NEW ZEALAND JOURNAL OF MATHEMATICS Volume 29 (2000), 203-209

ARITHMETIC 2-GENERATOR KLEINIAN GROUPS WITH QUADRATIC INVARIANT TRACE FIELD

C. M a c l a c h l a n a n d G.J. M a r t i n a n d J. M c K e n z i e(Received May 1999)

Abstract. We classify and describe all cocompact arithmetic Kleinian groups that are generated by two elements of finite order, and have quadratic invariant trace field. There are 26: 23 of these are Dehn fillings of knot or link complements and the others are generalised triangle groups.

1. Introduction

A generalised triangle group is a two-generator group with a presentation of the form

(a, b\ ap = bq — R(a, b)r — 1) where R(a, b) is a cyclically reduced word in the free product on a, b which involves both a and b and p, q, r are all integers which are at least 2. When R(a, b) = ab we have the classical triangle groups which are spherical, euclidean or hyperbolic when the sum 1 /p + 1 /q + 1/r is greater than 1, equal to 1 or less than 1 respectively.

In this paper those arithmetic generalised triangle groups T which are faithfully represented as discrete subgroups of PSL(2,C) with H3/ r compact and with quadratic invariant trace field (defined below) are obtained. There are just three such groups. More generally we determine those 2-generator Kleinian groups generated by two elements of finite order which are cocompact with the above restriction on the trace field. There are in fact 26 such Kleinian groups. Many, but not all, of these groups are Dehn fillings of 2-bridge knot and link complements as one might expect. However the three generalised triangle groups have orbit space S3 with a singular graph with 3 edges and two vertices. Also, not all the link complements are 2-bridge. There are surprisingly few number fields arising, just Q (V —d) for d = 1,2,3,6, 7. Our results here raise the following question, seemingly related to the meridean conjecture, Problem XXX in Kirby’s problem list [5].Question. Is it true that “generically” a cocompact Kleinian group generated by two elements of finite order p and q has orbit space given by (p, 0), (q, 0) Dehn filling either a 2-component link complement, a 2-bridge knot complement (p = q), or a Z 2-extension of a Dehn filling of a 2-bridge knot complement (q = 2).

We are unaware of a counterexample. In fact if p > q > 5, it might be possible that the orbit space is simply a Dehn filling of a 2-bridge knot or link.

If T is a cocompact Kleinian group generated by elements of finite order with imaginary quadratic trace field and if the traces of all elements are integral, then

1991 A M S Mathematics Subject Classification: Primary 30F40, 20H10.Research supported by grants from the N.Z. Marsden Fund (G JM +JM ) and an U .K ., E.P.S.R.C. Visiting Research Fellowship (GJM).

