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

Abstract We initiate the study of the groups ( l , m ∣ n , k ∣ p , q ) (l,m\mid n,k\mid p,q) defined by the presentation ⟨ a , b ∣ a l , b m , ( a ⁢ b ) n , ( a p ⁢ b q ) k ⟩ \langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle . When p = 1 p=1 and q = m - 1 q=m-1 , we obtain the group ( l , m ∣ n , k ) (l,m\mid n,k) , first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l = 2 l=2 and 1 n + 1 k ≤ 1 2 \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 π 2 ⁢ ( Z ) \pi_{2}(Z) , where 𝑍 is the space formed by attaching 2-cells corresponding to ( a ⁢ b ) n (ab)^{n} and ( a ⁢ b q ) k (ab^{q})^{k} to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and 𝑌, where π 1 ⁢ ( X ) ≅ C 2 \pi_{1}(X)\cong C_{2} and π 1 ⁢ ( Y ) ≅ C m \pi_{1}(Y)\cong C_{m} ; in particular, π 1 ( Z ) ≅ ( 2 , m ∣ n , k ∣ 1 , q ) \pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q) .