Character varieties for real forms

Abstract

Let $\Gamma$ be a finitely generated group and $G$ a real form of ${\mathrm{SL}}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the ${\mathrm{SL}}_n(\mathbb{C})$-character variety of $\Gamma$. We consider two anti-holomorphic involutions of the ${\mathrm{SL}}_n(\mathbb{C})$ character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of ${\mathrm{SL}}_n(\mathbb{C})$. We study in detail an example: the ${\mathrm{SL}}_n(\mathbb{C})$, $\mathrm{SU}(2,1)$ and $\mathrm{SU}(3)$ character varieties of the free product $\mathbb{Z}/3\mathbb{Z}\ast \mathbb{Z}/3\mathbb{Z}$.

20 pages, 4 figures, typos and minor mistakes corrected from v1

References in corpus (3)

• The topology of moduli spaces of free group representations
• A complex hyperbolic Riley slice
• Invariants of pairs in SL(4,C) and SU(3,1)

Cited by in corpus (2)

• Geometry of $\mathrm{SU}(3)$-character varieties of torus knots
• On the limit set of a spherical CR uniformization