Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory

TL;DR

This paper explores the central extensions of the Thompson group T within the context of quantized Teichmüller theory, revealing a specific extension related to the Euler class and providing explicit descriptions.

Contribution

It introduces a new central extension of the Thompson group T derived from the braided Ptolemy-Thompson group and offers explicit presentations of cyclic central extensions.

Findings

The central extension corresponds to 12 times the Euler class.

Explicit presentations of cyclic central extensions are provided.

The extension is obtained via abelianization of the braided Ptolemy-Thompson group.

Abstract

The central extension of the Thompson group $T$ that arises in the quantized Teichm\"uller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extensions of $T$ by means of explicit presentations.