Siberian Mathematical Journal, Vol. 40, No. 1, 1999

ISOMETRIES OF HYPERBOLIC FIBONACCI MANIFOLDSt)

A. Yu. Vesnin and A. A. Rasskazov UDC 512.817 + 515.162

w 1. In t roduc t ion

In the present paper we study the isometry groups of three-dimensional closed orientable hyper- bolic manifolds uniformized by the Fibonaeci groups. The Fibonacci groups F(2, m), re >__ 3, were introduced by J. Conway [1] and are defined by the foUowing presentation:

F(2, m) = (Xl, . . . , Xrn [ Xi2~i-I-1 ----- X i + 2 , i = 1 , . . . , m),

with all subscripts reduced mod m. Some survey of the results about finiteness, arithmeticity, and generalizations of the Fibonacci groups is given in [2]. Application of topological and geometric methods to studying the Fibonacci groups stems from the paper [3] by H. Helling, A. C. Kim and J. Mennicke wherein they constructed a family of three-dimensional manifolds Mn, n >_ 2, such that ~r~(Mn) ~ F(2,2n). Moreover, it was shown in [3] that these manifolds can be equipped with geometric structures of constant curvature. More exactly, the group F(2, 4) ~ Z5 acts by isometries on the spherical space S 3. In this case the manifold M2 is the lens space L(5, 2). The group F(2, 6) is isomorphic to the Euclidean crystallographic group acting on E r In this case the manifold M3 in the Euclidean Hantzsche-Wendt manifold [4]. For n > 4 the group F(2,2n) is isomorphic to a discrete co compact subgroup of PSL(2, C) acting by isometries on the LobachevskiY space I~ 3 without fixed points. For n >_ 4 the manifold Mn is hyperbolic, i.e. it has a metric of constant negative curvature. The manifolds Mn of [3] are ca~ed the Fibonacci manifolds.

We recaU some geometric and topological properties of the manifolds Mn. Exact formulae in terms of the Lobachevskil function for the volumes of hyperbolic Fibonacci manifolds were obtained in [5]. It was shown in [6] that for n >_ 2 the manifold Mn is the n-fold cyclic covering of the three-dimensional sphere, branched over the figure-eight knot. Moreover [7], for n >_ 2 the manifold M~ is the two-fold covering of the three-dimensional sphere, branched over the closed three-strings braid (~rx~21) n. In particular, M2 is two-fold branched over the figure-eight knot, and M3 is two-fold branched over the three-component link named Borromean rings. In [8] the Fibonacci manifolds were obtained by 1/0-Dehn surgery on the manifold A4(ff) that is a punctured torus bundle. In this case

1 In the present paper we develop a geometric approach to studding the Fibona~ci groups and

describe the isometry groups of the hyperbolic Fibona~ei manifolds. We recall that, by the rigidity theorem [9], the group Isom(Mn) of all isometries of the hyperbolic manifdd Mn coincides with the group Out(F(2, 2n)) of the outer automorphisms of the e~nd~eat~a gro,p of

F(2, 2n) = (Zl , . . . , z2, I xiZi+l = zi+2, i = 1 , . . . , 2n).

The obvious automorphism x/ ~ Xi+l, i = I,..., 2n, generates a cyclic subgroup 7.2n of the isometry group Isom(Mn). In [10] N. Kuiper asked whether the group Isom(Mn) is a cyclic group of order 2n. The negative answer follows from [7], where it was shown that Mn admits an involution that does

t) The research was supported by the Russian Foundation for Basic Reseaxch (Grants 98--01-00699 and 96--01-01523).

.... Novosibirsk. Translated from SibirskK MatematieheskK Zhurnal, Vol. 40, No. 1, pp. 14-29, January-February, 1999. Original article submitted February 22, 1997.

0037-4466/99/4001-0009 $22.00 ~ 1999 Kluwer Academic/Plenum Publishers

not belong to the group z2,. Moreover [8], all foliation-preserving homeomorphisms of A4(~) induce homeomorphisms of the Fibona~ci manifold. The group of foliation-preserving homeomorphisms of the manifold .M(~) can be found in [111 and is of order 8n.

