  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.
