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.

Orbifold expansion and entire functions with bounded Fatou components

Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

Torsion

This chapter deals with finite subgroups of the mapping class group. It first explains the distinction between finite-order mapping classes and finite-order homeomorphisms, focusing on the Nielsen realization theorem for cyclic groups and detection of torsion with the symplectic representation. It then considers the problem of finding an Euler characteristic for orbifolds, to prove a Gauss–Bonnet theorem for orbifolds, and to use these results to show that there is a universal lower bound of π‎/21 for the area of any 2-dimensional orientable hyperbolic orbifold. The chapter demonstrates that, when g is greater than or equal to 2, finite subgroups have order at most 84(g − 1) and cyclic subgroups have order at most 4g + 2. It also describes finitely many conjugacy classes of finite subgroups in Mod(S) and concludes by proving that Mod(Sɡ) is generated by finitely many elements of order 2.

Hyperbolic crystallography of two-periodic surfaces and associated structures

This paper describes the families of the simplest, two-periodic constant mean curvature surfaces, the genus-two HCB and SQL surfaces, and their isometries. All the discrete groups that contain the translations of the genus-two surfaces embedded in Euclidean three-space modulo the translation lattice are derived and enumerated. Using this information, the subgroup lattice graphs are constructed, which contain all of the group–subgroup relations of the aforementioned quotient groups. The resulting groups represent the two-dimensional representations of subperiodic layer groups with square and hexagonal supergroups, allowing exhaustive enumeration of tilings and associated patterns on these surfaces. Two examples are given: a two-periodic [3,7]-tiling with hyperbolic orbifold symbol {\sf {2223}} and a {\sf {22222}} surface decoration.

A minimal volume arithmetic cusped complex hyperbolic orbifold

The sister of Eisenstein–Picard modular group is described explicitly in [10], whose quotient is a noncompact arithmetic complex hyperbolic 2-orbifold of minimal volume (see [16]). We give a construction of a fundamental domain for this group. A presentation of that lattice can be obtained from that construction, which relates to one of Mostow's lattices.

ALL DISCRETE $\mathcal{RP}$ GROUPS WHOSE GENERATORS HAVE REAL TRACES

In this paper we give necessary and sufficient conditions for discreteness of a subgroup of PSL(2,ℂ) generated by a hyperbolic element and an elliptic one of odd order with non-orthogonally intersecting axes. Thus we completely determine two-generator non-elementary Kleinian groups without invariant plane with real traces of the generators and their commutator. We also give a list of all parameters that correspond to such groups. An interesting corollary of the result is that the group of the minimal known volume hyperbolic orbifold ℍ3/Γ353 has real parameters.

ON THE ARITHMETIC 2-BRIDGE KNOTS AND LINK ORBIFOLDS AND A NEW KNOT INVARIANT

Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve [Formula: see text] (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of [Formula: see text] in PSL [Formula: see text]. The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B[Formula: see text] are the images of canonical generators a, b of [Formula: see text]. Let (xn, zn) be the point of [Formula: see text] corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(xn, zn) has exactly one complex place and ϕ(xn)<ϕ(zn)<2 for every real embedding [Formula: see text]. Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot. Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q2≡1 mod p. We finish the paper with some open problems.