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DISCRETENESS OF SUBGROUPS OF SL ( 2 , C ) CONTAINING ELLIPTIC ELEMENTS

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DISCRETENESS OF SUBGROUPS OF SL ( 2 , C ) CONTAINING ELLIPTIC ELEMENTS

PEKKA TUKIA and XIANTAO WANG

Abstract

The following result is the main result of the paper. LetGSL(2,C)be non-elementary. IfG contains an elliptic element of order at least 3, thenGis discrete if and only if each non-elementary subgroup generated by two elliptic elements ofGis discrete.

1. Introduction

One of the consequences of Jørgensen’s inequality [7] is that a non-elementary subgroup of SL(2,C)is discrete if every two-generator subgroup is discrete (Jørgensen [7], [8]); if G ⊂ SL(2,R), the discreteness follows as soon as every cyclic subgroup is discrete (Jørgensen [8]). These results were extended by Wang and Yang who showed thatGis discrete if every subgroup generated by two loxodromic elements is discrete [11]; ifGcontains parabolic elements, thenGis discrete as soon as every subgroup generated by two parabolic ele- ments is discrete [12]. The main result of this paper is Theorem 3.1 showing that G is discrete as soon as every non-elementary subgroup generated by two elliptic elements is discrete; here we must assume thatGcontains an elliptic element of order at least 3. We will also show (Theorem 4.1) that if every subgroup generated by an elliptic and a loxodromic element is discrete, then Gis discrete, provided that there are elliptic elements of order at least 3. See the references [1], [2], [3], [5], [6], [9], [10] for further discussions of these theorems.

Finally, we complement these results for groups containing parabolic ele- ments. We will prove that ifGis non-elementary and contains parabolic ele- ments, thenGis discrete if every non-elementary subgroup generated by a parabolic and a loxodromic element is discrete (Theorem 5.1). The missing result would be that if non-elementaryGcontains parabolic and elliptic ele- ments, thenGis discrete if every subgroup generated by a parabolic and an elliptic element is discrete but this question is left open.

Received March 15, 2000.

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The reason for the assumption that there are elliptic elements of order at least 3 is that two elliptic elementsf and g of order 2 always generate an elementary groupG. This follows for algebraic reasons since the cyclic group generated byfgis of index 1 or 2 inG.

Acknowledgements. This work was completed during the second au- thor’s visit to the University of Helsinki. He wishes to express his thanks to the Department of Mathematics of the University of Helsinki for hospitality.

Research of the second author was partly supported by FNS of China, grant number 19801011.

2. Notations and preliminary results

We denote byH3= {(x, y, z)∈ R3: z >0}the hyperbolic 3-space and the hyperbolic space with boundary isH¯3=H3∪C. Iff ∈SL(2,C), we regard f as a Möbius transformation ofCand denote byf˜the Poincaré extension of f toH¯3and write

fix(f )= {x∈C=C∪ {∞}:f (x)=x},

ord(f )=the order off whenf is regarded as a Möbius transformation, Af = {z∈ ¯H3:f (z)˜ =z};

the notationAf is used only iff is elliptic andAf is the theaxisoff. The letterGalways denotes a subgroup of SL(2,C)unless otherwise stated.

The groupGis calledelementaryif there isz0∈ ¯H3such that the orbit Gz0= { ˜f (z0):fG}

is finite. In fact (cf. [1, p. 84]), ifGis elementary, then either there is a common fixed point inH3of the elements ofG(ifGis elliptic) or a one- or two-point orbit inCand this is the fixed point set of parabolic or loxodromic elements in the group. We callGisnon-elementaryifGis not elementary.

The following characterization of non-elementariness of a group generated by two elliptic elements is crucial for us. It follows easily from the facts that an elementary group has either a fixpoint inH¯3or there is a two-point orbit, cf. [1, p. 84]. We state this in

Lemma2.1.Two elliptic elementsf, g ∈ SL(2,C), whose orders are not both equal to2, generate a non-elementary group if and only if their Poincaré extensions toH¯3have no common fixed points.

The same conclusion is true if f is elliptic of order at least 3 andg is loxodromic.

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Lemma2.2.LetGbe non-elementary. IfgGis elliptic, then there are infinitely many elementsgi ofGsuch that eachgi is conjugate tog in G and no twog˜i’s have common fixpoints (i.e. their axes are disjoint).

Proof. SupposegGis elliptic. SinceGis non-elementary,Gcontains a loxodromic elementf (see [1, Theorem 5.1.3]) such that

fix(f )∩fix(g)=φ.

