Examples
For example, if and is free of rank two, then the character variety is since by the Fricke-Klein-Vogt theorem its coordinate ring is isomorphic to the complex polynomial ring in 3 variables . Restricting to gives a closed real three dimensional ball (semi-algebraic, but not algebraic).
More generally, the theory of Higgs bundles implies that the (primitive) twisted character varieties of closed surface groups (genus greater than 1) are generally smooth manifolds. This is a class of examples that has been much studied.
Read more about this topic: Character Variety
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