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Roots of Unity and the Character Variety of a Knot Complement.

Summary: Roots of Unity and the Character Variety of a
Knot Complement.
D. Cooper D.D. Long \Lambda
August 18, 1997
1 Introduction.
In [3] the following theorem is proved:
Theorem 1.1 Suppose that ae n is a sequence of representations of the funda­
mental group of a knot which blows up on the boundary torus T , and which con­
verge to a simplicial action on a tree. Suppose that there is an essential simple
closed curve C on T whose trace remains bounded. Then lim m!1 tr(ae m (C)) =
– + 1
– where – n = 1 whenever there is a component S of a reduced surface
associated to the degeneration so that S has n boundary components.
Precise definitions of the terms will be given below, but the rough description
is as follows. If one has a curve of characters of representations of a manifold
with a single torus boundary component, then the method of [5] for producing
boundary slopes is to go to some end of the character variety. Two things can
happen on the boundary torus when one does this; either all the characters
remain bounded and the surface produced from the resulting splitting can be
chosen to be closed, or there is a particular simple closed curve whose character


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics