# On the quantum Teichm\"uller invariants of fibred cusped 3-manifolds

## TL;DR

This paper demonstrates that reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms serve as intertwiners of local quantum Teichm"uller space representations, satisfying topological operations and defining invariants of mapping tori.

## Contribution

It characterizes these invariants as unique intertwiners fulfilling natural topological quantum field theory operations and trace conditions, linking quantum invariants to 3-manifold topology.

## Key findings

- Reduced quantum hyperbolic invariants act as intertwiners of quantum Teichm"uller space representations.
- These invariants are uniquely characterized by their topological and trace properties.
- The invariants define topological invariants of mapping tori.

## Abstract

We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichm\"uller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.