The hyperbolic lattice point problem in conjugacy classes

For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces ${{\Gamma\backslash{\mathbb{H}}}}$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound ${O(X^{2/3})}$, due to Good. For ${{\mathrm{SL}_{2}({\mathbb{Z}})}}$ we interpret our results in terms of indefinite quadratic forms.

Complete hyperbolic lattices derived from tessellations of type {4g,4g}

Regular tessellations of the hyperbolic plane play an important role in the design of signal constellations for digital communication systems. Self-dual tessellations of type [Formula: see text] with [Formula: see text], and [Formula: see text] have been considered where the corresponding arithmetic Fuchsian groups are derived from quaternion orders over quadratic extensions of the rational. The objectives of this work are to establish the maximal orders derived from [Formula: see text] tessellations for which the hyperbolic lattices are complete (the motivation for constructing complete hyperbolic lattices is their application to the design of hyperbolic lattice codes), and to identify the arithmetic Fuchsian group associated with a quaternion algebra and a quaternion order.