For O an imaginary quadratic ring, we compute a fundamental polyhedron of PE₂(O), the projective elementary subgroup of PSL₂(O). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE₂(O), show that it has infinite-index and is its own normalizer in PSL₂(O), and split PSL₂(O) into a free product with amalgamation that has PE₂(O) as one of its factors.

A family of fractal arrangements of circles is introduced for each imaginary quadratic field K. Collectively, these arrangements contain (up to an affine transformation) every set of circles in ℂ with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the almost local-global principle. Connections to the class group of K are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators.

Here is a lecture video from a conference at ICERM that explains the connection between this paper and the continued fractions paper. Here is the tool used to create the images in the paper. I mess with it sometimes to create specific images, so it may or may not be functioning properly.

Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for PSL₂(O) in the upper-half space model of hyperbolic space, where O is an imaginary quadratic ring of integers with discriminant Δ. We prove these bounds are asymptotically within (log Δ)^8.54 of one another. This improves on the previous best upper-bound, which is roughly off by a factor between Δ² and |Δ|^5/2 depending on the smallest prime dividing Δ. The gap between our upper and lower bounds is determined by an analog of Jacobsthal's function, introduced here for imaginary quadratic fields.

We propose and study a continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued fractions are shown to be retained, including exponential convergence, best-of-the-second-kind approximation quality (up to a constant), periodicity of quadratic irrational expansions, and polynomial time complexity.

Here is a C++ implementation of the continued fraction algorithm. Here is a C++ implementation of an algorithm that precomputes admissible parameters for a given ring (used to make Table 2 in the paper).

In 1982, Lagarias showed that solving the approximate Shortest Vector Problem also solves the problem of finding good simultaneous Diophantine approximations. Here we provide a deterministic, dimension-preserving reduction in the reverse direction. It has polynomial time and space complexity, and it is gap-preserving under the appropriate norms. We also give an alternative to the Lagarias algorithm by first reducing his version of simultaneous approximation to one with no explicit range in which a solution is sought.

Here is a short, general-audience introduction to the problem solved in the paper. It is the first half of the presentation at the 14th ANTS conference.