An extension of the Burau representation to a mapping class group associated to Thompson's group T

We study some aspects of the geometric representation theory of the Thompson and Neretin groups, suggested by their analogies with the diffeomorphism groups of the circle. We prove that the Burau representation of the Artin braid groups extends to a mapping class group $A_T$ related to Thompson's group $T$ by a short exact sequence $B_{\infty}\hookrightarrow A_T\to T$, where $B_{\infty}$ is the infinite braid group. This {\it non-commutative} extension abelianises to a central extension $0\to \Z\to A_T/[B_{\infty},B_{\infty}]\to T\to 1$ detecting the {\it discrete} version $\bar{gv}$ of the Bott-Virasoro-Godbillon-Vey class. A morphism from the above non-commutative extension to a reduced Pressley-Segal extension is then constructed, and the class $\bar{gv}$ is realised as a pull-back of the reduced Pressley-Segal class. A similar program is carried out for an extension of the Neretin group related to the {\it combinatorial} version of the Bott-Virasoro-Godbillon-Vey class.