The main aim of the present paper is to describe the group Isom(M.). Precisely, we show in Theorem 3.2 that for n > 4 the group Isom(M.) consists of 8n elements and has the presentation

Isom(M,) = (x, ylx 2" = y4= (yx)2 = (y--lx)2 = 1).

We recall [3, 6] that for n = 4, 5, 6, 8,12 the Fibonacci manifold Mn is arithmetic (in the sense of [12]). For these values of n the theorem was proven in [8] by comparing the volumes of the Fibonacci manifolds, calculated in [5], and the estimates of [12] for the volumes of arithmetic orbifolds over a given field. In the present paper the proof of the theorem is given for all n > 6. We use the geometric approach that is based on the results of [13,14] together with the ideas that appeared in [15-17]. Observe that the group Isom(M,) is the group (4,2n12,2) in the notations of [18, p. 159]. This group can also be presented as the semidirect product Z,M3~, where Z , is the cyclic group of order n and DS is the dihedral group of order 80

The structure of the paper is as follows. In Section 2 we recall the classical results by M. Dehn [19] and W. Magnus [20] on the symmetries of the figure-eight knot and show that these symmetries induce isometfies of the Fibonacci manifolds (Theorem 2.1). In Section 3 we study discrete extensions of the group F(2, 2n) and compute the normalizer of this group in the full group of isometries of the Lobachevskil space (Theorem 3.1). This in particular enables us to describe the group Isom(M,) (Theorem 3.2). In Section 4 we consider the orbifolds resulting from the action of isometries on the Fibonacci manifolds.

w 2. Symmetr ies of the Figure-Eight Knot

From the presentation of the Fibonacci group

F(2, 2n) = ( X l , . . . , Z2n I z iZ i+ l = zi+2, i = 1 , . . . , 2n)

we easily find an automorphism p of order n:

p " Zi "-'+ Zi+2, i = 1 , . . . , 2 n ,

where all subscripts are reduced mod 2n. Consider the extension 1", of F(2,2n) by p which is the semidirect product of F(2, 2n) and the cyclic group (p) of order n. Then

I',, ~ F(2,2n)X(p)

'~ ( X l , . . . , Z 2 n , P I P" = 1, p - l z i p = Xi+2, XiZi+l ----- Xi+2, i -- 1 , . . . , 2n) (XI, Z2, p [ ,O n = 1, p - l z l p = ZlZ2 , p - l z 2 p = Z2ZlZ2) .

In particular, the relation p - l z z p = z 2 p - l z l p holds. From this relation we can express Zl"

z l = p z ~ l p - l z 2 .

Thus~ Fn ~- (z2, p ip n = 1, z~lp- lx2p = .z lp-l=2z2>.

Introducing b such that z 2 = bp, we rewrite the last relation as

p - l b - l p - l b p = b-lp-lbpb.

Therefore, we come to the presentation

r . (b, p I p" b - 1, p-lib, p] [b, p]b),

10

where [b,p] = b-lp-Xbp. It is easy to check by a direct application of the Wirtinger algorithm [21] that

r ~ (b,p I p-S[b,p] = [b, p]b)

is the group of the figure-eight knot pictured in Fig. 1, wherein the generators b and p are marked. Thus [22], the group Fn is isomorphic to the fundamental group of the orbifold O(n) whose underlying space is the three-dimensional sphere and whose singular set is the figure-eight knot with the singularity index n.

I b I, P

; 0 "

Fig. 1. The fignre-eight knot.

Observe the following connection between the generators Z l , . . . , Z 2 n of the Fibonacci group F(2,2n) ~ lrl(Mn) and the generators b and p of the group Fn ~ ~rl(O(n)):

z2 = bp, Zl = p z 2 1 p - l z 2 = b - l p - l b p = [b,p].

Since p-lzip = zi+2, we obtain

z2i+2 = p-i(bp)pi, Z 2 / + l = p-i[b, p]pi,

where i = 0 , . . . , n - 1. We recall that the symmetries of the figure-eight knot were first studied by M. Dehn in [19], where

he demonstrated that the group F has eight outer automorphisms forming the dihedral group:

= (~, ~ i ~ = ~4 = ( ~ ) ~ = 1).

