On products of isometries of hyperbolic space

We show that for arbitrary fixed conjugacy classes C 1 , … , C l , l ⩾ 3 , of loxodromic isometries of the two-dimensional complex or quaternionic hyperbolic space there exist isometries g 1 , … , g l , where each g i ∈ 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.