ThusAgis disjoint from fix(f )and hencefk(Af)tend toward the attracting fixpoint off ask→ ∞. It easily follows that there is a sequencek1< k2. . . of integers such that the sets

fki(Ag)=Afkigfki

are pairwise disjoint. This proves the lemma.

Lemmas 2.1 and 2.2 have the

Corollary2.3.LetGbe non-elementary. ThenGcontains a non-elemen- tary subgroup generated by two elliptic elements if and only ifGcontains an elliptic element of order at least 3.

3. A discreteness criterion of subgroups of SL(2,C) with elliptic elements

In this section, we will prove the following main result of this paper.

Theorem3.1.LetGbe non-elementary. IfGcontain an elliptic element of order at least3, thenGis discrete if and only if each non-elementary subgroup generated by two elliptic elements ofGis discrete.

Remark. Two elliptic elements of order 2 always generate an elementary discrete group, cf. the Introduction. This is the reason for the assumption that there are elliptic elements of order at least 3.

We start with

Lemma3.2.LetGbe non-elementary. IfGcontains elliptic elements and each non.elementary subgroup generated by two elliptic elements ofGis dis- crete, thenGcontains no purely elliptic sequence{fn}such thatfnI as n→ ∞. HereIis the identity element.

Proof. SupposeGcontains a purely elliptic sequence{fn}such thatfnIasn→ ∞. We can obtain by passing to a subsequence (denoted in the same manner) that fix(fn)tends in the Hausdorff metric toward a one- or two-point

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setY. SinceGis non-elementary, there is a loxodromic elementhG(cf.

[1, Theorem 5.1.3]) such that

fix(h)Y =φ.

Thus, since fix(fn)Y, there are a neighborhoodVof the attracting fixpoint ofhand a neighborhoodW of the repelling fixpoint ofhsuch that for largen

AfnV =φ and AfnW =φ.

The latter of these formulae implies that there arepandn0such thathp(Afn)Vifn > n0. We fix suchpandn0. Now,hp(Afn)is the axis ofhpfnh−p. Hence the axes offn andgn = hpfnh−pdo not intersect and hencefn, gnis non- elementary by Lemma 2.1. However, bothfnIandgnIand hence the commutator [fn, gn]→Iand so Jørgensen’s inequality

|tr2(fn)−4| + |tr[fn, gn]−2| ≥1 is violated for largen. This contradiction proves the lemma.

Proof of Theorem3.1. Suppose that the non-elementaryGcontains an elliptic element of order at least 3 and that every non-elementary subgroup generated by two elliptic elements ofGis discrete butGis not discrete. Then Gcontains a sequence{fn}such thatfnI asn → ∞and where each fn = I. We can assume that fix(fn)tends in the Hausdorff metric toward a one- or two-point setY. We can find an elliptic elementgGof order at least 3 whose fixpoint set is disjoint fromY, cf. Lemma 2.2. Thus we can assume that fix(fn), fix(g)and fix(fngfn1)are disjoint for largen.

Lethn = fngfn1and setGn = g, hn. IfGnis non-elementary, thenGn

is discrete by the assumptions of the theorem and hence Jørgensen’s inequality

|tr2(ghn1)−4| + |tr[ghn1, hn]−2| ≥1

is true. However,hngand hence the left hand side of the above inequality tends to 0. This is a contradiction and soGnis elementary for largen. This is possible only if the axes ofgandhnintersect (Lemma 2.1) and since they do not have common fixpoints inC, they must have a common fixpointpninH3. It follows thathn1g also has the fixpointpnH3and hence{hn1g}is a purely elliptic sequence tending toI and this is impossible by Lemma 3.2.

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4. The elliptic-loxodromic case

As an application of theorem 3.1, we will prove a theorem which is midway between our theorem and the theorem of Wang and Yang in [11] where it was shown that ifGis non-elementary and every non-elementary subgroup generated by two loxodromic elements is discrete, thenGis discrete.

Theorem4.1.LetGbe non-elementary. IfGcontains an elliptic element of order at least3, thenGis discrete if and only if each non-elementary subgroup f, gofGis discrete wheref is elliptic andgis loxodromic.

Theorem 4.1 follows from theorem 3.1 and the following lemma.

Lemma 4.2. Let G be non-elementary such that G contains an elliptic element of order at least 3. Suppose that every non-elementary subgroup ofG generated by one elliptic and one loxodromic element ofGis discrete. Then every non-elementary subgroup of G generated by two elliptic elements is discrete.

Proof. Let f1 and g1 be elliptic elements of Gsuch that H = f1, g1 is non-elementary. We show thatH is discrete. If f1g1 is loxodromic, then H = f1, f1g1is discrete by assumption. Iff1g1 is non-loxodromic, then the proof of case (3) of Lemma 3 of [11] shows thatH is conjugate to a non- elementary subgroup of SL(2,R), cf. also Lemma 5.23 of Gehring and Martin [4].