In this case the action of the generators ~ and r on r is as follows:

~(b) = p, ~(p) = b, ~(b) = b- lpb , ~(p) = pbp -1

These studies were continued by W. Magnus [20]. He proved that r has no other outer automor- phisms. We note that the automorphism ~r induces an involution of the figure-eight knot whose axis is pictured by the dotted line in Fig. 1. The automorphism r induces an orientation-reversing symmetry of order four. The action of the automorphism group Ds on the generators b and p of F is given in Table 1. The action of D~ with respect to ~nother system of generators was described in [23].

TABLE 1

b p

cr p b r b-lpb pbp -1 r2 p-1 b-1

r 3 pb-lp-1 b-lp-lb

~r pbp -1 b-lpb ~rr 2 b-1 p-1

err s b-lp-1 b pb-lp-1

11

Comparing the presentations of F and Fn, we see that a and r induce automorphisms of I'n and consequently induce symmetries of the orbifold O(n). Connection between the symmetries of the figure-eight knot and the isometrics of the Fibonax:ci manifolds is described in the following

Theorem 2.1. Each symmetry of the figure-eight knot induces an isometry of every hyperbolic Fibonacci manifold.

PROOF. Consider the generators a and r of the symmetry group I~ which act on b and p as above. Since F(2, 2n) is the normal subgroup of Fn, the images

. (x2) = .(bp) = pb = x2x? 1,

~ ( ~ ) = . ( b p ) = b-~pbpbp -~ = p(z?~x2~?x)p -~

belong to F(2, 2n). Since for i = 0 , . . . , n - 1 we have the relations

x2i+2 = p-tz2p', x2i+l = p-i(pz21p-lx2)pi;

therefore, the images

o'(X2i+2) = b-ia(x2)b i, o'(x2i+l) = b-i(b~r-l(x2)b-la'(x2))b i, ~(x2i+2) = (pbp - 1 ) - l ~ ( x 2 ) ( p b p - 1 )

and T(X2i+I) -- (pbp-1)-i((pbp-1)T-l(x2)(pbp-1)-lT(z2))(pbp-1)i,

which can be obtained by conjugating the elements of F(2, 2n) by the elements of Pn, belong to the group F(2, 2n). Thus, a and r are automorphisms of F(2, 2n) and generate the group Ds of isometrics of the manifold Mn. []

To simplify notation, henceforth we use the same symbols for automorphisms of groups and the symmetries of manifolds and orbifolds induced by these automorphisms.

Consider the natural extension of r,, by the above automorphisms which has the semidirect product structure:

fl(n) = r,,ADs ~ f(2,2n)A(p,a, r).

In the next section we demonstrate the importance of the group f/(n) for our considerations.

w 3. Maximali ty of the Group f / (n)

Let Norm(F(2, 2n)) be the normalizer of the Fibonacci group F(2, 2n) in the group Isom(tP) of all isometries of the LobachevskiY space It 3 and let Isom(Mn) be the isometry group of the Fibonacci manifold M,t.

We recall [9] that the isometry group of a three-dimensional hyperbolic manifold is isomorphic to the group of outer automorphisms of its fundamental group. Therefore,

Isom(M~) ~ Norm(F(2,2n))/F(2,2n).

It was shown in Section 2 that the symmetries of the figure-eight knot induce isometries of the Fibonacci manifolds. Hence,

Norm(F(2,2n)) D f~(n) ~ F(2,2n)~(p,a,r).

Theorem 3.1. If n > 6 a n d A is a discrete group of isometrics of H s such that Fn C A then A c n(n) .

12

PROOF. Recall that F , is the fundamental group of the orbifold O(n). Consider a fundamental set for this group. The singular set of O(n) is the figure-eight knot which can be described as the two-bridge 5/2-knot. By [24], we can take a fundamental polyhedron for F , to be the combinatorial polyhedron P , pictured in Fig. 2. It is shown in [13] that for n > 4 the polyhedron ~n can be constructed in the Lobachevski~ space H a as a nonconvex fundamental polyhedron for F, . We describe its geometric parameters below.

