Volume of unit balls associated to quadratic differentials

TL;DR

This paper investigates the Thurston volume of the unit ball in measured lamination space associated with quadratic differentials, revealing its non-properness, conditions for divergence, and p-integrability properties.

Contribution

It characterizes the behavior of the volume function related to quadratic differentials, including its divergence conditions and integrability, advancing understanding of moduli space geometry.

Findings

Volume function is not proper on the moduli space.

Conditions identified for the volume function to tend to infinity.

Volume function is p-integrable for all 0<p<1.

Abstract

Associated to a holomorphic quadratic differential is a unit ball of the measured lamination space. The Thurston volume of the unit ball defines a function on the moduli space. We show that the volume function is not proper and characterize when it tends to infinity. We prove that the volume function is $p$-integrable for any $0<p<1$.