204 C. MACLACHLAN AND G.J. MARTIN AND J. MCKENZIE

p q 7 geometric description slope covolume3 3 - 3 - i (3,0),(3,0) Dehn fill 6^ 6(12,5) 1.22124 4 ( - 3 + V3i)/2 (4,0) Dehn fill 4i 6(5,2) 0.50744 4 (1 + 3V3i)/2 (4,0) Dehn fill 74 6(15,4) 3.04484 4 ( - 5 + s/7i)/2 (4,0),(4,0) Dehn fill 62 6(12,5) 2.66676 6 - 1 + * (6,0) Dehn fill 4i 6(5,2) 1.22126 6 — 2 -f- y/2i (6,0),(6,0) Dehn fill 6^ 6(12,5) 4.01536 6 1 + 3i (6,0),(6,0) Dehn fill 84 6(24,7) 6.10646 6 - 1 + V7i (6,0),(6,0) Dehn fill 8? 6(30,11) 7.11133 4 —2 + \/2 i (3,0),(4,0) Dehn fill 5f 6(8,3) 1.00384 6 V/6i (4,0),(6,0) Dehn fill 6j> 6(10,3) 3.12402 3 —2 -}- i (2,0),(3,0) Dehn fill 8? 6(24,7) 0.61062 3 — 1 — i (2,0),(3,0) Dehn fill 9 6(24,5) 0.61062 4 ( - 3 + V7i)/2 (2,0),(4,0) Dehn fill 8i 6(24,7) 1.33332 4 ( - l + V7i)/2 (2,0),(4,0) Dehn fill 91 6(24,5) 1.33332 4 ( - 3 - y/3i)/2 (2,0),(4,0) Dehn fill 6S 6(10,3) 0.25372 4 ( - 1 + V3i)/2 (a2 = b4 = ((a6_ 1 )^(a6)3) = 1) 0.25372 4 ( - 5 + Vr3*)/2 (2,0),(4,0) Dehn fill 8^ 6(30,11) 1.52242 4 (1 - V3i)/2 (a2 = bA = ( ( ( b - 1a)A(ba)A) 2ab~i y = 1) 1.52242 6 - 1 + s/2 i (2,0),(6,0) Dehn fill 8? 6(24,7) 2.00762 6 y/2i (2,0),(6,0) Dehn fill 9i 6(24,5) 2.00762 6 — 1 — i (2,0),(6,0) Dehn fill 62 6(10,3) 0.61062 6 i {a2 = b ° = ((a6- 1)*(a&):T = 1} 0.61062 6 - 2 - i (2,0),(6,0) Dehn fill 6(17,48) 6(48,17) 3.05322 6 1 + t (2,0),(6,0) Dehn fill 7j 3.05322 6 (1 + V7i)/2 (2,0),(6,0) Dehn fill 3.55562 6 ( - 3 - V7i)/2 (2,0),(6,0) Dehn fill 6(60,19) 3.5556

T a b l e 1. Arithmetic cocompact groups with quadratic invariant trace field

T is arithmetic. Hence this work fits into our wider program to determine all two generator arithmetic Kleinian groups.

When the two generators are either elliptic or parabolic, it has been shown that there are only finitely many arithmetic groups up to conjugacy. If the orbit space is noncompact, then the papers [1, 2, 6, 7] identify all of these groups. Since arithmeticity and noncompactness imply that the trace field is quadratic, our work here completes the identification of the 2-generator arithmetic Kleinian groups with quadratic invariant trace field. Much of what we do here follows from our paper[7]-

Note that for each k > 1 the generalised triangle groups with presentation (a, b | a3 '= b3 — ((ab)k(a~1b~1)k) ) all have a discrete and faithful cocompact representation in PSL(2,C). See [4] where a description of the underlying orb- ifolds are also given. Bounding the degree of the invariant trace field provides a mechanism for limiting the number of examples. The first case to consider is then the complex quadratic case.

The first table summarises our results. There are a couple of things to note on this table. First, as noted above, only three of the groups are generalised triangle groups. It is easy to see that a group obtained by Dehn filling a 2-bridge link complement is not a generalised triangle group since there are at most two conjugacy classes of torsion. Second, as we discuss a little later, the groups generated by elliptics of orders 2 and p come in pairs of equal covolume. This is because each

ARITHMETIC 2-GENERATOR KLEINIAN GROUPS 205

is a Z2-extension of a corresponding group generated by two elliptics of order p. These extensions are not in general distinct, but that turns out to be the case here as the orbit spaces are easy to distinguish. Thirdly, once p and q, the orders of the generators, are given there is a single complex parameter necessary to distinguish the group uniquely up to conjuagcy. This is the number 7 listed in thetable. The2-bridge knot and link complements given in the ta bles are from Rolfsen [11]. We also give Schubert’s normal form (or the slope) of the 2-bridge knots and links. Two entries in the table are 2-bridge links with at least 10 crossings and are not in Rolfsen’s tables. However the links are not too difficult to obtain as they come from the two-fold symmetries of 8 4 and 8 7 which interchange the components of the links. In fact the addition of these symmetries give the groups in question. Finally, for the links 7\ and 7§ (which are not 2-bridge), only one of the two possible fillings gives a hyperbolic orbit space.

The following figures illustrate the geometric relationships between the singular sets of orbifolds listed and their Z 2-quotients whose fundamental groups are generalised triangle groups.