Denote the vertices of ~,~ by Qo, Q1,Po, P1,...,t)9, where Q0 and Q1 are the midpoints of the edges PoPs and P2PT, respectively. The polyhedron P , has four faces each of which is nonflat in general:

QoPoP1P2PsP4Ps, QoPoPgPsPTP Ps, Q1P2PsP PsP PT, QxP2P1PoPgPsPT,

and has the edges PoPs, P2PT, and PiPi+l, i = 0 , . . . ,9, with all subscripts reduced rood 10. / . - ~ S

Fig. 2. The polyhedron ~ . .

The group r , is generated by the isometries S and T. The isometry S is the rotation of order n about the edge PoPs. This isometry identifies the faces QoPoPIP2P3P4Ps and QoPoPgPsPTPsPs. The isometry T is the rotation of order n about the edge P2P7. This isometry identifies the faces Q1P2PsP4PsPsP'r and Q1P2PIPoPgPsPT. By the Poincar~ theorem [25], we then have

F. ~- (S,T ] S" = T", T=I[S,T] = [S,T]S),

where [S,T] = S-1T-1ST. It is easy to check that the vertices P0 , . . . , P9 of P,, are divided into two equivalence classes with respect to the action of F,:

{P0, P6 = S-I (p0) , P4 = T-1S-I(po), P= = ST-IS-I (po) , Ps = TST-1S-I(po)}

and {Ps, P1 = S(Ps), P9 = TS(Ps), P7 = S-ITS(Ps) , 1'3 = T-1S-1TS(Ps)}.

Therefore, for each of the points P0,. .- , /)9 there is an elliptic element of order n in F, whose axis passes through this point.

We now describe the geometric parameters of the polyhedron ~n. Draw the additional edges QoPi and Q1Pi for i = 0 , . . . , 9. Then T~n can be represented as the union of the ten tetrahedra "lq = QoQ1PiPi+I, i = 0 , . . . ,9. It was shown in [13] that these tetrahedra can be divided into two classes that consist of pairwise isometric tetrahedra, {T1, T2, T~, Ts, ~ , 7~, ~ , To} and {~ , ~ } , which are pictured in Fig. 3.

13

Xn

Z~

Xn

Fig. 3

Z/t

Xn

Xn

By the formulae of non-Euclidean trigonometry, the lengths of edges and the dihedral angles of the tetrahedra in Fig. 3 can be determined from the following system of equations [13]:

sinh h. sin(Tn/2) = sinh(z,,/2), sinh ha sin 5n = sinh zn sin(It/n), cosh hn cosh(zn/2) - cosh zn, coth yn tanh(zn/2) = cot(It/n) tan 6~, cosh z,, = cosh z,, cosh y~, -/,, + 46,, = ~r.

Introducing the parameter

o . : 1,

we rewrite the solution of the system in the following form:

1 , / 2~ r = cosh h,, = cos( / n ),

a n - i ' V a n - 1 c o s h Ztt -~

0{. n - - 1

1 cosh y~ = a~ cosbr/n), cos 26~ = ~ = ~ - 46~.

~ n - 1'

As we see f rom Fig. 3, the faces of the tetrahedra are divided into two classes which consist of pairwise isometric triangles. With respect to this division, we define two sets of triangles which form the surface of the polyhedron 7~,,:

Wu 1 = { QoP1P2, QoP2P3, QoP3P4, QoPePT, QoP~'Ps, QoPsPg, O,1P3 P4 , Q x P4 Ps , Q x Ps Pe , Q1Ps P9 , QsPgPo, Q1PoP1},

with edge lengths zn, z,,, and zn, and

~"~ = {QoPoP1, QoPgPo, QoP4Ps, QoPsPe, Q1PIP2, Q1PuP3, Q1P6PT, Q1P, tPs},

with edge lengths z,,, yu, and z,,. Consider the extension of r,, by an isometry h E Isom(H s) such that the group (r~, h) is discrete.

