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
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)