On the algebraic components of the SL(2, ) character varieties of knot exteriors

We show that if a knot exterior satis"es certain conditions, then it has "nite cyclic coverings with arbitrarily large numbers of nontrivial algebraic components in their SL(2, ) character varieties (Theorem A). As an example, these conditions hold for hyperbolic punctured torus bundles over the circle (Theorem B). We investigate in more detail the "nite cyclic covers of the "gure-eight knot exterior and show that for every integer m there exists a "nite covering such that its SL(2, ) character variety contains curve components which have associated boundary slopes whose distance is larger than m (Theorem C). Lastly, we show that given an integer m then there exists a hyperbolic knot exterior in the 3-shpere which has a "nite cyclic covering such that its SL(2, ) character variety contains more than m norm curve components each of which contains the character of a discrete faithful presentation of the fundamental group of the covering space (Theorem D). 2002 Elsevier Science Ltd. All rights reserved.