  We study Lagrangian cobordism groups of oriented surfaces of genus greater
than two. We compute the immersed oriented Lagrangian cobordism group of these
surfaces. We show that a variant of this group, with relations given by
unobstructed immersed Lagrangian cobordisms computes the Grothendieck group of
the derived Fukaya category. The proof relies on an argument of Abouzaid.
