NEW ZEALAND JOURNAL OF MATHEMATICS Volume 26 (1997), 45-52

FAITH FU L C O M P L E X R E PR E SE N TA TIO N S OF ONE R E L A T O R G R O U PS

B e n j a m in F in e , F r a n k L e v in a n d G e r h a r d R o s e n b e r g e r

(Received September 1994)

Abstract. Given a one-relator product of cyclics G we begin a study of the following two questions. First, under what conditions on the relator R will a one-relator product admit a faithful two-dimensional complex representation?Second, under what conditions on the relator R will a one-relator product ad­mit a two-dimensional complex representation which is both faithful and with a discrete image? We first survey what is known about faithful discrete repre­sentations into P 5L 2(Q . We then prove a necessary condition on the relator, based on a classical result of Magnus, for a two generator one-relator group to admit a faithful discrete representation into P S I ^ Q - Using cancellation arguments in free groups we then describe conditions on words R(x, y) in a free group on two generators x , y which allow them to meet the above condition.

1. Introduction

A one-relator product of cyclics is a group G with a presentation of the form

G = .fln jflf = = R m = l), (1.1)

where ej = 0 or e* > 2 for i = 1 ,... , n, R is a cyclically reduced in the free product on {a i , . . . , an} involving all the generators and m > 1. R is called the relator and if m > 2, R is said to be a proper power. Such groups are natural generalizations of one-relator groups and provide natural algebraic generalizations of Fuchsian groups {see [4, 5]}. Crucial to the study of G is the use of essential representations which are representations p : G —> H, where H is a linear group over a some field, such that p{ai) has infinite order if ei = 0 and exact order e* if e* > 2 and p(R) has exact order m. Recall that a linear group over a field F is a subgroup of GLn(F) or PGLn(F ) for some n. Of particular importance have been essential representations into PSX2(C) [1, 3, 4, 5, 6, 7, 8, 9, 18]. If m > 2 these groups always admit essential representations into PSX2(C) and this has been used to show that under certain additional conditions they satisfy many properties in common with linear groups such as being virtually torsion-free and satisfying the Tits alternative. An n-dimensional complex representation of a group G is a representation p of G with image in GLn(C) or PGLn(C) for an integer n.

1991 AMS Mathematics Subject Classification: Primary 20E05,20E06,20E07; Secondary 20F06. Key words and phrases: one-relator products, one-relator group, complex representation.

46 BENJAMIN FINE, FRANK LEVIN AND GERHARD ROSENBERGER

In [5] the following two questions were posed :(1) Under what conditions on the relator R will a one-relator product admit a

faithful two-dimensional complex representation?(2) Under what conditions on the relator R will a one-relator product admit a

two-dimensional complex representation which is both faithful and with a discrete image?

In this note we begin a study of these questions. We first survey what is known about faithful discrete representations into PSL 2 (C). We then prove a necessary condition on the relator, based on a classical result of Magnus, for a two generator one-relator group to admit a faithful discrete representation into PSL 2 (C). We then use cancellation arguments in free groups to describe conditions on words R(x, y) in a free group on two generators x, y which allow them to meet the above condition.

2. Faithful Representations

A Fuchsian group F is a non-elementary discrete subgroup of PSL 2 (M) or a conjugate of such a group in PSL 2 (C). If F is finitely generated then F has a presentation - called the Poincare presentation - of the form

F = ( c i j . . . , 6p, h i, . . . , hi, <2i, bi, . . . , dg, bg, e 1, i 1 ,.. . ,p, R 1), (2.1)

where R = e i . . . eph i . . . ht [ai, &i]. . . [ag, bg] and p > 0 , t > 0 , g > 0 , p + t + g > 0, and mi > 2 for i = 1 ,... ,p. Fuchsian groups thus fall into the class of one-relator products of cyclics. The Euler Characteristic of F is given by x (F ) = where n(F) = 2g — 2 + 1 + £)(1 — 1 /mi). 2irp,(F) represents the hyperbolic area of a fundamental polygon for F.

