Jean Michel Institut de Mathématiques Hurwitz braid group action on n-tuples of reflections Wednesday 8th December, 12:05-12:55pm, Carslaw 157. Dubrovin and Mazocco (Inventiones 141 (2000), 55-147) have given a proof that, if the Hurwitz action of the braid group on a triple of Euclidean reflections in R3 has a finite orbit, then the group generated by these reflections is finite. The proof is rather long, and they ask if the analogous question has a positive answer in Rn. Humphries (J. Algebra 269 (2003), 556-588) has asserted such a result, but his proof is irremediably flawed. I have found a very short proof that if the Hurwitz orbit of an n-tuple of Euclidean reflections is finite, then the group generated is finite. As well as correcting Humphries' result, the proof is considerably simpler than Dubrovin's and Mazocco's argument.