Topological Entropy of Pseudo-Anosov Maps from a Typical Thurston’s Construction

In this paper, we develop a way to extract information about a random walk associated with a typical Thurston’s construction. We first observe that a typical Thurston’s construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston’s construction that have finite 2nd moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane’s estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston’s construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston’s construction.

Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds

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.

On the co-factors of degree 6 Salem number beta expansions

For [Formula: see text], a sequence [Formula: see text] with [Formula: see text] is the beta expansion of [Formula: see text] with respect to [Formula: see text] if [Formula: see text]. Defining [Formula: see text] to be the greedy beta expansion of [Formula: see text] with respect to [Formula: see text], it is known that [Formula: see text] is eventually periodic as long as [Formula: see text] is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with [Formula: see text]. We give examples of numbers with [Formula: see text] whose expansions have period and preperiod lengths exceeding [Formula: see text], yet are still eventually periodic.

Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable, and solvable. Then their centre is one- or two-dimensional. In both cases, a parametrisation of the set of such Lie groups is known. We give a necessary and sufficient condition for the existence of a lattice in terms of these parameters. For groups with one-dimensional centre this problem is related to Salem numbers.