The Tits alternative for generalized triangle groups of type (3,4,2)

A generalized triangle group is a group that can be presented in the form G=⟨x,y |xp=yq=w(x,y)r=1⟩ where p,q,r≥2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product Zp∗Zq=⟨x,y |xp=yq=1⟩. Rosenberger has conjectured that every generalized triangle group G satisfies the Ti...

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Datum:	2008
Hauptverfasser:	Howie, J., Williams, G.
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Sprache:	English
Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2008
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spelling	irk-123456789-1533572019-06-15T01:26:40Z The Tits alternative for generalized triangle groups of type (3,4,2) Howie, J. Williams, G. A generalized triangle group is a group that can be presented in the form G=⟨x,y \|xp=yq=w(x,y)r=1⟩ where p,q,r≥2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product Zp∗Zq=⟨x,y \|xp=yq=1⟩. Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p,q,r) is one of (2,3,2), (2,4,2), (2,5,2), (3,3,2), (3,4,2), or (3,5,2). Building on a result of Benyash-Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case (p,q,r)=(3,4,2). 2008 Article The Tits alternative for generalized triangle groups of type (3,4,2) / J. Howie, G. Williams // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 40–48. — Бібліогр.: 16 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20F05, 20E05, 57M07. http://dspace.nbuv.gov.ua/handle/123456789/153357 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A generalized triangle group is a group that can be presented in the form G=⟨x,y \|xp=yq=w(x,y)r=1⟩ where p,q,r≥2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product Zp∗Zq=⟨x,y \|xp=yq=1⟩. Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p,q,r) is one of (2,3,2), (2,4,2), (2,5,2), (3,3,2), (3,4,2), or (3,5,2). Building on a result of Benyash-Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case (p,q,r)=(3,4,2).
format	Article
author	Howie, J. Williams, G.
spellingShingle	Howie, J. Williams, G. The Tits alternative for generalized triangle groups of type (3,4,2) Algebra and Discrete Mathematics
author_facet	Howie, J. Williams, G.
author_sort	Howie, J.
title	The Tits alternative for generalized triangle groups of type (3,4,2)
title_short	The Tits alternative for generalized triangle groups of type (3,4,2)
title_full	The Tits alternative for generalized triangle groups of type (3,4,2)
title_fullStr	The Tits alternative for generalized triangle groups of type (3,4,2)
title_full_unstemmed	The Tits alternative for generalized triangle groups of type (3,4,2)
title_sort	tits alternative for generalized triangle groups of type (3,4,2)
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/153357
citation_txt	The Tits alternative for generalized triangle groups of type (3,4,2) / J. Howie, G. Williams // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 40–48. — Бібліогр.: 16 назв. — англ.
series	Algebra and Discrete Mathematics
