The groups \((2,3,p;q)\); asphericity and a conjecture of Coxeter
For any two positive integers \(p\) and \(q\), \((2,3,p,q)\) is the group \(\langle a, b: a^ 2 = b^ 3 = (ab)^ p = [a,b]^ q = 1\rangle\). Specific values of \((p,q)\) for which \((2,3,p;q)\) is finite or infinite are known. Beyond that, a conjecture of Coxeter about the closely related group \(G^{3,7,k}\) implies (if true) that \((2,3,7;q)\) is infinite for \(q \geq 9\). The authors of the paper under review advance the knowledge of finiteness of these groups by proving the following theorem: With the possible exceptions of \((p,q) = (13,4)\) and \((7,11)\), the group \((2,3,p;q)\) is infinite if and only if \(p\) and \(q\) satisfy one of the following conditions: (i) \(p = 7\), \(q \geq 9\); (ii) \(p = 8\) or 9, \(q \geq 6\); (iii) \(p = 10\) or 11, \(q \geq 5\); (iv) \(p \geq 12\), \(q \geq 4\). They report that M. Edjvet has very recently proved that (2,3,7;11) is infinite, leaving (2,3,13;4) as the only open case.
scientific article; zbMATH DE number 168583