨ R, S ⟩; ⟨ S, T ⟩; ⟨ RS, ST ⟩; ⟨ R, ST ⟩; ⟨ R, S, T ⟩; 3; 3; (3); (3); (3); Fig. 2. The lattice of the subgroups of G.]

We will formulate the above-mentioned property of the group G as the following Proposition 1. The lattice of subgroups of the group G up to conjugation is presented in Fig. 2.

Recall the well-known close relations between subgroups of the isometry group G of a manifold M = ℍ^{3}/Γ and its quotient spaces. Denote by φ the canonical epimorphism

φ : N(Γ) ⟶ N(Γ) / Γ ≅ G

defined by

φ : r → R, s → S, t → T.

We note that the exact sequence defined by the epimorphism φ:

1 ⟶ Γ ⟶ N(Γ) ⟶^{φ} G ⟶ 1

does not split.

Proposition 2. Assume that M, Γ, and G are as above. Then the following properties hold:

(1) Let H < G and Γ_{H} = φ^{−1}(H). Then the orbifold M/H is isometric to the quotient space ℍ^{3}/Γ_{H} and its orbifold group is π_{1}(M/H) ≅ Γ_{H}.

(2) Let H_{1} < H_{2} < G. Then the orbifold covering M/H_{1} → M/H_{2} induced by the group inclusion H_{1} < H_{2} is regular if and only if H_{1} ◁ H_{2}.

(3) Let Γ_{H} be a group such that Γ < Γ_{H} < N(Γ) and H = φ(Γ_{H}). Then the orbifold ℍ^{3}/Γ_{H} is isometric to the orbifold M/H.

4. The orbifolds arising under the group G action

4.1. The orbifold 9_{49}(2)

We recall [3], [13] that the Fomenko-Matveev-Weeks manifold M_{1} can be obtained by the Dehn surgeries on the components of the Whitehead link with surgery parameters (5, −1) and (5, −2). The covering properties of the manifolds obtained by Dehn surgeries on the Whitehead link were studied in [8], where these manifolds were described as the two-fold branched coverings of the 3-sphere. In particular, it was shown in [8] that the manifold M_{1} is the two-fold covering of the 3-sphere branched over the knot 9_{49} pictured in Fig. 3.

[figure: l; m; r; Fig. 3. The knot 9_{49}.]

Denote by 9_{49}(2) the orbifold whose underlying space is the 3-sphere and whose singular set is the knot 9_{49} with index 2. In this case one can say that the manifold M_{1} is the two-fold covering of the orbifold 9_{49}(2). As we see from the diagram in Fig. 2, there are three subgroups of order 2 in the group G. Therefore, the covering M_{1} ⟶^{2} 9_{49}(2) can be induced either by the involution R, by the involution S, or by the involution RS. In these cases we will get either the orbifold M_{1}/⟨ R ⟩ = ℍ^{3}/⟨ Γ, r ⟩, the orbifold M_{1}/⟨ S ⟩ = ℍ^{3}/⟨ Γ, s ⟩, or the orbifold M_{1}/⟨ RS ⟩ = ℍ^{3}/⟨ Γ, rs ⟩.

In virtue of the Armstrong theorem [1], the fundamental group of the underlying space of a orbifold ℍ^{3}/Δ is isomorphic to the factor group Δ/Δ_{0}, where Δ_{0} is the normal subgroup of Δ generated by all elements of finite order. Now we will find the fundamental groups of the underlying spaces of three above orbifolds. Firstly we consider the underlying space of the orbifold M_{1}/⟨ R ⟩ and its fundamental group ⟨ Γ, r ⟩ / Ncl(r), where Ncl(r) denotes the normal closure of r in the group ⟨ Γ, r ⟩. Analogously, we use notations Ncl(s) and Ncl(rs) for normal closures of elements s and rs in groups ⟨ Γ, s ⟩ and ⟨ Γ, rs ⟩, respectively. From(1) and (3) we have

⟨ Γ, r ⟩ / Ncl(r) = ⟨ a, b, r | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1,
r a r^{−1} = a^{−1}, r b r^{−1} = b^{−1}, r = 1 ⟩
≅ ⟨ a, b | a^{2} = b^{2} = 1, aba = bab = 1 ⟩ = ⟨ 1 ⟩. (7)