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2008). pp. 40 – 48 c© Journal “Algebra and Discrete Mathematics” The Tits alternative for generalized triangle groups of type (3, 4, 2) James Howie and Gerald Williams Communicated by V. I. Sushchansky Abstract. A generalized triangle group is a group that can be presented in the form G = 〈 x, y \| xp = yq = w(x, y)r = 1 〉 where p, q, r ≥ 2 and w(x, y) is a cyclically reduced word of length at least 2 in the free product Zp ∗ Zq = 〈 x, y \| xp = yq = 1 〉. Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p, q, r) is one of (2, 3, 2), (2, 4, 2), (2, 5, 2), (3, 3, 2), (3, 4, 2), or (3, 5, 2). Building on a result of Benyash-Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case (p, q, r) = (3, 4, 2). 1. Introduction A generalized triangle group is a group that can be presented in the form G = 〈 x, y \| xp = yq = w(x, y)r = 1 〉 where p, q, r ≥ 2 and w(x, y) is a cyclically reduced word of length at least 2 in the free product Zp ∗Zq = 〈 x, y \| xp = yq = 1 〉 that is not a proper power. It was conjectured by Rosenberger [16] that every generalized triangle group G satisfies the Tits alternative. That is, G either contains a non-abelian free subgroup or has a soluble subgroup of finite index. If 1/p+1/q+1/r < 1 then G contains a non-abelian free subgroup [2]; if r ≥ 3 then the Tits alternative holds, and in most cases G contains 2000 Mathematics Subject Classification: 20F05, 20E05, 57M07. Key words and phrases: Generalized triangle group, Tits alternative, free sub- group. J. Howie, G. Williams 41 a non-abelian free subgroup [9]. (These results are also described in the survey article [10] and in [11].) The cases r = 2, 1/p+1/q+1/r ≥ 1 have had to be treated on a case by case basis. The Tits alternative was shown to hold for the cases (3, 6, 2), (4, 4, 2) in [14], and for the cases (2, q, 2) (q ≥ 6) in [1],[4],[3],[5],[7],[15]. Thus the open cases of the conjecture are (p, q, r) = (2, 3, 2), (2, 4, 2), (2, 5, 2), (3, 3, 2), (3, 4, 2), and (3, 5, 2). In this paper we show that the conjecture holds for the case (3, 4, 2): Main Theorem. Let Γ = 〈 x, y \| x3 = y4 = w(x, y)2 = 1 〉 where w(x, y) = xα1yβ1 . . . xαkyβk , 1 ≤ αi ≤ 2, 1 ≤ βi ≤ 3 for each 1 ≤ i ≤ k where k ≥ 1. Then the Tits alternative holds for Γ. Benyash-Krivets and Barkovich [6],[7] have proved this result when k is even, and for this reason we focus on the case when k is odd. 2. Preliminaries We first recall some definitions and well-known facts concerning general- ized triangle groups; further details are available in (for example) [10]. Let G = 〈 x, y \| xℓ = ym = w(x, y)2 = 1 〉 where w(x, y) = xα1yβ1 . . . xαkyβk , 1 ≤ αi < ℓ, 1 ≤ βi < m for each 1 ≤ i ≤ k where k ≥ 1. A ho- momorphism ρ : G → H (for some group H) is said to be essential if ρ(x), ρ(y), ρ(w) are of orders ℓ,m, 2 respectively. By [2] G admits an essential representation into PSL(2,C). A projective matrix A ∈ PSL(2,C) is of order n if and only if tr(A) = 2 cos(qπ/n) for some (q, n) = 1. Note that in PSL(2,C) traces are only defined up to sign. A subgroup of PSL(2,C) is said to be elementary if it has a soluble subgroup of finite index, and is said to be non-elementary otherwise. Let ρ : 〈 x, y \| xℓ = ym = 1 〉 → PSL(2,C) be given by x 7→ X, y 7→ Y where X,Y have orders ℓ,m, respectively. Then w(x, y) 7→ w(X,Y ). By Horowitz [13] trw(X,Y ) is a polynomial with integer coefficients in trX, trY, trXY , of degree k in trXY . Since X,Y have orders ℓ,m, re- spectively, we may assume (by composing ρ with an automorphism of 〈 x, y \| xℓ = ym = 1 〉, if necessary), that trX = 2 cos(π/ℓ), trY = 2 cos(π/m). Moreover (again by [13]) X and Y can be any elements of PSL(2,C) with these traces. We refer to trw(X,Y ) as the trace poly- nomial of G. The representation