Asymptotic quadratic complexity of the fundamental-domain algorithm

Determine whether the asymptotic slope of the running time for computing a fundamental domain with Rickards’s algorithm is 2, equivalently whether the algorithm has asymptotically quadratic complexity in the relevant input-size parameter.

Background

The paper compares the running times of fundamental-domain computation, one-word-presentation computation, and Magma’s cohomology computation for families of arithmetic Fuchsian groups. In the reported log-scale experiments, the best linear-fit slope for the fundamental-domain computation is 1.54, while the authors note that the asymptotic slope is conjectured to be 2. Establishing this asymptotic behavior would clarify the complexity of the fundamental-domain stage on which the subsequent quasi-linear presentation and cohomology algorithms depend.