More generally any group G with a presentation of the form (2.1) is called an F-group. If G is an F-group with ^(G) > 0 then G actually presents a Fuchsian group. In this case the group G has a faithful, discrete representation.

A group of F-Type is a one-relator product of cyclics, with a presentation of the form

G = On; a ? = 1 ,U V = 1), (2.2)

where n > 2, e* = 0 or e* > 2 , 1 < p < n — 1, U = U (a i,. . . ,ap) is a cycli­cally reduced word in the free product on a i, . . . , ap which is of infinite order andV = V(ap+1 ,. an) is a cyclically reduced word in the free product on ap+1 , . . . , an which is of infinite order. In [5] it was shown that such groups in general sat­isfy many of the same algebraic properties as the more restricted F-groups. Also in [18] it was proved that when neither U nor V is a proper power in the free product on the generators which they involve, the group G admits a faithful two- dimensional complex representation. Whether the image group is discrete or not depends on the further exact structure of R. If both U and V are proper powers then there is no faithful representation [5]. In general a faithful, discrete representation p : G —► PSL 2 (C) of a group G is said to be of finite volume if H 3 /p(G) has finite volume where H 3 is hyperbolic 3-space.

FAITHFUL REPRESENTATIONS 47

Helling, Kim and Mennicke [13] (see also [19]) have shown that if m > 4 the group G = (a,b;am = b2 = ((a~1b)2 (ab)3)2 = 1) has a faithful, discrete image in PSL2(C). Further Helling, Mennicke and Vinberg [12] show that the groups G = (a,b\ak = bl = (aba~1bab~1)m = 1 ) with k,l,m = 0 or k ,l,m > 2 and with k < I have faithful, discrete representations if at most one of k,t,m is 2 and if (k,l,m ) 7 (2,3,3). Moreover this group G has a faithful discrete representation of finite volume if and only if 2 < k < I and (1/A:) + (1 /0 + (1 /m ) > 1 . In con­nection with these results it can be shown that the groups G = (a, 6; a3 = b3 = (aba~1bab~ 1 )2 = 1 ) and G = (a,b;a3 = b4 = (aba~1bab— l )2 = 1 ) are arithmetic. In a similar manner Hagelberg [10] and Hagelberg, Maclachlan and Rosenberger[11] showed that the groups G = (a,b;ak = b* = [a, b]m = 1 ) with k ,t,m > 2, and k < t have faithful, discrete representations of finite volume precisely when (k ,t,m ) = (3,3,3), (3 ,4 ,2) or (4,4,2) and that the groups G = (a,b;ak = bl = (a~1bab~1ab~1a~1b)m = 1 ) with k ,t,m > 2 and k < t have faithful discrete repre­sentations of finite volume if (1/A;) + (1/Ar) + (1/m) > 1 and (l/t) + (l/t) + (l/m) > 1 except for (k ,t,m ) = (2,2,m) and (2,3,2). In addition Hagelberg, Maclachlan and Rosenberger [11] proved that with (k ,t,m ) = (3,3,3), (3,4,2) or (4,4,2) the groups G = (a,b;ak = bl = [a, 6]m = 1) are arithmetic. If (k ,t,m ) = (3,3,3) the group G is a subgroup of finite index in the Bianchi group P S L ^O z) while if (k, t, m) = (3,4,2) or (4,4,2), G is commensurable with the Picard group P 5L 2( 0 \). On the other hand if (l/£) + (l/£) + (l/m ) < 1 with G — (a,b;ak = b1 = [a,b]m = 1) and 2 < k < t , 2 < m then G has a faithful, discrete representation into PSX2(C) of infinite volume.

The above examples seem to lead to a general necessary condition for a gener­alized triangle group G to have a faithful, discrete representation into PSL 2 (C ) of finite volume. In [11] there is the following partial result.