Analogously, we find

⟨ Γ, s ⟩ / Ncl(s) = ⟨ a, b, s | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1,
s a s^{−1} = b, s b s^{−1} = a, s = 1 ⟩
≅ ⟨ a | a^{5} = 1 ⟩ = ℤ_{5}.  (8)

By the same arguments, we get

⟨ Γ, rs ⟩ / Ncl(rs) = ⟨ a, b, rs | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1,

(rs) a (rs)^{−1} = b^{−1}, (rs) b (rs)^{−1} = a^{−1}, rs = 1 ⟩

≅ ⟨ a | a^{5} = 1 ⟩ = ℤ_{5}.  (9)

Because just one of the groups (7), (8) and (9) is trivial, we get that the covering M_{1} ⟶^{2} 9_{49}(2) is induced by the involution r.

Let us demonstrate the same by the direct calculation. More exactly, let us show that the orbifold groups of the orbifold M_{1}/⟨ R ⟩ and of the orbifold 9_{49}(2) are isomorphic. Indeed, from (1) and (3) we get

π_{1}(M_{1}/⟨ R ⟩) = ⟨ Γ, r ⟩ = ⟨ a, b, r | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1,
r a r^{−1} = a^{−1}, r b r^{−1} = b^{−1}, r^{2} = 1 ⟩ (10)

If we denote l = ar and m = rb, then from (10) we get

π_{1}(M_{1}/⟨ R ⟩) = ⟨ l, m, r | l^{2} = m^{2} = r^{2} = 1, (lm)^{2} (lr)^{2} (mr)^{2} lr = 1,
(rl)^{2} (rm)^{2} (lm)^{2} rm = 1 ⟩,  (11)

what is the group of the orbifold 9_{49}(2), where l, m and r correspond to the notations in Fig. 3. Because both compact orbifolds M_{1}/⟨ R ⟩ and 9_{49}(2) are hyperbolic and their groups are isomorphic, they are isometric by the rigidity theorem. Thus, the following statement is proved.

Lemma 1. In the above notations we have

⟨ Γ, r ⟩ ≅ π_{1}(9_{49}(2))

and

M_{1}/⟨ R ⟩ ≅ 9_{49}(2).

4.2. The orbifold 5_{2}(3)

The group G = Isom(M_{1}) contains only one subgroup of order three, which is generated by the element ST. Let us consider the orbifold group of the corresponding quotient space:

π_{1}(M_{1}/⟨ ST ⟩) = ⟨ Γ, st ⟩ = ⟨ a, b, st | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1,

(st) a (st)^{−1} = b, (st) b (st)^{−1} = b^{−1}a^{−1}, (st)^{3} = 1 ⟩

≅ ⟨ a, b, u | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1,
u a u^{−1} = b, u b u^{−1} = b^{−1}a^{−1}, u^{3} = 1 ⟩,

where u = st. Denote v = ua, then a = u^{−1}v and b = vu^{−1}. Hence,

π_{1}(M_{1}/⟨ ST ⟩) ≅ ⟨ u, v | uv^{−1}uv^{−1}u^{−1}vu^{−1}v^{−1}uv^{−1}uvu^{−1}v = 1,
u^{3} = v^{3} = 1 ⟩.  (12)

By direct calculations, using the Wirtinger algorithm, we see that the group (12) is the orbifold group of the orbifold whose underlying space is the 3-sphere and the singular set is the knot pictured in Fig. 4 with index equals three.

[figure: u; v; s; rs; r; Fig. 4. The knot 5_{2}.]

Moreover, generators u and v from (12) correspond to the loops in Fig. 4. The knot diagram in Fig. 4 is non-alternative. Using Reidemeister moves, one can simplify this diagram to the standard diagram of the knot 5_{2}. Let us denote by 5_{2}(3) the orbifold with the 3-sphere as its underlying space and with the knot 5_{2} with index 3 as its singular set. Because the orbifold 5_{2}(3) is hyperbolic [12], summarizing above considerations we get Lemma 2. In the above notations we have

⟨ Γ, st ⟩ ≅ π_{1}(5_{2}(3))

and

M_{1}/⟨ ST ⟩ ≅ 5_{2}(3).