Thus we can assume that H ⊂ SL(2,R). However, a non-discrete and non-elementary subgroup of SL(2,R)contains a sequence{hn}of hyperbolic elements such thathnI asn→ ∞, cf. [3, Corollary p. 199]. Pass first to a subsequence so that the sets fix(hn)have the Hausdorff limitXwhich is a one- or two-point set. SinceH is non-elementary, at least one of the elliptic elementsf1org1is of order at least 3, say ord(f1)≥3. Again, like in the proof of Lemma 3.2, the non-elementariness ofH implies that we can conjugatef1 inH to obtainfH so that fix(f )is disjoint fromX. Thus we can assume that for largen

fix(f )∩fix(hn)=φ.

It follows (Lemma 2.1) thatHn = f, hnis a non-elementary subgroup of H for large n. HenceHn is discrete as a subgroup ofH. However, this is a contradiction since now Jørgensen’s inequality

|tr2(hn)−4| + |tr[hn, f]−2| ≥1 is violated for largensincehnI asn→ ∞.

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5. The parabolic-loxodromic case We complement our results and prove

Theorem5.1.LetGbe a non-elementary group containing parabolic ele- ments. ThenGis discrete if and only if every non-elementary subgroup gen- erated by a parabolic and a loxodromic element ofGis discrete.

Proof. Suppose thatGis not discrete although every subgroup generated by a parabolic and a loxodromic element is discrete. We derive a contradiction as follow. Thus there is an infinite sequencefi ofG,fi =I, such thatfiI as i → ∞. Pass to a subsequence so that fix(fi)have the Hausdorff limit Xwhich is a one- or two-point set. Since Gis non-elementary and contains parabolic elements, we can find a parabolic elementgwhose fixpoint is not a point ofX. Thus, for largei, the fixpoint sets ofgandfiare disjoint. Using this fact, a simple calculation shows that for largeithere isni such thathi =figni is loxodromic. Sincefi andgdo not have common fixpoints, neither havehi

andgand soHi = g, hi = g, fiis non-elementary and hence discrete by assumption. However, for largei,

|tr2(fi)−4| + |tr[fi, gj]−2|<1

and soHi would have to be elementary by Jørgensen’s inequality. This con- tradiction proves the theorem.

REFERENCES

1. Beardon, A. F.,The geometry of discrete groups, Springer-Verlag, 1983.

2. Beardon, A. F.,Some remarks on nondiscrete Möbius groups, Ann. Acad. Sci. Fenn Ser. A I Math. 21 (1996), 69–79.

3. Doyle, C. and James, D.,Discreteness criteria and higher order generations for subgroups ofSL(2, R), Illinois J. Math. 25 (1981), 191–200.

4. Gehring, F. W. and Martin, G. J.,Inequalities for Möbius transformations, J. Reine Angew.

Math. 418 (1991), 31–76.

5. Gilman, J.,Inequalities in discrete subgroups ofPSL(2, R), Canad. J. Math. 40 (1988), 114–

130.

6. Isachenko, N. A.,Systems of generators of subgroups ofPSL(2, C), Siberian Math. J. 31 (1990), 162–165.

7. Jørgensen, T.,On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), 739–749.

8. Jørgensen, T.,A note on subgroups ofSL(2, C), Quart. J. Math. Oxford 28 (1977), 209–212.

9. Rosenberger, G., Some remarks on a paper of C. Doyle and D. James on subgroups of SL(2, R), Illinois J. Math. 28 (1984), 348–351.

10. Rosenberger, G.,Minimal generating systems of a subgroup ofSL(2, C), Proc. Edinburgh Math. Soc. 31 (1988), 261–265.

11. Wang, X. and Yang, W.,Discreteness criterion for subgroups inSL(2, C), Math. Proc. Cam- bridge Philos. Soc. 124 (1998), 51–55.

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12. Wang, X. and Yang, W.,Dense subgroups and discrete subgroups inSL(2, C), Quart. J. Math.

Oxford 50 (1999), 517–521.

PEKKA TUKIA

DEPARTMENT OF MATHEMATICS P. O. BOX 4 (YLIOPISTONKATU 5) FIN-00014 UNIVERSITY OF HELSINKI FINLAND

E-mail:pekka.tukia@helsinki.fi

XIANTAO WANG

DEPARTMENT OF MATHEMATICS HUNAN UNIVERSITY

CHANGSHA, HUNAN 410082 P. R. OF CHINA

E-mail:xtwang@mail.hunu.edu.cn

Referencer

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