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Triangulated surfaces are compact Riemann surfaces equipped with a conformaltriangulation by equilateral triangles. In 2004, Brooks and Makover asked howtriangulated surfaces are distributed in the moduli space of Riemann surfacesas the genus tends to infinity. Mirzakhani raised this question in her 2010 ICMaddress. We show that in the large genus case, triangulated surfaces are welldistributed in moduli space in a fairly strong sense. We do this by provingupper and lower bounds for the number of triangulated surfaces lying in aTeichm\"uller ball in moduli space. In particular, we show that the number oftriangulated surfaces lying in a Teichm\"uller unit ball is at most exponentialin the number of triangles, independent of the genus.

Large genus bounds for the distribution of triangulated surfaces in moduli space