Theorem 2 .1 ([11]). Let G = (a, 6;ap = bq = Rm(a,b) = 1 ) with p = 0 o r p > 2, q = 0 or q > 2 and m > 2 and R(a, b) a cyclically reduced word, not a proper power, in the free product on {a, b} which involves both a and b. Suppose one of the following holds:

(1) m > 4,(2) m = 3, p is odd if p > 2 and q is odd if q > 2.

Suppose further that G has a faithful, discrete representation into PSL 2 (C) of finite volume. Then p > 2, q > 2 and (1 /p) + (1 /q) -I- (1/m ) > 1.

A generalization of this theorem would be of great interest. In the special case of finitely generated one-relator groups we can directly obtain such a generalization using results of Chiswell [2] and Ratcliffe [17] on Euler characteristic.

Theorem 2.2. Let G = G = (ai , . . . ,an; Rm(ai, . . . ,an) = 1) with n > 2, m > 1 and R(a\,. . . ,an) a cyclically reduced word, not a proper power in the free group on { a i , . . . ,an} involving all the generators. Suppose that G has a faithful, discrete representation into PSL 2 (C) of finite volume then n = 2 and m = 1 - that is G is a torsion-free two-generator, one-relator group.

48 BENJAMIN FINE, FRANK LEVIN AND GERHARD ROSENBERGER

Proof. From [2] G has an Euler characteristic given by x(G) — I — n + (1/m). From [17] if G has a faithful, discrete representation into PSL2(<C) of finite volume then the Euler characteristic must be non-negative. It follows then that n = 2 and

The result of Theorem 2.2 leads us to consider the problem of classifying all the torsion-free two-generator one-relator groups which admit a faithful, discrete representation into PSL2(<C) of finite volume. There are many known examples of such groups with this property, for instance G = (a, 6; aba~1bab~l = 1 ).

3. Two-Generator One-Relator GroupsWe now consider the case of a two-generator one-relator group G with presenta­

tion

where R(a, b) is a non-trivial cyclically reduced word in the free group on {a, 6} involving both a and b and m > 1. In general there are no faithful representations into P5L2(<C). For example it can be shown that the group H = (a, 6; a1 bs = 1) with s,t > 2 has no faithful representation [5]. However from certain special properties of complex projective matrices coupled with a property of conjugates in free groups due to Magnus (see [15]) we get the following necessary condition for such a group to admit a faithful two-dimensional complex representation. This result was proved in a slightly different manner by Magnus [16].

Theorem 3.1. Let G be a two-generator one-relator group with form (3.1) and suppose G is non-metabelian. If G admits a faithful representation into PSL2(C) then the relator R(a, b) must satisfy the property that the word R(a, b) is conjugate in the free group on {a, b} to the word R±1(a~1,b~1).

Proof. Assume that there is a faithful representation p : G —> P5L2(C) with a —► A,b —► B. From the properties of complex projective matrices it is known that there is a projective matrix C with C A C ~l = A ~l and C B C ~1 = B ~l . Therefore we must have that Rm(A~1, B ~ l) = 1. Because p is faithful we then must also have Rm(a~1,b~1) = 1 in G. Since G has a faithful representation in P5L2(<C) it can be considered as a finitely generated linear group and hence it is residually finite and Hopfian. Therefore the map a —> a-1 , b —> 6_1 defines an automorphism of G which is induced by a Nielsen transformation. From a result of Magnus (see_[15, Theorem 4.11], also [16]) R(a,b) must be conjugate in the free group to R±1(a~1,b~1). Recall that in a free group F if X m is conjugate to Y ±m then X is conjugate to Y ± l . □

We note that Theorem 3.1 does not hold if G is metabelian. The group G(n) = (a, b; bab~l = an), where n > 2 has a faithful representation into P5L2(C) given by

But bab la n is not conjugate to (bab xa n)±1 in the free group on {a, b}.

m = l. □

G = (a, 6; Rm(a, b) = 1 ), (3.1)

FAITHFUL REPRESENTATIONS 49

From standard cancellation arguments in free groups we can further describe conditions on words R(x,y) in a free group on two generators x, y which allows them to be conjugate to R±1(x~1 ,y ~ x) and thus can be permissible relators in a two generator one-relator group which admits a faithful two dimensional complex representation.

