2024/08/26 by Fitzhugh, Nick, Schondorf, Aaron, Shrestha, Sunrose +2
#FOS: Mathematics #Geometric Topology (math.GT) #Probability (math.PR)
paper · doi:10.48550/arxiv.2408.14041
A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. After a labelling of the squares by \1, …, n\, we can describe an STS with n squares using two permutations σ, τ∈ Sn, where σ encodes how the squares are glued horizontally and τ encodes how the squares are glued vertically. Hence, a previously considered natural model for STSs with n squares is Sn × Sn with the uniform distribution. We modify this model to obtain a new one: We fix α∈ [0,1] and let Kμn be a conjugacy class of Sn with at most nα cycles. Then Kμn × Sn with the uniform distribution is a model for STSs with restricted horizontal gluings. We deduce the asymptotic (as n grows) number of components, genus distribution, most likely stratum and set of holonomy vectors of saddle connections for random STSs in this new model.