Abstract
It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines c Q^{1/2} + O(Q^{1/4}) square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to frac{4}{3}Q^{3/2}+O(Q). Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3.
Highlights
A Salem number is a real algebraic integer λ > 1 such that all of its Galois conjugates except λ−1 have absolute value equal to 1
It has been known for some time that the exponential lengths of the closed geodesics of an arithmetic hyperbolic n-dimensional manifold or orbifold are given by Salem numbers
We will say that an arithmetic hyperbolic n-orbifold O which has a closed geodesic of length generates a Salem number λ = e((n mod 2)+1)
Summary
A Salem number is a real algebraic integer λ > 1 such that all of its Galois conjugates except λ−1 have absolute value equal to 1. It was elaborated upon and generalized to higher dimensions by Emery et al (2019) Their Theorem 1.1 implies that, for a non-compact arithmetic hyperbolic n-orbifold O, a closed geodesic of length corresponds to a Salem number λ = e if the dimension n is even, and to a so called square-rootable Salem number λ = e2 if n is odd. For proving Theorem 2 we take advantage of some special properties of Salem numbers of degree 4 Extending these results to higher degrees would require an extension of Marklof’s length spectrum asymptotic to arithmetic orbifolds of dimension greater than 3 and an analogue of the Götze–Gusakova theorem for square-rootable Salem numbers of higher degree.
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