Deforming reducible representations of surface and 2-orbifold groups

TL;DR

This paper investigates how the space of representations of surface and 2-orbifold groups into special linear groups deforms, revealing singularities when representations are embedded into higher dimensions.

Contribution

It demonstrates that the character variety becomes singular when a $ ext{SL}_n( eal)$ representation is viewed in $ ext{SL}_{n+1}( eal)$, providing a detailed description of these singularities.

Findings

Character variety is smooth at $ ext{SL}_n( eal)$-irreducible points.

Embedding into higher dimension introduces singularities in the character variety.

The paper characterizes the nature of these singularities.

Abstract

For a compact 2-orbifold with negative Euler characteristic $\mathcal O^2$, the variety of characters of $\pi_1(\mathcal O^2)$ in $\mathrm{SL}_{n}(\mathbb R)$ is a non-singular manifold at $\mathbb C$-irreducible representations. In this paper we prove that when a $\mathbb C$-irreducible representation of $\pi_1(\mathcal O^2)$ in $\mathrm{SL}_{n}(\mathbb R)$ is viewed in $\mathrm{SL}_{n+1}(\mathbb R)$, then the variety of characters is singular, and we describe the singularity.