Pants decompositions of random surfaces

TL;DR

This paper demonstrates that random surfaces, both hyperbolic and Euclidean, typically require large total length pants decompositions, with bounds growing faster than the genus, highlighting the complexity of their geometric structure.

Contribution

It establishes that most hyperbolic surfaces and randomly glued Euclidean triangle surfaces have large pants decompositions, providing new bounds and probabilistic insights.

Findings

Hyperbolic surfaces require pants decompositions of length at least g^{7/6 - ε}.

Most metrics in the moduli space exhibit this large decomposition property.

Randomly glued Euclidean surfaces also have large pants decompositions similar to hyperbolic cases.

Abstract

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus $g$ for which any pants decomposition requires curves of total length at least $g^{7/6 - \epsilon}$. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.