The Groups (2, M Vertical Bar N, K Vertical Bar 1, Q): Finiteness And Homotopy

We initiate the study of the groups (l, m, vertical bar n, k vertical bar p, q) defined by the presentation < a; b vertical bar a(l) , b(m) ,(ab)(n) , (a(p)b(q))(k)>. When p = 1 and q = m - 1, we obtain the group (l, m, vertical bar n, k) , first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l = 2 and <= 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) congruent to C-2 and pi(1)(Y) congruent to C-m; in particular, pi(1)(Z) congruent to (2, m vertical bar n, k vertical bar l, q).