Recall that 5' E r,, is an elliptic element of order n and denote the axis of 5' by 1. Since ~ is a fundamental polyhedron for rn, without loss of generality we can assume that h(1) t3 7~n # r In this case we have two possibilities: either h(l) passes through a vertex of some triangle of the set W~ U W~ or h(l) intersects some triangle of W~ U W~ in an interior point. We show that the second possibility cannot occur.

Indeed, h(l) is the axis of an element of order n which is conjugate with 5'. On the other hand, as was pointed out above, each of the vertices of the polyhedra ~ , . . . , ~ and so each of the vertices of the triangles in W~ U W~ belong to the axis of an element of order n. F. Gehring and G. Martin in [14] estimated the distances between the axes of elliptic elements in a discrete group. Namely, the following estimate holds for the distance p6 between two axes of dements of order 6:

sinh p6 ~ x/~.

14

The estimate for the distance p, between the two axes of elements of order n )__ 7 is given by

sinhp. > v /2c~ 2 sin 2 Ir/n

Propos i t ion 3.1. If n _> 6 and T / s a triangle in PY~ then for every point of T there exists a vertex of T such that the distance between the point and this vertex is less then p,.

PROOF. Rewrite the Gehring-Martin estimates as follows:

i coshpe > ce -- 2, coshp. _> c. -- zsm{, / )~:~'-2"lr'n" -- 1

for n > 7. Whence we infer the asymptotic estimate

n 2 C1~ ~ . . . .

2z.2 '

Consider a triangle T = A B C in }d) 1 with edges B C = z . , A B = z , , and AC = z, . This triangle is pictured in Fig. 4. Denote by R~ the radius of the circumscribed circle with center O. Then OA = OB = R~. Drop the perpendiculars OD and ON from O to the edges A B and BC. Denote by 7~. the value of the angle DAO.

A

B E z . C

C

O Xn

A ' B

Fig. 4 Fig. 5

With the above notations, from the right-angled triangle A B E we find

sin ~. sinh z . = sinh(z./2),

and from the fight-angled triangle AOD we see that

= t = h

Hence, cosh z. - 1

tanh R 1

Using the above expressions for cosh z. and cosh z. in terms of an, after obvious simplifications we obtain

+ i cosh R 1 ~ / a , 1 - v / 3 a . + 4an coSC~r/n ) A- 2"

This formula easily implies the following asymptotic estimate:

cosh " V~; ' n ~ oo.

Thus, c~ has quadratic growth in n and cosh R 1 has linear growth in n. Computing the values of the functions c. and cosh R~ for small n and comparing their ~raphs, we deduce the following inequality for all n ~ 6: cosh R~ < c~. The values of c~ and cosh R~ for n = 6 , . . . , 16 are presented in Table 2.

15

TABLE 2

n c~ cosh P~ cosh/~

6 2 1.4812... 1.4362...

7 1.6559... 1.6;]23... 1.5782...

8 2.4142... 1.7867... 1.7245... 9 3.2743... 1.9447... 1.8748... 10 4.2360... 2.1060... 2.0287... 11 5.2993... 2.2700... 2.1855... 12 6.4641... 2.4365... 2, 3449... 13 7.7302... 2.6050... 2.5063... 14 9.0978... 2.7753... 2.6696... 15 10.5667... 2.9471... 2.8343... 16 12.1370... 3.1201... 3.0003...

This completes the proof of Proposition 3.1. [::]

Propos i t ion 3.2. I f n > 6 and T is a triangle in 14/~ then for every point of T there exists a vertex of T such that the distance between the point and this vertex is less then pn.

PROOF. To find the radius of the circumscribed circle of T, we use the following property.

