We study the algebraic property of the representation of the
mapping class group of a closed oriented surface of genus
constructed by V F R Jones [Annals of Math. 126 (1987) 335-388]. It arises
from the Iwahori–Hecke algebra representations of Artin’s braid group of
strings, and is defined over integral Laurent polynomials
. We substitute
the parameter
with , and then
expand the powers
in their Taylor series. This expansion naturally induces a filtration on the Torelli
group which is coarser than its lower central series. We present some results on
the structure of the associated graded quotients, which include that the
second Johnson homomorphism factors through the representation. As an
application, we also discuss the relation with the Casson invariant of homology
–spheres.
Keywords
Jones representation, mapping class group, Torelli group,
Johnson homomorphism
