Geometric Group Theory

A surface is called tangle-free when it has no complicated topology on a small scale. This property is useful in applications such as Benjamini-Schramm convergence, strong covergence of representations, and spectral gaps. Consequently, there was much recent interest in tangle-freeness of random surfaces, primarily in random models induced by the Weil-Petersson measure, counting finite coverings, and Brooks-Makover model of Belyi surfaces. I will review these results, and discuss the ongoing work to extend them to branched coverings of surfaces with cone points.