ρ induces an essential representation G → PSL(2,C) if and only if trρ(w) = 0; that is, if and only if trXY is 42 Tits alternative for generalized triangle groups a root of trw(X,Y ). By [13] the leading coefficient of trw(X,Y ) is given by c = k∏ i=1 sin(αiπ/ℓ) sin(βiπ/m) sin(π/ℓ) sin(π/m) . (1) Now if X,Y generate a non-elementary subgroup of PSL(2,C) then ρ(G) (and hence G) contains a non-abelian free subgroup. Thus in prov- ing that G contains a non-abelian free subgroup we may assume thatX,Y generate an elementary subgroup of PSL(2,C). By Corollary 2.4 of [16] there are then three possibilities: (i) X,Y generate a finite subgroup of PSL(2,C); (ii) tr[X,Y ] = 2; or (iii) trXY = 0. The finite subgroups of PSL(2,C) are the alternating groups A4 and A5, the symmetric group S4, cyclic and dihedral groups (see for example [8]). The Fricke identity tr[X,Y ] = (trX)2 + (trY )2 + (trXY )2 − (trX)(trY )(trXY ) − 2 implies that (ii) is equivalent to trXY = 2 cos(π/ℓ± π/m). These values occur as roots of trw(X,Y ) if and only if G admits an essential cyclic representation. Such a representation can be realized as x 7→ A, y 7→ B where A = ( eiπ/ℓ 0 0 e−iπ/ℓ ) , B = ( e±iπ/m 0 0 e∓iπ/m ) . We summarize the above as Lemma 1. Let G = 〈 x, y \| xℓ = ym = w(x, y)2 = 1 〉. Suppose G → PSL(2,C) is an essential representation given by x 7→ X, y 7→ Y , where trX = 2 cos(π/ℓ), trY = 2 cos(π/m). If G does not contain a non-abelian free subgroup then one of the following occurs: 1. X,Y generate A4, S4, A5 or a finite dihedral group; 2. trXY = 2 cos(π/ℓ± π/m); 3. trXY = 0. Case 2 occurs if and only if G admits an essential cyclic representation. 3. Proof of Main Theorem Throughout this section Γ will be the group defined in the Main Theorem. Lemma 2. If Γ admits an essential cyclic representation then Γ contains a non-abelian free subgroup. J. Howie, G. Williams 43 Proof. Let ρ : Γ → Z12 be an essential representation. Then K = kerρ has a deficiency zero presentation with generators a1 = yxy−1x−1, a2 = y2xy−2x−1, a3 = y3xy−3x−1, a4 = xyxy−1x−2, a5 = xy2xy−2x−2, a6 = xy3xy−3x−2, and with relators W ′(ai, . . . , a6, a1, . . . , ai−1)W ′(y2aiy 2, . . . , y2a6y 2, y2a1y 2, . . . , y2ai−1y 2) (1 ≤ i ≤ 6) where W ′ is a rewrite of W . Let S = { [ai, aj ], ai(y 2aiy 2) (1 ≤ i, j ≤ 6) }, and let L,N respectively be the normal closures of S and S ∪ {a6} in K. Noting that y2a1y 2 = a3a −1 2 , y2a2y 2 = a−1 2 , y2a3y 2 = a1a −1 2 , y2a4y 2 = a2a6a −1 5 a−1 2 , y2a5y 2 = a2a −1 5 a−1 2 , y2a6y 2 = a2a4a −1 5 a−1 2 , we have that K/L ∼= Z 4 and K/N ∼= Z 3, and hence that N/N ′ 6= 0. Let φ : K → K be given by ai 7→ y2aiy 2 (1 ≤ i ≤ 6). It is clear from the presentation of K that φ is an automorphism of K; furthermore φ(N) = N . In the abelian group K/N , φ(ai) = y2aiy 2 = a−1 i (1 ≤ i ≤ 6). That is, φ induces the antipodal automorphism α 7→ −α on K/N . By Corollary 3.2 of [14], K contains a non-abelian free subgroup. We will write the trace polynomial of Γ as τ(λ) = trw(X,Y ), where tr(X) = 1, tr(Y ) = √ 2, λ = tr(XY ). By Lemmas 1 and 2 we may assume that trXY = 0 or X,Y generate A4, S4, or A5. But Y has order 4 so X,Y cannot generate A4 or A5. If X,Y generate S4 then the product XY has order 2 or 4 so trXY = 0,± √ 2. Suppose trXY = − √ 2. It follows from the identity trXY + trX−1Y = (trX)(trY ) that trX−1Y = 2 √ 2. Replacing X by X−1 in Lemma 1 shows that Γ contains a non-abelian