L e m m a 3.1. Let A B C be a triangle in H 2 with respective angles a, fl, and 7 at the vertices A, B, and C. Assume that the edge subtending the angle fl has the length b. Then the following formula holds for the radius R of the circumscribed circle

cosh b - 1 tanh R =

sinh b cos(a-~+')"

PROOF. Denote the center of the circumscribed circle by O and consider the triangles ABO, BCO, and CAO. Introduce the following notations: ~ - Z O A C - ZOCA, ~ - Z O B C = ZOCB, and (r__ Z O A B -- Z O B A . Suppose that O lies outside the triangle A B C (see Fig. 5). In this case the following relations hold:

From them we can express ~, 7/, and ( through a,/3, and 7- By the cosine theorem for hyperbolic triangles [25] we have

cosh R - cosh R cosh b - sinh R sinh b cos ~;

hence, cosh b - 1

ta hR = b os '

where ~ = ( - a + 13 7)/2. A similar result holds in the case when the center O of the circumscribed circle is inside A B C . Thus, the lemma is proven. D

We now return to the proof of Proposition 3.2. Consider T - A B C in Wn 2 with edges A B = yn, AC = Zn, and B C = Xn. It is pictured in Fig. 5. Denote the radius of the circumscribed circle by R~. By Lemma 3.1,

t nh = . . . . . .

sinh zn Cos(ZA+~U-r/2)"

16

Using the formulae of hyperbolic trigonometry [25] and the above expressions for zn, yn, and z , through the parameter a . , we obtain

cosh = cos( /n) § 1 ~ / ~ cos(~r/n) + 3a . - 2 - 2~,~ cos(~r/n)"

It is easy to see from this that the following asymptotic formula holds:

n c o s h ~ . o o .

Thus, the function c~ has quadratic in n and the function cosh R~ has linear growth in n. cosh p~growth

Computing the values of cn and for small n and comparing their graphs, we deduce that the following inequality holds for all n _> 6: cosh ]?~ < c~. The values of c~ and cosh R~ for n = 6,..., 16 are presented in Table 2. Proposition 3.2 is thus proven. O

Since for each vertex of an arbitrary triangle in pp1 U PV~ there is an elliptic element of order n whose axis passes through this vertex; in virtue of Propositions 3.1 and 3.2, the axis h(1) cannot pass through interior points of these triangles.

We now consider the case when h(1) passes through a vertex of the polyhedron Pn. As was mentioned above, the vertices P0,..., Ps are divided into two classes with respect to the action of the group rn: those equivalent to P0 and those equivalent to P1- Considering h = hh l, where h I G rn, we therefore derive that h(Q0) coincides with one of the following four vertices: Q0, Q1, P0, and PI. Observe that there exists an isometry of H 3 which is induced by an automorphism of the group r.A(r) such that the image of Q0 is Px and the image of QI is P0. Indeed, according to the action of the automorphism r on Fn we have

r(S) = S- ITS , r(T) = T S T -1.

Denote by g the automorphism that results from conjugating r by the element T S -1. Then

g(S) = ST-X(S -1TS)TS -1 = T -1S -1T = (T-1ST) -1,

g(T) = ST -X(TST-1 )TS -1 = S.

Therefore, g induces an isometry of H 3 such that the image of the axis of S is the axis of the element T-1ST , and the image of the axis of T is the axis of S. Observe [13] that the segment [Q0, Qx] is a common perpendicular to the axis of S (at Q0) and the axis of T (at Q1). The segment [P0, P1] is a common perpendicular to the axis of T-1ST (at P]) and the axis of S (at P0).. Thus, g(Qo) = P1 and g(Q1) = P0. Considering h = hh', h" E r.A(r), we therefore conclude that h(Qo) coincides with Q0 or Q1. Recall that the group r . has an automorphism a such that

a(S) = T, a(T) = S.

This automorphism induces an isometry of H s which carries the axis of S to the axis of T and vice versa. Since the segment [Q0, Q1] is a common perpendicular to these axes, we obtain a(Qo) = Q1 and a(Q1) = Qo. Considering h = hh m, h n' G rnA(a, r) = n(n), we then obtain h(Q0) = Q0. Thus, the axes l and h(l) of order n > 6 have the common point Q0 and so coincide by [15]. Under the action of h the axis of S goes t o itself. Multiplying h by an appropriate power of S, we may assume that h(Q1) E :Pn. Therefore, h(Q1) = Q1 and under the action of the isometry h the axis of T goes to itself. Thus, the isometry h has two invariant skew lines and their common perpendicular [Q0, Q1] is fixed. Multiplying h, if necessary, by a symmetry from D8, we infer that h = id; hence, h E f~(n). The theorem is proven. C]

We recall that a discrete group is said to be a maximally discrete group if it is not a proper subgroup of another discrete group.

