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The triangle groups. (English) Zbl 0818.20032

The triangle group \(\Delta (\ell, m,n)\) is defined by the presentation \(\langle a,b,c:a^ 2= b^ 2= c^ 2= (ab)^ \ell= (bc)^ m= (ca)^ n= 1\rangle\). It is finite if \({1\over \ell}+ {1\over m}+ {1\over n}>1\) and infinite if \({1\over \ell}+ {1\over m}+ {1\over n}< 1\). A fairly detailed description of these groups is available in H. S. M. Coxeter and W. O. J. Moser’s well known book [Generators and relations for discrete groups (4th ed. Springer 1980; Zbl 0422.20001)].
It is known that \(\Delta (\ell, m,n)\) is parabolic, elliptic or hyperbolic depending upon whether \({1\over \ell}+ {1\over m}+ {1\over n}= 1\), \({1\over \ell}+ {1\over m}+ {1\over n}>1\) or \({1\over\ell}+ {1\over m}+ {1\over n}<1\). If we let \(x=ab\), \(y=bc\) and \(H= \langle x,y\rangle\) then it is straightforward to see that \(H\) is a normal subgroup of index 2 in \(\Delta (\ell, m,n)\). The group \(H\) is isomorphic to the van Dyck group \(D(\ell, m,n)= \langle x,y\): \(x^ \ell= y^ m= (xy)^ n= 1\rangle\). It has been shown in the paper under review that \(D(\ell, m,n)/ D'(\ell, m,n)\) is finite iff at most one of \(\ell\), \(m\), \(n\) is zero; \(D(\ell, m,n)\) is perfect iff \(\ell\), \(m\), \(n\) are relatively prime; and \(D(n, n,n)\) is infinite iff \(n\geq 3\). The description of the various triangle groups in the three categories, viz. parabolic, elliptic and hyperbolic is also given.
Reviewer’s remark: Strangely, six out of thirteen references are not cited in the text and some relevant, and latest references such as [M. D. E. Conder, Q. J. Math., Oxf. II. Ser. 38, 427-447 (1987; Zbl 0628.20030); Q. Mushtaq and H. Servatius, J. Lond. Math. Soc., II. Ser. 48, 77-86 (1993; Zbl 0737.20020); Q. Mushtaq, Commun. Algebra 18, No. 11, 3857-3888 (1990; Zbl 0721.20033); and Q. Mushtaq and F. Shaheen, Ars Comb. 23 A, 187-193 (1987; Zbl 0621.20019)] are missing.

MSC:

20F05 Generators, relations, and presentations of groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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References:

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