The Johnson-Morita theory for the handlebody group

Abstract

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.