We show that for arbitrary fixed conjugacy classes C1, ..., Cl, l ≥ 3, of loxodromic isometries of the two-dimensional complex or quaternionic hyperbolic space there exist isometries g1, ..., gl, where each gi∈ Ci, 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. © 2009 Elsevier B.V.
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