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Deforming reducible representations of surface and 2-orbifold groups

Joan Porti

TL;DR

This work analyzes how deforming a $\mathbb{C}$-irreducible representation of a surface/2-orbifold group inside $\mathrm{SL}_{n}(\mathbb{R})$ behaves when the representation is embedded into $\mathrm{SL}_{n+1}(\mathbb{R})$. It establishes that the associated point in the real character variety becomes singular, with a precise local model: the germ is a cone over a unit tangent bundle (or its projective/antipodal variants) determined by the dimension $d=\dim H^1(\mathcal{O}^2,\mathbb{R}^{n}_{\rho})$ and the parity of $n+1$, and by boundary data. The results extend to non-orientable orbifolds and have concrete consequences for convex projective structures and projective deformations of hyperbolic 3-orbifolds, including a quadratic-type singularity description via Goldman–Millson theory. By combining slice techniques, Lie-algebra decompositions, and cohomological obstructions, the paper provides a detailed, topological understanding of how higher-rank deformations interact with reducibility and end structures. These findings illuminate the geometry of higher-rank representation spaces and their applications to projective geometry on orbifolds and 3-manifolds.

Abstract

For a compact 2-orbifold with negative Euler characteristic $\mathcal O^2$, the variety of characters of $π_1(\mathcal O^2)$ 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 $π_1(\mathcal O^2)$ 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.

Deforming reducible representations of surface and 2-orbifold groups

TL;DR

This work analyzes how deforming a -irreducible representation of a surface/2-orbifold group inside behaves when the representation is embedded into . It establishes that the associated point in the real character variety becomes singular, with a precise local model: the germ is a cone over a unit tangent bundle (or its projective/antipodal variants) determined by the dimension and the parity of , and by boundary data. The results extend to non-orientable orbifolds and have concrete consequences for convex projective structures and projective deformations of hyperbolic 3-orbifolds, including a quadratic-type singularity description via Goldman–Millson theory. By combining slice techniques, Lie-algebra decompositions, and cohomological obstructions, the paper provides a detailed, topological understanding of how higher-rank deformations interact with reducibility and end structures. These findings illuminate the geometry of higher-rank representation spaces and their applications to projective geometry on orbifolds and 3-manifolds.

Abstract

For a compact 2-orbifold with negative Euler characteristic , the variety of characters of in is a non-singular manifold at -irreducible representations. In this paper we prove that when a -irreducible representation of in is viewed in , then the variety of characters is singular, and we describe the singularity.
Paper Structure (25 sections, 38 theorems, 131 equations, 1 figure, 1 table)

This paper contains 25 sections, 38 theorems, 131 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgements
  4. Preliminaries on varieties of characters
  5. Variety of characters for real reductive groups
  6. Group cohomology
  7. Products in cohomology
  8. Tangent space to the variety of representations
  9. Orbifolds
  10. Obstructions to integrability
  11. Deformation space
  12. An analytic slice at semisimple representations
  13. Decomposing the Lie algebra
  14. Prof of Theorem \ref{['Theorem:orientable']}
  15. Non-orientable orbifolds
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\mathcal{O}^2$ be a compact and orientable 2-orbifold, satisfying $\chi(\mathcal{O}^2)<0$. Let $\chi_\rho\in X( \mathcal{O}^2, \mathrm{SL}_{n}(\mathbb R) )$ be the character of a $\mathbb C$-irreducible representation $\rho$, $d= \dim(H^1(\mathcal{O}^2,\mathbb R^n_\rho))$, and $b= \dim(H^1(\ma where $X$ is as in Table Table:Table1 and $\mathbb R^p\times \{0\}\times\{0\}$ corresponds to a smo

Figures (1)

  • Figure 1: The four examples in Section \ref{['section:Examples']}. The orbifold $\mathcal{O}^2_1$ is the orientation covering of $\mathcal{O}^2_2$, and $\mathcal{O}^2_3$, of $\mathcal{O}^2_4$ (for a suitable choice of labels).

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: LunaRichardsonSlodowy
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4: RichardsonDuke
  • Definition 2.5
  • Theorem 2.6: JohnsonMillson
  • Lemma 2.7
  • proof
  • ...and 67 more