Singularities of free group character varieties

TL;DR

This paper classifies the algebraic and topological singularities of moduli spaces of free group representations into classical groups, showing that singularities correspond to reducible representations and that these spaces are generally not topological manifolds.

Contribution

It provides a complete classification of the singular stratification of free group character varieties and links algebraic singularities to topological properties.

Findings

Singular locus corresponds exactly to reducible representations.

Moduli spaces are generally not topological manifolds.

Explicit examples where the spaces are manifolds.

Abstract

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating algebraic singularities to topological singularities, we show the moduli spaces X generally are not topological manifolds, except for a few examples we explicitly describe.