4.3. The isometries of the orbifold 5_{2}(3)

It was shown in [5] and [7] that the symmetry group of the complement 𝕊^{3} \ 5_{2} of the knot 5_{2} in the 3-sphere consists of four elements, and all nontrivial symmetries are involutions. We will show that these involutions induce the involutions of the orbifold 5_{2}(3) which can be lifted to the involutions r, s and rs on the universal covering space ℍ^{3}. We recall that the group π_{1}(5_{2}(3)) of the orbifold 5_{2}(3) is isomorphic to the group π_{1}(M_{1}/⟨ ST ⟩) and by (12) it has the following presentation:

π_{1}(5_{2}(3)) = ⟨ u, v, | uv^{−1}uv^{−1}u^{−1}vu^{−1}v^{−1}uv^{−1}uvu^{−1}v = 1, u^{3} = v^{3} = 1 ⟩,

(13)

where u = st and v = ua = sta. By (4) we have t^{2} = a^{−1}, so we get v = st^{−1}. Considering the action of the involutions r, s and rs on the generators u and v, we will get:

r u r^{−1} = r st r^{−1} = st^{−1} = v,  r v r^{−1} = r st^{−1} r^{−1} = st = u, (14)
s u s^{−1} = s st s^{−1} = ts^{−1} = v^{−1}, s v s^{−1} = s st^{−1} s^{−1} = t^{−1}s^{−1} = u^{−1}, (15)
(rs) u (rs)^{−1} = r v^{−1} r^{−1} = u^{−1},  (rs) v (rs)^{−1} = r u^{−1} r^{−1} = v^{−1}, (16)

where we used (4). Denote isometries of 5_{2}(3) defined by (14), (15) and (16) also by r, s and rs in the correspondence to the notations of the isometries of ℍ^{3}. Axes of the involutions s and rs are pictured in Fig. 4, and the axis of the involution r is perpendicular to both above axes.

We will consider quotient spaces of the orbifold 5_{2}(3) by the isometries r, s and rs, and denote these orbifolds by

7_{1}^{2}(2, 3) = 5_{2}(3) / ⟨ r ⟩,  (17)
Θ_{1}(2, 2, 3) = 5_{2}(3) / ⟨ s ⟩,  (18)
Θ_{2}(2, 2, 3) = 5_{2}(3) / ⟨ rs ⟩.  (19)

4.4. The orbifold 7_{1}^{2}(2, 3)

By the definition, the orbifold 7_{1}^{2}(2, 3) is the quotient space of the orbifold 5_{2}(3) by the action of the involution r. Therefore, the underlying space of 7_{1}^{2}(2, 3) is the 3-sphere. Because the axis of the involution r and the singular set of 5_{2}(3) are disjoint, the singular set of the quotient space is a two-component link, one component of which has the branch index 2, and other has the branch index 3. One can check directly that this two-component singular set is the two-component link with seven cross-points which is noted by 7_{1}^{2} in the table of knots and links [10]. This explains the notation 7_{1}^{2}(2, 3) used for this orbifold.

Lemma 3. In the above notations we have

⟨ Γ, r, st ⟩ ≅ π_{1}(7_{1}^{2}(2, 3))

and

M_{1}/⟨ R, ST ⟩ ≅ 7_{1}^{2}(2, 3).

Proof. According to the action of the automorphisms r and st on the group Γ, we have:

⟨ Γ, r, st ⟩ = ⟨ a, b, r, st | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1, (st)^{3} = 1,
(st) a (st)^{−1} = b, (st) b (st)^{−1} = b^{−1}a^{−1},
r a r^{−1} = a^{−1}, r b r^{−1} = b^{−1}, r^{2} = 1 ⟩
≅ ⟨ u, v, r | uv^{−1}uv^{−1}u^{−1}vu^{−1}v^{−1}uv^{−1}uvu^{−1}v = 1, u^{3} = v^{3} = 1,
r u r^{−1} = v, r v r^{−1} = u, r^{2} = 1 ⟩,

where u = st, v = ua = st^{−1}. Whence

⟨ Γ, r, st ⟩ = ⟨ u, r | uru^{−1}ruru^{−1}ru^{−1}ruru^{−1}ru^{−1}ruru^{−1}rururu^{−1}rur = 1,
u^{3} = r^{2} = 1 ⟩.  (20)