17

Corol la ry 3.1. For n > 6 the group fl(n) is a maximally discrete group.

PROOF. Indeed, assume that A is a discrete group such that fl(n) C A. Then F~ C fl(n) C A. On the other hand, by Theorem 3.1 we have A C fl(n). Thus, fl(n) is a maximally discrete group. []

REMARK. For n = 4, 5 the groups fl(n) are maximally discrete groups as well [8]. This fact follows from their arithmeticity and the comparison of hyperbolic volumes. Observe tha t the Fibonacci group F(2,10) is commensurable with the Coxeter group with the following Coxeter scheme:

0 0 (1 0

This Coxeter group is generated by reflections in the faces of a compact hyperbolic tetrahedron.

T h e o r e m 3.2. The isometry group Isom(M,) of the hyperbolic Fibonacci manifold Mn, n > 4, consists of 8n elements and has the following presentation

Isom(Mn) = (x,y I z2'z = y4 = (yz)2 = (y-ix)2 = 1).

PROOF. In the cases n = 4,5,6, 8, 12 when the Fibonacci group is arithmetic, the claim was proven by C. Maclachlan and A. Reid [8]. We prove the claim for all n > 6. We recall that

Isom(M.) ~ Norm(F(2,2n))/F(2,2n),

where Norm(F(2, 2n)) is the normalizer of F(2, 2n) in the isometry group of H 3. On the one hand, by Theorem 2.1 we have

Norm(F(2,2n)) D fl(n).

On the other hand, the group Norm(F(2, 2n)) is discrete as a finite index extension of I ' , . Therefore, by Corollary 3.1 we have

Sorm(F(2,2n)) C ~(n).

Thus,

Norm(F(2,2n)) = n(n) ~ F(2, 2n))~(Z,Q~)s),

and therefore

(F(2 )) /F(2 ) " ADs Norm ,2n ,2n = Z~ .

The claimed 2-generator presentation of the group follows from [11]. []

w 4. Quotient Orbifolds

In this section we consider some quotient orbifolds resulting from the action of isometrics on the hyperbolic Fibonacci manifolds. As was indicated in the previous section, the quotient space of the manifold M,~ by the action of the isometry p is the orbifold O(n) whose underlying space is the three- dimensional sphere and whose singular set is the figure-eight knot with the singularity index n. Thus, O(n) = M./(p).

Consider the involution ~.2 E Ds. The quotient space of the orbifold O(n) by the action of this involution was discussed, for example, in [7, 26].

P ropos i t ion 4.1. The quotient space M./(p, r 2) of the hyperbolic Fibonacci manlfold Mn by the action of the group of isometrics generated by p and 7 -2 is the orbifold whose underlying space is the three-dimensionM sphere and whose singular set is the two-component link 6 5 pictured in Fig. 6 with singularity indices 2 and n on its components.

18

Fig. 6. The singular set of the orbifold M./(p, r2).

PROOF. The claim follows from the fact that M./(p, r 2) ~' O( , ) / ( r2) . C] Consider the involution a E I1% inducing the symmetry of the figure-eight knot with the axis

pictured by the dotted line in Fig. 1.

Propos i t ion 4.2. The quotient space M./(p, a) of the hyperbolic Fibonacci manifold M. by the action of the group of isometrics generated by p and a is the orbifold whose underlying space is the three-dimensional sphere and whose singular set is the spatial graph pictured in Fig. 7.

PROOF. The claim follows from the fact that M./(p, a) "~ O(n)/(a). yl *Ct

f

I

I I

I

I '(__ ' !

' I !

.- p O"

A" , "B !

Fig. 7. The singular set of the orbifold M./(p, a).