free subgroup. Thus we may assume that the only roots λ = trXY of τ are λ = 0, √ 2. Using (1) the leading coefficient of τ is given by c = ±( √ 2)κ where κ denotes the number of values of i for which βi = 2. Hence τ(λ) takes the form τ(λ) = ( √ 2)κλs(λ− √ 2)k−s (2) where s ≥ 0. Moreover, Theorem 2 of [7] implies that the Main Theorem holds when k is even, so we may assume that k is odd. 44 Tits alternative for generalized triangle groups Let A = ( eiπ/3 0 1 e−iπ/3 ) , B = ( eiπ/4 z 0 e−iπ/4 ) . Then trA = 1, trB = √ 2, trAB = z − ( √ 6 − √ 2)/2. Consider the representation ρ : 〈 x, y \| x3 = y4 = 1 〉 → PSL(2,C) given by x 7→ A, y 7→ B. Then trρ(xα1yβ1 . . . xαkyβk) = τ(z − ( √ 6 − √ 2)/2) whose constant term (by (2)) is ±( √ 2)κ(( √ 6 − √ 2)/2)s(( √ 6 + √ 2)/2)k−s which simplifies to ±( √ 2)κ(( √ 6 + √ 2)/2)k−2s. Now the constant term in tr(Aα1Bβ1 . . . AαkBβk) is equal to 2 cos ( (4 ∑k i=1 αi + 3 ∑k i=1 βi)π 12 ) . Thus ( √ 2)κ(( √ 6 + √ 2)/2))k−2s = 2 cos ( (4 ∑ k i=1 αi+3 ∑ k i=1 βi)π 12 ) and since k is odd, this only happens if κ = 0 and k − 2s = ±1. It follows that 4 k∑ i=1 αi + 3 k∑ i=1 βi = 1, 5, 7, 11 mod 12. (3) Since κ = 0 there is no value of i for which βi = 2 and hence Γ maps homomorphically onto the group Γ̄ = 〈 x, y \| x3 = y2 = w̄(x, y)2 = 1 〉 (4) where w̄(x, y) = xα1y . . . xαky. If w̄ is a proper power then Γ̄ contains a non-abelian free subgroup by [2]. Thus we may assume that w̄ is not a proper power, and so (4) is a presentation of Γ̄ as a generalized triangle group. We will write the trace polynomial of Γ̄ as σ(µ) = trw̄(X̄, Ȳ ), where tr(X̄) = 1, tr(Ȳ ) = 0, µ = tr(X̄Ȳ ). It follows from (3) that ∑k i=1 αi 6= 0 mod 3 so Γ̄ admits no essential cyclic representation. By Lemma 1 we may assume that µ = 0 or X̄, Ȳ generate A4, S4, A5 or a finite dihe- dral group, in which case X̄Ȳ has order 2, 3, 4, or 5 and hence µ = 0,±1,± √ 2, (±1± √ 5)/2. Moreover X̄ is of order 4 in SL(2,C) so X̄−1 = −X̄ and thus tr(X̄−1Ȳ ) = −µ and trw̄(X̄, Ȳ ) = (−1)ktrw̄(X̄−1, Ȳ ), so J. Howie, G. Williams 45 σw(µ) = ±σw(−µ). Thus µ and −µ occur as roots of σ with equal mul- tiplicity. By (1) the leading coefficient of σ is ±1 so σ(µ) = ±µu1(µ2 − 1)u2(µ2 − 2)u3(µ2 − (3 + √ 5)/2)u4(µ2 − (3− √ 5)/2)u5 where u1, u2, u3, u4, u5 ≥ 0 and u1 + 2u2 + 2u3 + 2u4 + 2u5 = k. Since trw̄(X̄Ȳ ) is a polynomial with integer coefficients in trX̄ = 1, trȲ = 0, µ we have that u5 = u4 so σ(µ) = ±µu1(µ2 − 1)u2(µ2 − 2)u3(µ4 − 3µ2 + 1)u4 (5) and u1 + 2u2 + 2u3 + 4u4 = k. Let Ã = ( eiπ/3 0 1 e−iπ/3 ) , B̃ = ( i z 0 −i ) . Then trÃ = 1, trB̃ = 0, trÃB̃ = z − √ 3. Now the constant term in σ(z − √ 3) is (− √ 3)u1 · 2u2 . But the constant term in tr(Ãα1B̃ . . . ÃαkB̃) is 2 cos((2 ∑k i=1 αi + 3k)π/3) = ± √ 3 so u1 = 1, u2 = 0 and thus k = 1 + 2u3 + 4u4. Lemma 3. If √ 2 is a repeated root of σ(µ) then Γ contains a non-abelian free subgroup. Proof. Let q : Γ → Γ̄ denote the canonical epimorphism. By hypothesis, there is an essential representation ρ : Γ̄ → PSL(2,C[µ]/(µ − √ 2)2). Indeed, we can construct ρ explicitly via: ρ(x) = ( eiπ/3 µ 0 e−iπ/3 ) , ρ(y) = ( 0 −1 1 0 ) . Composing this with the canonical epimorphism ψ : PSL(2,C[µ]/(µ− √ 2)2) → PSL(2,C[µ]/(µ− √ 2)) ∼= PSL(2,C) gives an essential representation ρ̃ = ψ ◦ ρ : Γ̄ → PSL(2,C) with image S4, corresponding to the root √ 2 of the trace polynomial. Let K̄ denote the kernel of ρ̃, V the kernel of ψ, and K the kernel of the composite