Homology cobordism group of homology cylinders and invariants related to lower central series

The homology cobordism group of homology cylinders is a generalization of both the mapping class group of surfaces and the string link concordance group. We consider extensions of Johnson homomorphisms of a mapping class group, Milnor invariants and Orr invariants of links to homology cylinders, all of which are related to free nilpotent groups. We establish a combined filtration via kernels of extended Johnson homomorphisms and extended Milnor invariants. We determine its image under the three invariants, and investigate relations among the invariants, and relations of the filtration to automorphism groups of free nilpotent groups and to graded free Lie algebras. We obtain the number of linearly independent invariants by examining the successive quotients of the filtration.