First we give some notation. Suppose F is a free group on {a, b}. If s,t — ±1 then C/(as, 6*) means U(as,bt) = 1 or U{as,bt) = aseib1 .. . aser6 r for some natural number r and with e*, fi non-zero integers. If U (as, bl) ^ 1 then t/_ 1 (as, b*) clearly means that U~1(as, tf) = b~t ra~ser . . . b~t 1a~sei. Analagously we use the notation U(bs,al) ,s ,t = ±1. The first two propositions characterize the situation where U(a,b) is conjugate to U~1(a~1, 6-1 ) within the free group F on {a, b}.

Proposition 3.2. Suppose F is a free group on {a,b} and let 1 ^ U = U(a,b) be an element of F or 1 ^ U = U(b, a) be an element of F . If U = U(a,b) assume that there is a V in F with V ~ lU{a,b)V = U~1(a~1, 6_1). If U = U(b,a) assume that there is a V in F with V’_ 1 C/(6, a)V = U~l{b~l , a-1 ). Then, possibly after replacing U by ,b~l) = U*(b, a) if U = U(a,b) or U by U~1(b~1 ,a~1) —U*(a,b) if U = U(b,a) - one of the following cases holds:

(1 ) U — U(a,b) and a~tU(a,b)at = £/- 1 (a- 1 , 6_1) for some non-zero integer t,(2) U = U(a,b) = atUi(b,a)bsatU2(b,a)bq for some non-zero integers t,s,q and

b~sUi(b, a)bs = C/1- 1 (6_1, a-1 ) and b~qU2{b,a)bq = U2 l (b~l ,a~l),(3 ) U = U(b,a) and b^Uib^a)^ = C/_ 1 (6_1 , a- 1 ) for some non-zero integer t,(4) U — U(b,a) = btUi(a,b)asbtU2{a,b)aq for some non-zero integers t,s,q and

a~sUi(a,b)as = C/1_ 1 (a- 1 , 6_1) and a~qU2(a,b)aq = J7 -1 (a_ 1 ,6-1 ).

Proof. Suppose, without loss of generality that 1 ^ U = U(a, b) = aei b?1 . . . aerb r, r > 1 with e*, fi non-zero integers. Let V = adlbkl . . . adnbknadn+1 with all ki non­zero integers, di a non-zero integer for i = 2 ,... , n and d\ and dn+1 integers. Let L denote free product length in the free product decomposition F = (a) * (b) of F.

Cancellation arguments with respect to L give immediately that d\ ^ 0 if and only if dn+1 ^ 0. If d\ = dn+1 = 0 then V = bkl . . .a dnbkn and since U(a,b)V = VU ~1(a~1, 6-1 ) we have U~1(a~1, 6_ 1 )V’_1 = V ~ 1U(a,b) and so bfraer b^aexb~kn . .. b~kl = b~kn . . . b~klaeib^ . . . aerb r. Therefore here we may replace U(a,b) by U~1(a~1,b~1) = W(b,a) (and V by V--1). Hence we may assume that d\ ^ 0 ^ dn+1 . Then

a€lbfl . . . aerbfradlbkl... adnbknadn+1 = adl6fcl.. . adnbkn adn+1bfr aBr... b^aei

with d\ ^ 0 ^ dn+i- If n > r then from the above equation V = V1V2 with V\ = U(a,b) and C/(a, 6)V2 = V2?7 - 1 (o_ 1 , 6_1). Here we can replace V by V2. Therefore we may assume that r > n .