As one can see by direct calculations, the group ⟨ Γ, r, st ⟩ is isomorphic to the group of the orbifold 7_{1}^{2}(2, 3), where generators u and r correspond to loops of the same name in Fig. 5.

[figure: 2; 3; α; u; β; γ; δ; ϵ; r; Fig. 5. The link 7_{1}^{2}.]

Indeed, by the Wirtinger algorithm, α = r^{−1}u^{−1}r, β = α^{−1}uα, γ = β^{−1}αβ, δ = rγr^{−1}, ϵ = r^{−1}βr, and from the relation ϵ^{−1} = β^{−1}δβ we will get the nontrivial relation from the presentation (20) for the group ⟨ Γ, r, st ⟩.

Corollary. The following diagram of the regular orbifold coverings is commutative:

[figure: M_{1}; 3; 2; 9_{49}(2); 5_{2}(3); 3; 2; 7_{1}^{2}(2, 3)]

Proof. We recall that by virtue of lemmata 1 – 3, we have the following descriptions of the orbifolds: 9_{49}(2) ≅ M_{1}/⟨ R ⟩, 5_{2}(3) ≅ M_{1}/⟨ ST ⟩ and 7_{1}^{2}(2, 3) ≅ M_{1}/⟨ R, ST ⟩. The commutativity of the above diagram follows from the fact that the groups ⟨ R ⟩ ≅ ℤ_{2} and ⟨ ST ⟩ ≅ ℤ_{3} are normal subgroups of the group ⟨ R, ST ⟩ ≅ ℤ_{2} ⊕ ℤ_{3} of indices 3 and 2 respectively.

4.5. The orbifold Θ_{1}(2, 2, 3)

Let us consider the orbifold Θ_{1}(2, 2, 3) = 5_{2}(3)/⟨ s ⟩.

Lemma 4. In the above notations we have

⟨ Γ, s, t ⟩ ≅ π_{1}(Θ_{1}(2, 2, 3))

and

M_{1}/⟨ S, T ⟩ ≅ Θ_{1}(2, 2, 3).

Proof. The proof of the lemma follows from the definition of the orbifold Θ_{1}(2, 2, 3) and Proposition 2.

Because the underlying space of the orbifold 5_{2}(3) is the 3-sphere and the isometry s is an involution, we get that the underlying space of the orbifold Θ_{1}(2, 2, 3) is the 3-sphere also. The singular set Θ_{1} of Θ_{1}(2, 2, 3) is pictured in Fig. 6.

[figure: s; p; f; Fig. 6. The Θ_{1}-curve.]

From(1), (3) and (4) we get the following presentation:

⟨ Γ, s, t ⟩ =  ⟨ a, b, s, t | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1, s^{2} = (st)^{3} = 1,

t^{2} = a^{−1}, s a s^{−1} = b, s b s^{−1} = a, t a t^{−1} = a. t b t^{−1} = a^{−1}b^{−1} ⟩.

Whence

⟨ Γ, s, t ⟩ = ⟨ p, s | p^{3} = s^{2} = [(sp^{−1})^{3} (sp)^{4}]^{2} = 1 ⟩,  (21)

where p = st. The elements s, p and f = (sp^{−1})^{3} (sp)^{4} correspond to the loops in Θ_{1}(2, 2, 3) which are shown in Fig. 6. We remark that the singular set of the orbifold Θ_{1}(2, 2, 3) is a Θ-curve in the terminology of [14], that is a spatial Θgraph with 2 vertices and 3 edges. If we delete the edge corresponding to the loop f, then we will get the torus knot 7_{1}. Hence, we can regard the singular set of the orbifold Θ_{1}(2, 2, 3) as the knot 7_{1} with a bridge. The relations p^{2} = 1, s^{3} = 1, and f^{2} = 1 in the group of the orbifold Θ_{1}(2, 2, 3) correspond to the loops around the three edges of the spatial Θ-graph.

