The compatibility of the filtration of mapping class groups of two surfaces pasted along the boundaries

Let Σng be an orientable surface of genus g⩾0 with n⩾0 punctures and Γng be its pure mapping class group. The group Γng has a filtration {Γng[m]}m⩾1 induced from its action on the fundamental group of Σng. If g=g1+g2 and n=n1+n2−2, we have a homomorphism Γn1g1[3]×Γn2g2[3]→Γng [Math. Ann. 304 (1996) 99], which is induced from pasting two surfaces with one boundary component along their boundaries. That this homomorphism preserves the filtration strictly has been shown by Nakamura in the case that n⩾1. We shall show that this holds also in the case that n=0. As an application, we obtain a lower bound of the rank of the graded module associated with the filtration of Γg(= Γg0).