Free subgroups in certain generalized triangle groups of type (2, m, 2)

A generalized triangle group is a group that can be presented in the form G = < x, y vertical bar x(p) = y(q) = w( x, y)(r) = 1 > where p, q, r >= 2 and w(x, y) is a cyclically reduced word of length at least 2 in the free product Z(p)*Z(q) = < x, y vertical bar x(p) = y(q) = 1 >. Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple ( p, q, r) is one of (3, 3, 2), (3, 4, 2), (3, 5, 2), or (2, m, 2) where m = 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. In this paper, we show that the Tits alternative holds in the cases ( p, q, r) = ( 2, m, 2) where m = 6, 10, 12, 15, 20, 30, 60.