Considering the symmetry tr of Fig. 1, we obtain the graph that is pictured on the left of Fig. ? and has two vertices and three edges connecting these vertices. Therefore, this graph is the e-graph according to the terminology of spatial graph theory [27]. This graph can also be described as the toms knot 51 with a bridge AB (see the right side of Fig. 7). The points A and B are the images of the intersection points of the singular set of the orbifold O(n) with the axis of the involution a. Observe that the arc AB is the image of the tunnel of the figure-eight knot in the sense of [28]. In this case the two arcs of the singular set of the orbifold M,/{p, a), which are images of the axis of the involution a, have singularity indices 2. The third arc, which is the image of the singular set of the orbifold O(n), has the singularity index n. The group of the orbifold M./(p, a) has the presentation

A(n) = F(2, 2n)A{p, a) ~- (p, b, a I P" = b" = a 2 = 1, p-l[b, p] = [b, p]b, trpa = b),

where the generators p, b, and a correspond to Fig. 7. Indeed, the relation p-~[b, p] - [b, p]b follows from the fact that the loop around the bridge AB is an element of order two. The relation apa = b follows by the Wirtinger algorithm [21].

It is easy to see from Fig. 7 that the spatial e-graph has an involution c~ whose axis is pictured by the dotted line. The involution a induces a symmetry of the orbifold M,/(p, a) From Fig. 7 we

19

see that . (b) = b -~, a(p) - bp-lb -1.

Thus, the involution a is conjugate to the involution a r 2 (see Table 1). Thus, the quotient orbifolds Mn/(p, a, a) and Mr,/(p, a, r z) axe isometric. [3

Propos i t ion 4.3. The quotient space Mn/(p, a, r 2) of the hyperbolic Fibonacci manifold Mn by the action of the group of isometrics generated by p, a, and r 2 is the orbifold whose underlying space is the three-dimensionM sphere and whose singular set is pictured in Fig. 8.

f d �9 v --

k j w E

j

Fig. 8. The singular set of the orbifold Mn/(p, a, r2).

PROOF. The claim follows from the fact that M,,/(p, a, r 2) ~ (o(~)/(~))/(~2). o The graph pictured in Fig. 8 has four vertices and six edges. As an abstract graph it is isomorphic

to the complete graph on four vertices. On the other hand, this graph can be described as the figure- eight knot with two arcs CD and EF which axe his tunnels. The vertices C, E, and F are the images of the intersection points of the singular set of the orbifold O(n)/(a) with the axis of the involution a. The vertex D is the image of the point A in Fig. 7. One edge of the graph has singularity index n and the other edges have singularity index 2. The group of the orbifold Mn/(p, a, r z) has the following presentation:

II(n) - F(2, 2n)A(p, a, r 2)

(p, b, ~, ~ I f = b" = ~2 = ~2 = t, p-~[b, p] = [b, p]b, apa = b, ct-lba = b -1, ct-lpot = bp-lb-1),

where the elements p, b, a, and a correspond to Fig. 8. Another presentation of the group H(n) can be obtained by the Wirtinger algorithm:

( ~ p ~ p - ~ ) 2 = ( p - ~ ) 2 ( ~ p - ~ - ~ ) 2 = t).

As we can see from Fig. 8, the singular set of the orbifold M./(p, a, r 2) has an antipodal involution X such that x(C) = F, x(D) = E, x(E) = D, and x(F) = C. Theorem 3.2 readily yields (p, a, r 2, X) ~ Isom(M,~).

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Fig. 9. The quotient orbifo!d/14~ / Isom(Mn).

Since the underlying space of the orbifold Mn/(p, a, r 2) is the three-dimensional sphere, the under- lying space of the quotient orbifold M,~/Isom(Mn) is the pseudo-manifold p3 which can be represented as the union of two cones whose common base is the projective plane. This pseudo-manifold and the singular set of the orbifold are schematically pictured in Fig. 9, where one singularity index equals n and the others equal 2.

Proposition 4.4. The quotient space M./ Isom(M.) of the hyperbolic Fibonacci manifold M. by the action of its isometry group Isom(M,) is the orbifold whose underlying space is the pseudo- manifold p3 and whose singular set in pictured in Fig. 9.

The authors are grateful to A. D. Mednykh for helpful discussion of the results and interest in the article.

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