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The variety of characters in $$\text{PSL}_2(\mathbb{C})$$. (English) Zbl 1100.57014
The $$\text{SL}_ 2(\mathbb C)$$- and $$\text{PSL}_ 2(\mathbb C)$$-character varieties of the fundamental groups of hyperbolic $$3$$-manifolds have played an important role in $$3$$-dimensional topology. In this article, the authors give a careful and fully referenced development of the algebraic theory of $$\text{PSL}_ 2(\mathbb C)$$-representation varieties and their associated character varieties. As an application, they show that for every $$n$$, there is a hyperbolic punctured-torus bundle over $$S^1$$ whose $$\text{PSL}_ 2(\mathbb C)$$-character variety has at least $$n$$ irreducible one-dimensional components whose characters do not lift to $$\text{SL}_ 2(\mathbb C)$$. Additional results about singular sets of character varieties are obtained, in particular, the singular set is computed for the $$\text{PSL}_ 2(\mathbb C)$$-character variety of a free group $$F_n$$, $$n\geq 3$$.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 20C15 Ordinary representations and characters 57M05 Fundamental group, presentations, free differential calculus
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