Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group (1) : , (3), and ³. An axis of a hyperbolic element of (1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of (1) was assumed by the Hecke triangle group G_q for q 3. (For q = 3, we have (1) = G₃.) These works employed the analog of ³, denoted _q.In the context of the G_q, the present paper gives the analog of , which we denote _q. As in the case q = 3, we have [_q:_q] = 2. A rather full discussion of geometry of _q\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on _q\ H and _q\ H does not hold for q 7.As of this writing, we have not been able to obtain an appropriate analog of (3).     

Fields with large Kronecker constants

For any algebraic number field K there is a positive number ? such that if α is a nonzero integer of K other than a root of unity, then at least one conjugate of α has absolute value ≥ 1 + ?. It has been conjectured that ? can be taken as $\text{2}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{? 1}$, where n is the degree of K over the field of rationals. In this paper various conditions are discussed under which the validity of this conjecture can be established.     

Two algorithms for closed horocycles on the modular surface

We track the trajectories of individual directed horocycles on the modular surface. Our tracking is constructive, and we thus effectively establish topological transitivity and even line-transitivity for the horocyclic flow.     

McShane’s identity,using elliptic elements

We introduce a new method to establish McShane’s Identity. Elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics. The highest points of simple closed geodesics are already given as the appropriate complement of the regions excluded by those elements of order two that factor hyperbolic elements whose axis projects to be simple. The widths of the intersection with an appropriate horocycle of the excluded regions sum to give McShane’s value of 1/2. The remaining points on the horocycle are highest points of simple open geodesics, we show that this set has zero Hausdorff dimension.      

Length spectra of the Hecke triangle groups

Recently, the study of (singular) surfaces with λ<2, but not of Hecke's form has been undertaken by C. M. [Judge] in connection with the Lax-Phillips work on the Roelcke-Selberg Conjecture