### 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 R^{3} 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 R^{n}.
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.