Pants complex, TQFT and hyperbolic geometry

TL;DR

This paper explores the deep connections between quantum topology, hyperbolic geometry, and Teichmüller theory, introducing a quantum intersection number that links surface decompositions to hyperbolic 3-manifold volumes.

Contribution

It defines a quantum intersection number from skein theory, establishes bounds relating it to geometric intersection, and relates it to hyperbolic volume and mapping class group dynamics.

Findings

Quantum intersection number bounds geometric intersection

Pants graph with quantum metric is quasi-isometric to Teichmüller space

Translation length bounds hyperbolic volume of fibered 3-manifolds

Abstract

We introduce a coarse perspective on relations of the $SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the $SU(2)$-TQFT, we define the quantum intersection number between pants decompositions of a closed surface. We show that the quantum intersection number admits two sided bounds in terms of the geometric intersection number and we use it to obtain a metric on the pants graph of surfaces. Using work of Brock we show that the pants graph equipped with this metric is quasi-isometric to the Teichm\"uller space with the Weil-Petersson metric and that the translation length of our metric provides two sided linear bounds on the volume of hyperbolic fibered manifolds. We briefly discuss how these relations are interpeted from the view point of $SU(2)$-character varieties of 3-manifolds. We also obtain a characterization of pseudo-Anosov mapping classes in terms of asymptotics of the quantum intersection number under iteration in the mapping class group and relate these asymptotics with stretch factors. We also discuss how these results fit with a conjecture of Andersen, Masbaum and Ueno about quantum representations of mapping class groups.