On Products of Isometries of Hyperbolic Space

E. Falbel and R. Wentworth

We show that for arbitrary fixed conjugacy classes C_1,..., C_l, l>2, of loxodromic isometries of the two dimensional complex or quaternionic hyperbolic space there exist isometries g_1,..., g_l, where each g_i is in C_i, and whose product is the identity. The result follows from the properness, up to conjugation, of the multiplication map on a pair of conjugacy classes in rank 1 groups.