Chow--Kuenneth decomposition for special varieties

In this paper we investigate Murre’s conjecture on the Chow–K"unneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space ${\text{cM}}_g$, in genus at most 8 and show existence of a Chow–K"unneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow–K"unneth decompositions.