None of the generalised triangle groups in the table are of the form given in [4]. However, there is one symmetry of 4i, the figure-8 knot, that gives a singular set of the form they use (Figure 1). The bold curve in the singular set of the quotient corresponds to the knot’s axis of symmetry.

Figure 2 describes the symmetry of the figure-8 knot corresponding to the first and third of the generalised triangle groups in the table. Figure 3 shows the symmetry of 74 that gives the second one.

2. Kleinian Groups

A Kleinian group G is a discrete subgroup of PSL(2, C), the group of all orientation- preserving isometries of H3. All groups considered will be non-elementary, i.e. not virtually abelian. The quotient space H3/G is a hyperbolic orbifold or, if G is

F i g u r e 1 . A singular graph of the type in [4]

206 C. MACLACHLAN AND G.J. MARTIN AND J. MCKENZIE

F i g u r e 2. The singular graph for the 1st and 3rd generalised triangle groups.

F i g u r e 3 . The singular graph for the 2nd generalised triangle group.

torsion-free, a hyperbolic manifold. The linear fractional action of a matrix

* = ( c d ) e S i (2- C >

on C extends via the Poincare extension to give an isometry / of the upper-half space model of H3, where

ARITHMETIC 2-GENERATOR KLEINIAN GROUPS 207

3. Two Generator Groups

Suppose G is generated by two primitive elliptic elements / , g of orders p, q where p < q and that G is a cocompact Kleinian group. Then

tr2( / ) — 4 = —4 sin2(7r/p), tr2( 3, since for the groups with p = 2 the subgroup (g ,fg f ) is of index 2 in (f,g ). Conversely, each group generated by a pair of elements gi,g2 of the same order can be extended by elements of order 2 which conjugate g\ to g f 1. In these cases, kG = Q ( t r /V ) (e.g. [2]).

We now normalise our choice of matrix representatives for the elements / , g. Thus (f,g ) is conjugate in PSL(2,C ) to a subgroup generated by the images of

cos7x/p isimr/p \ Y — ( COS7T/(l it sin n/q isimr/p cos7r/p J ’ y it-1 simr/q cosn/q

Here t is a complex parameter and by further conjugation, if necessary, we can assume that \t\ < 1.

Thus for fixed p , q the space of all such discrete groups is determined by the single complex parameter

l i f t 9) = tr[f, g \ - 2 = (t - l/ t f sin2(7r fp) sin2(7r/g) (2)

and it is this value which we seek (see Table 1). We have chosen 7 here rather than t for a number of reasons, the most obvious of which is that 7 ( / , g) G kG.

If G is to be compact it must have Euler characteristic zero and thus cannot be a free product of cyclic groups. However, if the isometric circles of g lie inside the intersection of the isometric circles for / , then the Klein combination theorem shows that ( / , g) is a free product. Thus, with the normalisation above, if the inequalities

| sin(7r/g) cos(7r/p) ± t cos(7r/g) sin(7r/_p) | + |i|sin(7r/p) < sin(7r/g) (3)

hold for both choices of sign, then the group G = (f) * (g). This bounds the region in the complex plane in which t and hence 7 ( / , g) may lie and proves, when combined with the bounds on p and q obtained above, that there are only finitely many candidates. We now search through this region in the complex plane for all the quadratic integers giving a (long) list of all possible values for 7 (f,g ).

4. Candidates

Refinements of the above argument concerning isometric circles can be made by cutting and pasting together fundamental domains for / and g in such a way so as to achieve a combinatorial structure that implies the group splits as a free product via the combination theorems as in [7].

In practise, to further refine the possibilities for 7 (f,g ), we have implemented a version of the Dirichlet subroutine in Jeff Weeks’ program SNAPPEA, modified by by the third author, to decide if the group acted discontinuously on the sphere at infinity. Our input was the following tables extracted from [7]. For the noncompact groups the invariant quaternion algebra must be the 2 x 2 matrices over kG. This is an arithmetic criterion for 7 ( / , g) and these cases have been eliminated from the tables of [7] since they are of no interest to us here. The following tables then

(1)

208 C. MACLACHLAN AND G.J. MARTIN AND J. MCKENZIE

contain all remaining values of 7 which are either not obviously free or are not noncompact, together with the orders of the generators.

