The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy

We initiate the study of the groups (l,m\mid n,k\mid p,q) defined by the presentation \langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle. When p=1 and q=m-1, we obtain the group (l,m\mid n,k), first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l=2 and \frac{1}{n}+\frac{1}{k}\leq\frac{1}{2} and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group \pi_{2}(Z), where 𝑍 is the space formed by attaching 2-cells corresponding to (ab)^{n} and (ab^{q})^{k} to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and 𝑌, where \pi_{1}(X)\cong C_{2} and \pi_{1}(Y)\cong C_{m}; in particular, \pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q).