42 research outputs found
Salem numbers and arithmetic hyperbolic groups
In this paper we prove that there is a direct relationship between Salem
numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic
groups that are determined by a quadratic form over a totally real number
field. As an application we determine a sharp lower bound for the length of a
closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each
dimension n. We also discuss a "short geodesic conjecture", and prove its
equivalence with "Lehmer's conjecture" for Salem numbers.Comment: The exposition in version 3 is more compact; this shortens the paper:
26 pages now instead of 37. A discussion on Lehmer's problem has been added
in Section 1.2. Final version, to appear is Trans. AM