THE GROUPS (L,M-VERTICAL-BAR-N,K)

The groups (I, m\n, k) defined by the presentations [a, b : a(l) = b(m ) = (ab)(n) = (ab(-1))(k) = 1], were first studied systematically by C oxeter in 1939, and have been a subject of interest ever since, partic ularly with regard to the question as to which of them are finite. The finiteness question has been completely determined for l = 2 and l = 3, and there are some other partial results. In this paper, we give a complete determination as to which of the groups (l, m\n, k) are finit e.The proof of this result essentially splits into two parts. When I, m, n and k are ''large'' (in a sense to be made precise in the paper), we can use arguments in terms of pictures to show that (l, m\n, k) is infinite; this will involve finding generators for the second homotop y modules of the presentations. For small values of I, m, n and k, the groups are finite, and we can quote previously established results. F or intermediate values, the groups can still be infinite, even though the arguments in terms of pictures do not apply. In these cases, where the status of the group was previously open, we produce a series of i ndividual arguments to show that the groups are infinite; many of thes e are based on computational results.