Centrally extended mapping class groups from quantum Teichmuller theory

TL;DR

This paper explores the structure of central extensions of punctured surface mapping class groups in quantum Teichmüller theory, revealing their relation to Meyer and Euler classes, and drawing parallels with Thompson groups.

Contribution

It identifies the specific form of the central extension in quantum Teichmüller theory, linking it to known cohomological classes and analogous results in Thompson groups.

Findings

Central extension is 12 times the Meyer class plus puncture Euler classes.

Establishes an analogy with Thompson group results.

Provides a cohomological characterization of the extension.

Abstract

The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichm\"uller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in \cite{FS} for the Thompson groups.