Higher-order Alexander invariants of plane algebraic curves

TL;DR

This paper introduces new higher-order Alexander invariants for plane algebraic curves, extending tools from knot and 3-manifold theory to study the fundamental groups of curve complements.

Contribution

It defines higher-order Alexander modules and degrees for algebraic curves, providing new obstructions on possible fundamental groups of affine curve complements.

Findings

Higher-order degrees are finite for curves in general position at infinity.

New invariants distinguish different types of fundamental groups.

Obstructions to certain groups arising as curve complement groups.

Abstract

We define new higher-order Alexander modules $\mathcal{A}_n(C)$ and higher-order degrees $\delta_n(C)$ which are invariants of the algebraic planar curve $C$. These come from analyzing the module structure of the homology of certain solvable covers of the complement of the curve $C$. These invariants are in the spirit of those developed by T. Cochran in \cite{C} and S. Harvey in \cite{H} and \cite{Har}, which were used to study knots, 3-manifolds, and finitely presented groups, respectively. We show that for curves in general position at infinity, the higher-order degrees are finite. This provides new obstructions on the type of groups that can arise as fundamental groups of complements to affine curves in general position at infinity.