Salem numbers and arithmetic hyperbolic groups
Geometric Topology
2018-07-03 v3 Group Theory
Number Theory
Abstract
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.
Cite
@article{arxiv.1506.03727,
title = {Salem numbers and arithmetic hyperbolic groups},
author = {Vincent Emery and John G. Ratcliffe and Steven T. Tschantz},
journal= {arXiv preprint arXiv:1506.03727},
year = {2018}
}
Comments
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. AMS