Finite Hurwitz braid group actions on sequences of Euclidean reflections. (Q1414676)

The author proves that if \(r_1,\dots, r_n\) are Euclidean reflections corresponding to a linearly independent set of vectors, then the group \(\langle r_1,\dots, r_n\rangle\) is finite if and only if the natural Hurwitz braid group action on such ordered sets of reflections has finite orbit; then he characterizes the orbits for this action. The author applies 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.