Characterization of Alexander polynomials of Hurwitz C-groups

Let $P(t)=(-1)^m t^m+\sum_{i=0}^{m-1}a_i t^i ein\mathbb Z[t]$. A Hurwitz $C$-group is a group arising as the fundamental group of the complement of an affine Hurwitz curve. The roots of $P(t)$ and their multiplicities are taken over the complex numbers.

Alexander-polynomial characterization. The polynomial $P(t)$ is the Alexander polynomial of some Hurwitz $C$-group $G$ if and only if: (i) the roots of $P(t)$ are roots of unity; and (ii) whenever $\zeta$ is a primitive $p^k$-th root of unity, with $p$ prime, the multiplicity of the root $t=\zeta$ of $P(t)$ is no greater than the multiplicity of the root $t=1$.

This gives a necessary and sufficient algebraic characterization of the Alexander polynomials realized by Hurwitz $C$-groups, extending the known restrictions on Alexander polynomials of Hurwitz curves. The supplied text presents the characterization as a theorem, so it is solved rather than an open conjecture.