By the other hand, as we see from the presentation (21), the group ⟨ Γ, s, t ⟩ is a generalized triangle group in the terminology of [2]. We recall that some connections between generalized triangle groups and 3-orbifolds with a two-bridge knot with a bridge as the singular set were studied in [4].

4.6. The orbifold Θ_{2}(2, 2, 3)

Let us consider the orbifold Θ_{2}(2, 2, 3) = 5_{2}(3)/⟨ rs ⟩.

Lemma 5. In the above notations we have

⟨ Γ, rs, st ⟩ ≅ π_{1}(Θ_{2}(2, 2, 3))

and

M_{1}/⟨ RS, ST ⟩ ≅ Θ_{2}(2, 2, 3).

Proof. The proof of the lemma follows from the definition of the orbifold Θ_{2}(2, 2, 3) and Proposition 2.

Because the underlying space of the orbifold 5_{2}(3) is the 3-sphere and the isometry rs is an involution, we get that the underlying space of the orbifold Θ_{2}(2, 2, 3) is the 3-sphere also. The singular set Θ_{2} of Θ_{2}(2, 2, 3) is pictured in Fig. 7.

[figure: α; β; h; Fig. 7. The Θ_{2}-curve.]

From(1), (3) and (4) we get the following presentation:

⟨ Γ, rs, st ⟩ = ⟨ a, b, rs, st | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1, (rs)^{2} = 1,
(rs) a (rs)^{−1} = b^{−1}, (rs) b (rs)^{−1} = a^{−1},
(st) a (st)^{−1} = b, (st) b (st)^{−1} = b^{−1}a^{−1}, (st)^{3} = 1 ⟩. (22)

In respect to generators α and β pictured in Fig. 7, this group has presentation:

⟨ Γ, rs, st ⟩ = ⟨ α, β | α^{3} = β^{2} = [(αβ)^{2} (α^{−1}β) (αβ)^{2} (α^{−1}β)^{2}]^{2} = 1 ⟩, (23)

where h = (αβ)^{2} (α^{−1}β) (αβ)^{2} (α^{−1}β)^{2}. We remark that the singular set of the orbifold Θ_{2}(2, 2, 3) is the two-bridge knot 5_{2} with a bridge; the orbifold group with the presentation (23) is a generalized triangle group; and the relations α^{3} = 1, β^{2} = 1 and h^{2} = 1 correspond to the loops around the edges of the spatial Θ-graph.

4.7. The orbifold Θ_{3}(2, 2, 2)

According to [5] and [7], the symmetry group of the knot 9_{49} is the dihedral group 𝔻_{3} of order 6. This group consists of three elements which form the cyclic group of order 3 and of the three mutually conjugated involutions. We will show that one of these involutions is induced by the involution rs of the space ℍ^{3}.

Indeed, from (1), (3) and (4), using the relations l = ar and m = rb from subsection 4.1, we have:

(rs) l (rs)^{−1} = m, (rs) r (rs)^{−1} = r, (rs) m (rs)^{−1} = l,
l^{2} = m^{2} = r^{2} = (rs)^{2} = 1.

Therefore, the isometry rs induces an isometry of the orbifold 9_{49}(2) which interchange the arcs corresponding to l and m, and preserves the arc corresponding to r invariant (see Fig. 3).

The quotient space of the orbifold 9_{49}(2) = ℍ^{3}/⟨ Γ, r ⟩ by the action of the isometry rs is the orbifold Θ_{3}(2, 2, 2) whose underlying space is the 3-sphere and whose singular set, pictured in Fig. 8, is the Θ-curve, each arc of which has index 2 (that is so called π-orbifold).

[figure: l; r; s; rs; Fig. 8. The Θ_{3}-curve.]

