Dimension of representation and character varieties for two and three-orbifolds

TL;DR

This paper investigates the dimensions of representation and character varieties for 2- and 3-orbifolds, providing formulas and tools for their computation, especially in hyperbolic cases, and exploring dimension growth in SL(n,C).

Contribution

It introduces new methods for computing dimensions of character varieties of orbifolds and establishes a relationship between boundary and orbifold dimensions in hyperbolic 3-orbifolds.

Findings

Dimension of character variety component equals half the boundary's character variety dimension.

Provides tools for computing dimensions of 2-orbifold character varieties.

Analyzes dimension growth of 3-manifold character varieties in SL(n,C).

Abstract

We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3-orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the dimension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing dimensions of varieties of characters of 2-orbifolds, including the Hitchin component. We apply this computation to the dimension growth of varieties of characters of some 3-dimensional manifolds in SL(n,C).