If n = 0 then U(a,b)adl = adlC/_ 1 (a_1, b"1) the first of the two possibilities. Now assume that r > n > 1. If r = n then again from the above equationV = U{a,b)adn+1 and therefore U(a, b)adn+1=a<in+lu 1,6 Now assume that r > n > 1. Then U(a,b) = W\(a, b)W2(a, b) with V = W\(a, b)adn+1, dn+1 = er+i and we get b)Wi(a,b)adn+1 = adn+1 W;f 1 (a- 1 , 6- 1 )iyi_ 1 (a_ 1 ,6-1 )

50 BENJAMIN FINE, FRANK LEVIN AND GERHARD ROSENBERGER

Let Wi(a,b) = aeUi(b,a)bk and W2{a, b) = adU2(b,a)bf. We then get Ui(b, a)bkadn+1 = 6feL^"1 (6_ 1 ,a_ 1 )ae. Hence e = dn+1 and Ui(b,a)bk = 6fcC7f1 (6 -1 ,a -1). Analagously d = dn+i and U2{b,a)bf = 6^C "1 (6- 1 ,a_1). This completes the second possibility in the proposition. □

After possibly exchanging a and b if U(a,b) is conjugate to U~1(a~1, 6_1) we are left with the situation 1 ^ U = U(a, b) = aeib^ . . .a er6 r, r > 1 with e*,/* non-zero integers and U(a,b)ad = adt/- 1 (a- 1 , 6_1) for some non-zero integer d. If r = 1 then we have the equation aeib iei = aeib lSl so U{a, 6)ad = adU~1(a~1, b~1) with d = e\. Now let r > 2. Comparing the exponents in the above equation leads to the next propostion.Proposition 3.3. Let 1 ,£ U = U(a, b) = aeib^ . . .a erb r, r > 2 with ei,fi non­zero integers. Suppose U(a,b)ad — adU~1(a~1 ,b~l ) for some non-zero integer d.

(1) If r = 2s > 2 is even then

U(a,b) = a eib^ .. .aesb aaes+1b saesb s~1 . ..a 62 1

(2) If r = 2s — 1 > 3 is odd then

U(a, b) = aeibS1 . . . ae“b saeab a~1... ae26^ .The final proposition handles the case where U(a,b) is conjugate to U(a~1,b~1).

Proposition 3.4. Let 1 ^ U = U(a,b) be in F or 1 ^ U = U{b, a) be in F. If U — U(a,b) assume that there is a V in F with V ~ 1U(a,b)V = C/(o- 1 , 6-1 ). If U = U(b,a) assume that there is a V in F with V - 1 {7(6,a)V = U{b~x,a~l). Then possibly after replacing U by U(a~1,b~1) = U*(b,a) if U = U(a,b) or U by C/(6_ 1 ,a_1) = U*(a,b) if U = U(b,a) one of the following cases holds:

(1) U = U(a,b) and U(a,b) = (S(a,b)S(a~1,b~1))d for some natural number d and some element S(a,b) in F.

(2) U = U(b,a) and U(b,a) = (T(6, a)T(6_ 1 ,a_1))d for some natural number d and some element T(b, a) in F.

Proof. Suppose, without loss of generality that 1 ^ U — U(a, b) = aei b?1 . . . aerb r, r > 1 with ei,fi non-zero integers. Let V = adlbkl. . . adnbknadn+1 with all ki non­zero integers, di a non-zero integer for i = 2 ,... , n and d\ and dn+1 integers. Again let L denote the free product length in the free product decomposition F = (a) * (b) of F. In this case cancellation arguments with respect to L give that d\ / 0 if and only if dn+1 = 0 . If d\ = 0 and dn+1 ± 0 then V = bkl. ■ • ddnbknddn+l and a~eib~f1 . . . a~erb~fra~dn+1b~kn . . . a~dlb~kl = a~dn+1b~kn . .. bklaeib ... a€rb r, and we may replace U(a,b) by U(a~l,b~l) = W(b, a) {and V by V-1 }. Therefore we may assume that d\ ^ 0 and dn+1 = 0. Then