Because of (1), (3) and (4), using that

Θ_{3}(2, 2, 2) = 9_{49}(2) / ⟨ rs ⟩ = ℍ^{3} / ⟨ Γ, r, rs ⟩ = ℍ^{3} / ⟨ Γ, r, s ⟩, the group π_{1}(Θ_{3}(2, 2, 2)) is isomorphic to the group ⟨ Γ, r, s ⟩ and has the following presentation:

⟨ Γ, r, s ⟩ = ⟨ a, b, r, s | abab^{2}a^{−2}b^{2} = baba^{2}b^{−2}a^{2} = 1, r^{2} = s^{2} = (rs)^{2} = 1,
r a r^{−1} = a^{−1}, r b r^{−1} = b^{−1}, s a s^{−1} = b, s b s^{−1} = a ⟩
≅ ⟨ l, r, s | l^{2} = r^{2} = s^{2} = (rs)^{2} = (srl)^{2}(rlsl)r(lslr)(lrs)^{2}l = 1 ⟩, (24)

where generators l = ar, r and s correspond to Fig. 8.

Summarizing above considerations, we get

Lemma 6. In the above notations we have

⟨ Γ, r, s ⟩ ≅ π_{1}(Θ_{3}(2, 2, 2))

and

M_{1}/⟨ R, S ⟩ ≅ Θ_{3}(2, 2, 2).

4.8. The orbifolds L_{5,−1}(2) and L_{5,−2}(2)

It is clear that the group π_{1}(Θ_{3}(2, 2, 2)) = ⟨ Γ, r, s ⟩ with the presentation (24) has three subgroups of index 2, which can be derived as kernels of the three epimorphisms φ_{s}, φ_{lr} and φ_{lrs} of the group ⟨ Γ, r, s ⟩ to ℤ_{2} = ⟨ α | α^{2} = 1 ⟩. These epimorphisms are defined by

φ_{s}(s) = α, φ_{s}(l) = φ_{s}(r) = 1,  (25)
φ_{lr}(s) = 1, φ_{lr}(l) = φ_{lr}(r) = α,  (26)
φ_{lrs}(l) = φ_{lrs}(r) = φ_{lrs}(s) = α.  (27)

By the Reidemeister–Schreier method, using formulae (1), (3) and (4), we have:

Kerφ_{s} ≅ ⟨ l, r, sls ⟩ ≅ ⟨ ar, r, br ⟩ ≅ ⟨ Γ, r ⟩,  (28)
Kerφ_{lr} ≅ ⟨ lr, s ⟩ ≅ ⟨ a, s ⟩ ≅ ⟨ Γ, s ⟩,  (29)
Kerφ_{lrs} ≅ ⟨ lr, rs ⟩ ≅ ⟨ a, rs ⟩ ≅ ⟨ Γ, rs ⟩.  (30)

The singular set Θ_{3} of the orbifold Θ_{3}(2, 2, 2) is pictured in Fig. 8. Similar to [14], we can correspond to the Θ-curve Θ_{3} three constituent knots K_{1}, K_{2} and K_{3}. The first knot K_{1}, which we will get after deleting from Θ_{3} the edge corresponding to r (see Fig. 8), is the trivial knot. The second knot K_{2}, which corresponds to the deleting of the edge s, is the torus knot 5_{1}, that is the twobridge knot K(−5/1). The third knot K_{3}, which corresponds to the deleting of the edge rs, is the figure-eight knot 4_{1}, that is the two-bridge knot K(−5/2).

The corresponding orbifolds ℍ^{3}/Kerφ_{s} = ℍ^{3}/⟨ Γ, r ⟩, ℍ^{3}/Kerφ_{lr} = ℍ^{3}/⟨ Γ, s ⟩ and ℍ^{3}/Kerφ_{lrs} = ℍ^{3}/⟨ Γ, rs ⟩ can be obtained as the two-fold coverings of the orbifold Θ_{3}(2, 2, 2) branched over the constituent knots K_{1}, K_{2} and K_{3}, respectively. Therefore, the underlying space of the orbifold ℍ^{3}/⟨ Γ, r ⟩ is the 3-sphere, the underlying space of the orbifold ℍ^{3}/⟨ Γ, s ⟩ is the lens space L_{5,−1} and the underlying space of the orbifold ℍ^{3}/⟨ Γ, rs ⟩ is the lens space L_{5,−2}.

