Counting tangles in coverings of closed 2-orbifolds via ribbon categories

Source abstract

We describe what a typical covering of a closed 2-orbifold looks like on a small scale. Specifically, we prove that a uniformly random covering of degree n contains in expectation $Θ(n^0)$ short closed geodesics, potentially $Θ(n^0)$ pairs of nearby order-2 cone points, in general $Θ(n^{\frac{1}{m}})$ cone points of order m, and with high probability no small regions with more complicated topology. This generalises many results of Magee and Puder (2023) from orientable surfaces to possibly non-orientable orbifolds, and answers some of the questions raised by Puder and Zimhoni (2024).