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Contemporary Mathematics
Eigenvalues of hyperbolic elements in Kleinian groups
D. D. Long* and A. W. Reid
1. Introduction
Let be a torsionfree Kleinian group, so that M = H3
/ is an orientable hy
perbolic 3manifold. The nontrivial elements of are classified as either parabolic
or hyperbolic. If is hyperbolic, then has an axis in H3
which projects to
a closed geodesic g in M (which depends only on the conjugacy class of in ).
The element acts on its axis by translating and possibly rotating around the axis.
In terms of eigenvalues, if is hyperbolic, we let
= = rei
be the eigenvalue of (more accurately of a preimage of in SL(2, C)) for which
 > 1. The angle takes values in [0, 2), and is the rotation angle mentioned
above. We will usually suppress the subscripts. A hyperbolic element is called
purely hyperbolic if and only if = 0, or equivalently, if tr() R.
The length of the closed geodesic g is given by 2 ln  and the collections of
these lengths counted with multiplicities is a wellknown important geometric in
