  The Johnson-Morita theory is an algebraic approach to the mapping class group
of a surface, in which one considers its action on the successive nilpotent
quotients of the fundamental group of the surface. In this paper, we develop an
analogue of this theory for the handlebody group, i.e. the mapping class group
of a 3-dimensional handlebody. Thus, we obtain a filtration on the handlebody
group, prove that its associated graded embeds into a Lie algebra of "special
derivations", and give an explicit diagrammatic description of this graded Lie
algebra in terms of "oriented trees with beads". Our new diagrammatic method
reveals part of the richness of the algebraic structure of the handlebody
group, which lies mainly in the subgroup generated by Dehn twists along
meridians: the so-called "twist group". As an application, we obtain that each
term of the associated graded of the lower central series of the twist group is
infinitely generated.