aei6 .. .aerb*radlbkl . . . adnbkn = adlbkl . .. adnbkna~eib~^ . .. a~erb~fr

with d\ 7 0. If n > r then V = V\V2 with V\ = U(a,b) and U(a,b)V2 = V2U(a~1,b~1) and here we may replace V by V2. Therefore we may assume that r > n . If r = n then V = U(a,b) = U(a~1,b~1) which is impossible. Thus we have r > n. Then t/(a ,b) = Ui(a,b)U2{a,b) with Ui(a,b) = V and we get the equa­tion U2{a,b)Ul {a,b) = U\(a~l ,b~l)U2{a~l ,b~l). If L(E/i(a,b)) = L(U2{a,b)) then U2(a,b) — £/i(a- 1 , 6-1 ) and therefore U(a,b) = U\(a, b)U\(a~x, 6_1).

FAITHFUL REPRESENTATIONS 51

If L(Ui(a, b)) < L(U2(a,b)) then 1 /2(0, b) — U\(a 1,b 1)Wi(a,b) and W i(a ,6)tfi(a ,6) = Ui{a, b)Wi(a~l , b~l). If L(U2{a,b)) < L(C /i(a,6)) then Ui(a,b) = W2(a,b)U2{a -\ b -1) and U2(a,b)W2{a,b) = W2( a -\ b -1)U2(a,b) that is W2 (a, b)U2(a, b) = U2(a,b)W2 (a~1 ,b~x) with W2 (a, b) = ^ ( a -1 , &_1). In both these last two cases the desired result follows by induction. □

References

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2. I. Chiswell, Euler characteristics of groups, Math. Z. 147 (1976), 1-11.3. B. Fine, J. Howie and G. Rosenberger, One-relator quotients and free products

of cyclics, Proc. Amer. Math. Soc. 102 (1988) 1-6.4. B. Fine and G. Rosenberger, Complex representations and one-relator products

of cy clics, in Geometry of Group Representations, Contemporary Math. 74(1987), 131-149.

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6. B. Fine, F. Levin and G. Rosenberger, Free subgroups and decompositions of one-relator products of cy clics: Part 1: The Tits alternative, Arch. Math (50)(1988), 97-109.

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8. B. Fine, J. Howie and G. Rosenberger, Ree-Mendelsohn pairs in generalized triangle groups, Comm, in Algebra 17 (2) (1989), 251-258.

9. B. Fine, G. Rosenberger and M. Stille, Euler characteristic for one-relator products of cy clics, to appear.

10. M. Hagelberg, Generalized triangle groups and three-dimensional orbifolds, to appear.

11. M. Hagelberg, C. Maclachlan and G. Rosenberger , On discrete generalized triangle groups, to appear.

12. H. Helling, J. Mennicke and E.B. Vinberg, On some general triangle groups and three-dimensional orbifolds, to appear.

13. H. Helling, A. Kim and J. Mennicke, On the Fibonacci Groups, preprint.14. R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, 1977.15. W. Magnus, A. Karass and D. Solitar, Combinatorial Group Theory, Wiley,

Interscience, 1968.16. W. Magnus, Two generator subgroups of PSL2(C), Nachrichten der Academie

der Wissenschaften in Gottingen 7 (1975), 81-94.17. J. Ratcliffe, Euler Characteristics and Discrete Subgroups of SL(2, C), preprint.

52 BENJAMIN FINE, FRANK LEVIN AND GERHARD ROSENBERGER

18. G. Rosenberger, Faithful linear representations and residual finiteness of certain one-relator products of cyclics, J. Siberian Math. Soc. (1990).

19. R. Thomas, On Fibonacci Groups, preprint.

Prank LevinFakultat und Institut fur Mathematik Universitat Bochum Universitatsstrafie 150 4630 Bochum 1FEDERAL REPUBLIC OF GERMANY

Gerhard RosenbergerFachbereich Mathematik UniversitatDortmundPostfach 50 05 004600 Dortmund 50FEDERAL REPUBLIC OF GERMANY gerhardrosenberger@mathematik.uni-dortmund.de

Benjamin Fine Department of Mathematics Fairfield University FairfieldConnecticut 06430 U.S.A.fine@fairl.fairfield.edu