Generalised Triangle Groups of Type (2,4,2)

A conjecture of Rosenberger says that a group of the form x, y|xᵖ=yq=W (x, y) ʳ=1 (with r>1) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in the case (p, q, r) = (2, 4, 2). The results obtained are only partial, but nevertheless provide strong evidence in support of the conjecture in the case in question, in that the word W in any counterexample is shown to satisfy some strong restrictions. The exponent-sums of x and y in W must be even and odd respectively, while its free-product (or syllable) length must be at least 68. There is also a report of computer investigations which yield a stronger lower bound of 196 for the free-product length.