map ρ̃ ◦ q : Γ → PSL(2,C). Then V is a complex vector space, since its elements have the form ±(I+(µ− √ 2)A) for various 2×2 matrices A, with multiplication [±(I + (µ− √ 2)A)][±(I + (µ− √ 2)B)] = ±(I + (µ− √ 2)(A+B)). Now K̄ is generated by conjugates of (xy)4 and ρ((xy)4) = −I+(µ− √ 2)M where M = ( 2 √ 2 −2(1 + i √ 3) 2(1 − i √ 3) −2 √ 2 ) . Since M is non-zero, 46 Tits alternative for generalized triangle groups K̄ (and hence K) maps onto the free abelian group of rank 1. Let N be a normal subgroup of K such that K/N ∼= Z. Note that K arises as the fundamental group of a 2-dimensional CW- complex X arising from the given presentation of Γ. This complex X has 24 cells of dimension 0, 48 cells of dimension 1, and 24(1 4+ 1 3+ 1 2) = 26 cells of dimension 2. Here, 24/4 = 6 of the 2-cells (call them α1, . . . , α6, say) arise from the relator y4, 24/3 = 8 (α7, . . . , α14, say) arise from the relator x3, and 24/2 = 12 (α15, . . . , α26, say) arise from the relator w(x, y)2. Moreover, α1, . . . , α6 are attached by maps which are 2nd powers. Let X̂ be the regular covering complex of X corresponding to the normal subgroup N of K and let α̂i denote a lift of the 2-cell αi. Then each of α̂1, . . . , α̂6 is a 2-cell attached by a map which is a 2nd power. Let GF2 denote the field of 2 elements. Now H2(X̂,GF2) is a sub- group of the 2-chain group C2(X̂,GF2) and since K/N freely permutes the cells of X̂, C2(X̂,GF2) is a free GF2(K/N)-module on the basis α̂1, . . . , α̂26. Let Q be the free GF2(K/N)-submodule of C2(X̂,GF2) of rank 6 generated by α̂1, . . . , α̂6. Since these 2-cells are attached by maps which are 2nd powers, their boundaries in the 1-chain group C1(X̂,GF2) are zero. Thus Q is a subgroup of H2(X̂,GF2). Since the rank of Q is greater than χ(X) = 2, Theorem A of [14] implies that K, and hence Γ, contains a non-abelian free subgroup Lemma 4. If (1 + √ 5)/2 is a repeated root of σ(µ) then Γ contains a non-abelian free subgroup. Proof. The proof is similar to that of Lemma 3. In this case ρ̃ has image A5, corresponding to the root (1+ √ 5)/2. The complex X has 60 0-cells, 120 1-cells, and 60(1 4 + 1 3 + 1 2) = 65 2-cells (so χ(X) = 5). Moreover, 60/4 = 15 of the 2-cells (call them α1, . . . , α15, say) are attached by maps which are 2nd powers. As before, the free GF2(K/N)-submodule, Q, of C2(X̂,GF2) of rank 15 generated by α̂1, . . . , α̂15 is a subgroup of H2(X̂,GF2). Since the rank of Q is greater than χ(X), Theorem A of [14] again implies that K contains a non-abelian free subgroup. By Lemmas 3 and 4 we may assume u3, u4 ≤ 1 so k ≤ 7. A computer search reveals that if k = 3 or 7 then there is no word w(x, y) such that τ(λ) is of the form (2). If k = 5 then (up to cyclic permutation, inversion, and automorphisms of 〈 x \| x3 〉 and 〈 y \| y4 〉) the only word w(x, y) with τ(λ) of the form (2) is w = xyxyx2y3x2yxy3. In this case, a computer search using GAP [12] shows that Γ contains a subgroup of index 4 which maps onto the free group of rank 2. If k = 1 then either Γ = 〈 x, y \| x3 = y4 = (xy)2 = 1 〉 or Γ = 〈 x, y \| x3 = y4 = (xy2)2 = 1 〉. J. Howie, G. Williams 47 In the first case Γ ∼= S4, and in the second Γ can be written as an amalgamated free product Γ = 〈 x, y2 \| x3 = y4 = (xy2)2 = 1 〉 ∗ 〈 y2 \| y4 〉〈 y \| y4 〉 in which the amalgamated subgroup has index 3 in the first factor and index 2 in the second, and thus Γ contains a non-abelian