An advantage in using this numerical approach was that we were able to estimate the covolume of the orbit space for the groups which did not act discontinuously on the sphere. This greatly aided our search for a geometric description of the orbit spaces as presented in Table 1. This search was conducted by looking at all the various Dehn fillings of the 2-bridge knots and links up to nine crossings using SNAPPEA. Given the underlying orbifold and a presentation of the fundamental group, it is routine to check that one has the correct group (that is there are no numerical errors).

P q 7 Eliminatedp q 7 Eliminated 6 6 - 1 + i3 3 - 1 + 3 i discontinuous 6 6 5 + 5 i discontinuous3 3 - 3 - i3 3 - 4 + V2i discontinuous

b b 3v2 i discontinuous

4 4 1 + 4\/2 i discontinuous b b - 2 + V2 ifi fi - 2 discontinuous4 4 - 2 + 2s/2i discontinuous fi fi - 1 + V7i4 4 (1 + 3\/3i)/2 fi fi 2 + 2\/7 i discontinuous4 4 ( - 3 + v/3’i)/2 fi fi - 3 discontinuous4 4 —3 + 2y/3i discontinuous 3 4 — 2 + \/2i4 4 ( -1 + 3 ^ 7 0 /2 discontinuous 3 4 2 + 3\/2 i discontinuous4 4 ( - 5 + V7i)/2 3 6 — 2 -f- \/6i discontinuous4 4 ( - 3 + 3vT li)/2 discontinuous 3 6 1 + v/l5i discontinuous4 4 ( - 7 W l l i ) / 2 discontinuous 4 6 \/6 i4 4 ( - 9 + v /l5i)/2 discontinuous 4 6 —3 + 2\/6 i discontinuous

4 6 3 + 2VT5 i discontinuous

References

1. M.D.E. Conder, C. Maclachlan, G.J. Martin and E. O ’Brien, Two-generator arithmetic Kleinian groups III, to appear.

2. F.W. Gehring, C. Maclachlan and G.J. Martin, Two-generator arithmetic Kleinian groups 77, Bull. London Math. Soc. 30 (1998), 258 - 266.

3. F.W. Gehring, C. Maclachlan, G.J. Martin and A.W. Reid, Arithmeticity, discreteness and volume, Trans. Amer. Math. Soc. 349 (1997), 3611 - 3643.

4. K.N. Jones and A.W. Reid, Minimal index torsion-free subgroups of Kleinian groups, Math. Ann. 310 (1998), 235-250.

5. R.C. Kirby, Problems in Low-Dimensional Topology, Berkley, 1995.6 . C. Maclachlan and G.J. Martin, Two-generator arithmetic Kleinian groups, to

appear.7. C. Maclachlan and G.J. Martin, The noncompact arithmetic generalised triangle

groups, to appear.8 . C. Maclachlan and A.W. Reid, Commensurability classes of arithmetic Kleinian

groups and their Fuchsian subgroups, Math. Proc. Camb. Phil. Soc. 102 (1987), 251-257.

9. B. Maskit, Kleinian Groups, Springer-Verlag, 1988.

ARITHMETIC 2-GENERATOR KLEINIAN GROUPS 209

10. A.W. Reid, A note on trace fields of Kleinian groups, Bull. London Math. Soc. 22 (1990), 349-352.

11. D. Rolfsen, Knots and Links, Publish or Perish, 1976.

Gaven Martin The University of Auckland Mathematics Department Private Bag 92019 Auckland N E W ZEALAND mart in@mat h. auckland. ac. nz

John W . McKenzieThe University of AucklandMathematics DepartmentPrivate Bag 92019AucklandN E W ZEALAN D

jmck@math. auckland. ac.nz

Colin MacLachlanDepartment of Mathematical SciencesMeston BuildingUniversity of AberdeenAberdeen AB24 3UESCO TLAN DUN ITED KIN G D O M

c.maclachlan@maths.abdn. ac.nz