Bad irreducible subgroups and singular locus for character varieties in $PSL(p,\mathbb{C})$

TL;DR

This paper characterizes the centralizers of irreducible representations into $PSL(p,\mathbb{C})$, describing the singular locus of the irreducible character variety for certain groups, and identifies these as algebraic singularities.

Contribution

It provides a detailed description of the singular locus of irreducible character varieties for free groups and surface groups in $PSL(p,\mathbb{C})$, linking it to algebraic singularities.

Findings

The singular locus corresponds exactly to algebraic singularities.

Complete description of the singular locus for free and surface groups.

Centralizers of irreducible representations are characterized explicitly.

Abstract

We describe the centralizer of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $\chi^i(\Gamma,PSL(p,\mathbb{C}))$. When $\Gamma$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of algebraic singularities of the irreducible part of the character variety.