Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds

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 $$c Q^{1/2} + O(Q^{1/4})$$ 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 $$\frac{4}{3}Q^{3/2}+O(Q)$$ 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.