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:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“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)