Families of group presentations related to topology (Q1771332)

The authors consider a class of cyclically presented groups \(G_n^\varepsilon (m,k,h)\), where \(\varepsilon =(a,b,r,s)\in\mathbb Z^4\) for \(n\geq 2\) and the integers \(m,k,h\) are taken modulo \(n\). Then \(G_n^\varepsilon (m,k,h)\) has generators \(x_1,\ldots ,x_n\) and defining relations \[ x_i^a x_{i+k}^b x_{i+h+m}^a = (x_{i+h}^r x_{i+m}^r)^s \] for \(i=1,\ldots ,n\) (subscripts mod \(n\)). This class of groups contains many well known classes of groups like some Fibonacci-groups, Sieradski groups, Gilbert-Howie groups and others. Under certain conditions on the parameters some of these groups are shown to be isomorphic. The main result is the following: Theorem 3.1 Suppose that \(n\) and \(b\) are odd and \(n\) is coprime with \(2k-h-m\). Then the group \(G_n^\varepsilon (m,k,h)\) cannot be the fundamental group of any hyperbolic 3-orbifold of finite volume. In the last part of the paper the authors study a certain subclass of the groups \(G_n^\varepsilon (m,k,h)\), where \(a=b=1\) and \(s=0\). Under certain conditions these groups are shown to be aspherical.