Finite Hurwitz braid group actions on sequences of Euclidean reflections

We prove that if r1,…,rn are Euclidean reflections corresponding to a linearly independent set of vectors, then the group r1,…,rn is finite if and only if the natural Hurwitz braid group action on such ordered sets of reflections has finite orbit and we characterise the orbits for this action. We apply this to give a representation of the braid group on n strands onto the alternating or symmetric groups of degree (n+1)n−2 (for most n) which is related to the Morse theory of polynomials, as studied by Catanese, Paluszny and Wajnryb.