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

### Joan Porti

Algebraic & Geometric Topology 22 (2022) 1905–1967
##### 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, and 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 $\mathrm{SL}\left(n,ℂ\right)$.

##### Keywords
variety of representations, variety of characters, orbifold
##### Mathematical Subject Classification
Primary: 14D20, 57K32
Secondary: 57K20
##### Publication
Received: 7 September 2020
Revised: 10 March 2021
Accepted: 3 May 2021
Published: 10 October 2022
##### Authors
 Joan Porti Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona Spain Centre de Recerca Matemàtica Barcelona Spain https://mat.uab.cat/~porti