free subgroup. This completes the proof of the Main Theorem. References [1] O.A. Barkovich and V.V. Benyash-Krivets. On Tits alternative for generalized triangular groups of (2,6,2) type (Russian). Dokl. Nat. Akad. Nauk. Belarusi, 48(3):28–33, 2003. [2] Gilbert Baumslag, John W. Morgan, and Peter B. Shalen. Generalized triangle groups. Math. Proc. Cambridge Philos. Soc., 102(1):25–31, 1987. [3] V.V. Benyash-Krivets. On the Tits alternative for some finitely generated groups (Russian). Dokl. Nat. Akad. Nauk. Belarusi, 47(2):14–17, 2003. [4] V.V. Benyash-Krivets. On free subgroups of certain generalised triangle groups (Russian). Dokl. Nat. Akad. Nauk. Belarusi, 47(3):29–32, 2003. [5] V.V. Benyash-Krivets. On Rosenberger’s conjecture for generalized triangle groups of types (2, 10, 2) and (2, 20, 2). In Shyam L. Kalla et al., editor, Proceed- ings of the international conference on mathematics and its applications, pages 59–74. Kuwait Foundation for the Advancement of Sciences, 2005. [6] V.V Benyash-Krivets and O.A. Barkovich. On the Tits alternative for some generalized triangle groups of type (3, 4, 2) (Russian). Dokl. Nat. Akad. Nauk. Belarusi, 47(6):24–27, 2003. [7] V.V Benyash-Krivets and O.A. Barkovich. On the Tits alternative for some generalized triangle groups. Algebra Discrete Math., 2004(2):23–43, 2004. [8] H.S.M. Coxeter and W.O.J. Moser. Generators and relations for discrete groups. Ergeb. Math. Grenzgebiete. Springer-Verlag, Berlin-Heidelberg-New York, 1972. [9] Benjamin Fine, Frank Levin, and Gerhard Rosenberger. Free subgroups and decompositions of one-relator products of cyclics. I. The Tits alternative. Arch. Math. (Basel), 50(2):97–109, 1988. [10] Benjamin Fine, Frank Roehl, and Gerhard Rosenberger. The Tits alternative for generalized triangle groups. In Young Gheel Baik et al., editor, Groups - Korea ’98. Proceedings of the 4th international conference, Pusan, Korea, August 10-16, 1998, pages 95–131. Berlin: Walter de Gruyter, 2000. [11] Benjamin Fine and Gerhard Rosenberger. Algebraic generalizations of discrete groups: a path to combinatorial group theory through one-relator products. New York: Marcel Dekker, 1999. [12] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4, 2004. (http://www.gap-system.org). [13] Robert D. Horowitz. Characters of free groups represented in the two-dimensional special linear group. Comm. Pure Appl. Math., 25:635–649, 1972. [14] James Howie. Free subgroups in groups of small deficiency. J. Group Theory, 1(1):95–112, 1998. 48 Tits alternative for generalized triangle groups [15] James Howie and Gerald Williams. Free subgroups in certain generalized triangle groups of type (2, m, 2). Geometriae Dedicata, 119 (2006), 181–197. [16] Gerhard Rosenberger. On free subgroups of generalized triangle groups. Algebra i Logika, 28(2):227–240, 245, 1989. Contact information J. Howie Maxwell Institute of Mathematical Sciences Heriot-Watt University Edinburgh EH14 4AS United Kingdom E-Mail: J.Howie@hw.ac.uk URL: www.ma.hw.ac.uk/maths/People/ Frontpages/jim.html G. Williams Department of Mathematical Sciences University of Essex Colchester CO4 3SQ United Kingdom E-Mail: gwill@essex.ac.uk URL: www.essex.ac.uk/maths/staff/williams/ Received by the editors: 15.05.2007 and in final form 16.10.2007.

The Tits alternative for generalized triangle groups of type (3,4,2)

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