Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the genus tends to infinity. Mirzakhani raised this question in her 2010 ICM address. We show that in the large genus case, triangulated surfaces are well distributed in moduli space in a fairly strong sense. We do this by proving upper and lower bounds for the number of triangulated surfaces lying in a Teichmüller ball in moduli space. In particular, we show that the number of triangulated surfaces lying in a Teichmüller unit ball is at most exponential in the number of triangles, independent of the genus.

三角曲面是配备有等边三角形等角三角剖分的紧凑黎曼曲面。 2004 年，Brooks 和 Makover 提出了当亏格趋于无穷大时三角曲面如何分布在黎曼曲面模空间中的问题。 Mirzakhani 在 2010 年 ICM 演讲中提出了这个问题。我们证明，在大亏格的情况下，三角曲面在模空间中具有相当强的分布。我们通过证明模空间中 Teichmüller 球中三角曲面数量的上限和下限来实现这一点。特别是，我们证明了位于 Teichmüller 单位球中的三角曲面的数量至多是三角形数量的指数，与属无关。