The singular set of the orbifold ℍ^{3}/⟨ Γ, r ⟩ is the knot 9_{49} with index 2. Hence ℍ^{3}/Kerφ_{s} = 9_{49}(2) (see Lemma 1) and this orbifold was discussed in subsection 4.1. The singular sets of the orbifolds ℍ^{3}/⟨ Γ, s ⟩ and ℍ^{3}/⟨ Γ, rs ⟩ have branch index 2, so we denote these orbifolds by L_{5,−1}(2) and L_{5,−2}(2), respectively. M. Sakuma kindly informed the authors that using his approach, one can describe the singular sets of the orbifolds L_{5,−1}(2) and L_{5,−2}(2) by lifting them on the 5-fold coverings of L_{5,−1} and L_{5,−2} which are already the 3-sphere.

Using Proposition 2, from above discussions we get the following statements.

Lemma 7. In the above notations we have

⟨ Γ, s ⟩ ≅ π_{1}(L_{5,−1}(2))

and

M_{1}/⟨ S ⟩ ≅ L_{5,−1}(2).

Lemma 8. In the above notations we have

⟨ Γ, rs ⟩ ≅ π_{1}(L_{5,−2}(2))

and

M_{1}/⟨ RS ⟩ ≅ L_{5,−2}(2).

4.9. The orbifold T(2, 2, 2, 2, 2, 3)

We remark that the orbifold Θ_{1}(2, 2, 3) has the non-trivial symmetry of order two. This is the involution which interchanges two vertices of the spatial Θ_{1}-curve shown in Fig. 6.

We will prove that one of the liftings of this involution on the universal covering ℍ^{3} is the isometry r. Indeed, from (1), (3), (4) and (21), using the notation p = st, we get

rpr^{−1} = st^{−1} = sp^{−1}s^{−1}, r^{2} = 1.  (31)

Whence, the group ⟨ Γ, s, t ⟩ with the presentation (21) is invariant under the conjugation by r. Therefore, r induces the isometry of the orbifold Θ_{1}(2, 2, 3) with the quotient space

T (2, 2, 2, 2, 2, 3) = Θ_{1}(2, 2, 3) / ⟨ r ⟩ = ⟨ Γ, r, s, t ⟩.

Because r is the involution and the underlying space of the orbifold Θ_{1}(2, 2, 3) is the 3-sphere, the underlying space of the quotient orbifold is also the 3-sphere. The singular set T of the quotient orbifold is the knotted 1-skeleton of the tetrahedron, pictured in Fig. 9. One of its edges has the index 3, and all other have indices 2. This explains the notation T (2, 2, 2, 2, 2, 3) for this quotient orbifold.

[figure: r; p; s; Fig. 9. The spatial graph T.]

According to the definition of the orbifold T (2, 2, 2, 2, 2, 3) and by Proposition 2 we get

Lemma 9. In the above notations we have

⟨ Γ, r, s, t ⟩ ≅ π_{1}(T (2, 2, 2, 2, 2, 3))

and

M_{1}/⟨ R, S, T ⟩ ≅ T (2, 2, 2, 2, 2, 3).

Using (4), (21), and (31), one can derive the following presentation for the fundamental group of the orbifold T (2, 2, 2, 2, 2, 3):

⟨ Γ, r, s, t ⟩ = ⟨ p, r, s | s^{2} = r^{2} = p^{3} = [(sp)^{4}(sp^{−1})^{3}]^{2} = 1,

rsr^{−1} = s. rpr^{−1} = sp^{−1}s^{−1} ⟩,

where the generators p, r and s of the group π_{1}(T (2, 2, 2, 2, 2, 3)) are correspond to Fig. 9.

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Alexander Mednykh Sobolev Institute of Mathematics Siberian Branch Russian Academy of Sciences Novosibirsk 630090, Russia mednykh@math.nsc.ru Andrei Vesnin Sobolev Institute of Mathematics Siberian Branch Russian Academy of Sciences Novosibirsk 630090, Russia vesnin@math.nsc.ru

Received February 28, 1997 and in final form March 6, 1997